lsd0003

draft-summermatter-set-union.xml (194157B)

---

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 <rfc xmlns:xi="http://www.w3.org/2001/XInclude" category="info" docName="draft-summermatter-set-union-01" ipr="trust200902"
      obsoletes="" updates="" submissionType="IETF" xml:lang="en" version="3">
     <!-- xml2rfc v2v3 conversion 2.26.0 -->
     <front>
         <title abbrev="Set Union">
             Byzantine Fault Tolerant Set Reconciliation
         </title>
         <seriesInfo name="Internet-Draft" value="draft-summermatter-set-union-01"/>
         <author fullname="Elias Summermatter" initials="E." surname="Summermatter">
             <organization>Seccom GmbH</organization>
             <address>
                 <postal>
                     <street>Brunnmattstrasse 44</street>
                     <city>Bern</city>
                     <code>3007</code>
                     <country>CH</country>
                 </postal>
                 <email>elias.summermatter@seccom.ch</email>
             </address>
         </author>
         <author fullname="Christian Grothoff" initials="C." surname="Grothoff">
             <organization>Berner Fachhochschule</organization>
             <address>
                 <postal>
                     <street>Hoeheweg 80</street>
                     <city>Biel/Bienne</city>
                     <code>2501</code>
                     <country>CH</country>
                 </postal>
                 <email>grothoff@gnunet.org</email>
             </address>
         </author>
 
         <!-- Meta-data Declarations -->
         <area>General</area>
         <workgroup>Independent Stream</workgroup>
         <keyword>name systems</keyword>
         <abstract>
             <t>This document contains a protocol specification for Byzantine fault-tolerant
                 Set Reconciliation.
             </t>
         </abstract>
     </front>
     <middle>
         <section anchor="introduction" numbered="true" toc="default">
             <name>Introduction</name>
             <t>
               This document describes a byzantine fault tolerant set reconciliation protocol used to efficient and securely
               compute the union of two sets across a network.
             </t>
             <t>
               This byzantine fault tolerant set reconciliation
               protocol can be used in a variety of applications.
 
               Our primary envisioned application domain is the
               distribution of revocation messages in the GNU Name
               System (GNS) <xref target="GNS" format="default" />. In GNS,
               key revocation messages are usually flooded across the
               peer-to-peer overlay network to all connected peers
               whenever a key is revoked. However, as peers may be
               offline or the network might have been partitioned,
               there is a need to reconcile revocation lists whenever
               network partitions are healed or peers go online.  The
               GNU Name System uses the protocol described in this
               specification to efficiently distribute revocation
               messages whenever network partitions are healed.
 
               Another application domain for the protocol described
               in this specification are Byzantine fault-tolerant
               bulletin boards, like those required in some secure
               multiparty computations.  A well-known example for
               secure multiparty computations are various E-voting
               protocols <xref target="CryptographicallySecureVoting" format="default"/> which
               use a bulletin board to share the votes and intermediate
               computational results. We note that for such systems,
               the set reconciliation protocol is merely a component of
               a multiparty consensus protocol, such as the one
               described in Dold's "Byzantine set-union consensus using
               efficient set reconciliation" <xref target="ByzantineSetUnionConsensusUsingEfficientSetReconciliation" format="default"/>.
             </t>
             <t>
               The protocol described in this report is generic and
               suitable for a wide range of applications. As a result,
               the internal structure of the elements in the sets MUST
               be defined and verified by the application using the
               protocol.  This document thus does not cover the element
               structure, except for imposing a limit on the maximum
               size of an element.
             </t>
             <t>
               The protocol faces an inherent trade-off between minimizing
               the number of network round-trips and the number of bytes
               sent over the network.  Thus, for the protocol to choose
               the right parameters for a given situation, applications
               using an implementation of the protocol SHOULD provide a
               parameter that specifies
               the cost-ratio of round-trips vs. bandwidth usage.  Given
               this trade-off factor, an implementation CAN then choose parameters
               that minimize total execution cost.  In particular, there
               is one major choice to be made, namely between sending the
               complete set of elements, or computing the set differences and
               transmitting only the elements in the set differences.
               In the latter case, our design is basically a concrete
               implementation of a proposal by Eppstein.<xref target="Eppstein" format="default" />
             </t>
 
             <t>
               We say that our set reconciliation protocol is Byzantine
               fault-tolerant because it provides cryptographic and
               probabilistic methods to discover if the other peer
               is dishonest or misbehaving.
               Here, the security objective is to limit resources wasted on
               malicious actors. Malicious actors could send malformed
               messages, including malformed set elements, claim to
               have much larger numbers of valid set elements than they
               actually hold, or request the retransmission of elements
               that they have already received in previous
               interactions.  Bounding resources consumed by malicous
               actors is important to ensure that higher-level protocols
               can use set reconciliation and still meet their resource
               targets.  This can be particularly critical in multi-round
               synchronous consensus protocols where peers that cannot
               answer in a timely fashion would have to be treated as
               failed or malicious.
             </t>
             <t>
               To defend against some of these attacks, applications
               SHOULD remember the number of elements previously
               shared with a peer, and SHOULD provide a way to check that
               elements are well-formed. Applications MAY also
               provide an upper bound on the total number of valid
               elements that exist. For example, in E-voting, the
               number of eligible voters MAY be used to provide such
               an upper bound.
             </t>
             <t>
               A first draft of this RFC is part of Elias Summermatter's
               bachelor thesis. Many of the algorithms and parameters
               documented in this RFC are derived from experiments
               detailed in this thesis.
               <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/>
             </t>
 
             <t>
               This document defines the normative wire format of resource records, resolution processes,
               cryptographic routines and security considerations for use by implementors.
               SETU requires a bidirectional secure communication channel between the two parties.
               Specification of the communication channel is out of scope of this document.
             </t>
             <t>
                 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL
                 NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and
                 "OPTIONAL" in this document are to be interpreted as described
                 in <xref target="RFC2119"/>.
             </t>
         </section>
 
         <section anchor="background" numbered="true" toc="default">
             <name>Background</name>
             <section anchor="bf" numbered="true" toc="default">
                 <name>Bloom Filter</name>
                 <t>
                   A Bloom filter (BF) is a space-efficient probabilistic
                   datastructure to test if an element is part of a set of elements.
                   Elements are identified by an element ID.
                   Since a BF is a probabilistic datastructure, it is possible to have false-positives: when asked
                   if an element is in the set, the answer from a BF is either "no" or "maybe".
                 </t>
                 <t>
                     A BF consists of L buckets. Every bucket is a binary value that can be either 0 or 1. All buckets are initialized
                     to 0.  A mapping function M is used to map each ID of each element from the set to a subset of k buckets.  In the original proposal by Bloom, M is non-injective
                     and can thus map the same element multiple times to the same bucket.
                     The type of the mapping function can thus be described by the following mathematical notation:
                 </t>
                 <figure anchor="bf_mapping_function_math">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
             ------------------------------------
             # M: E->B^k
             ------------------------------------
             # L = Number of buckets
             # B = 0,1,2,3,4,...L-1 (the buckets)
             # k = Number of buckets per element
             # E = Set of elements
             ------------------------------------
             Example: L=256, k=3
             M('element-data') = {4,6,255}
 
                      ]]></artwork>
                 </figure>
                 <t>
                     A typical mapping function is constructed by hashing the element, for example
                     using the well-known <relref  section="2" target="RFC5869" displayFormat="of">HKDF construction</relref>.
                 </t>
                 <t>
                     To add an element to the BF, the corresponding buckets under the map M are set to 1.
                     To check if an element may be in the set, one tests if all buckets under the map M are set to 1.
                 </t>
                 <t>
                     In the BF the buckets are set to 1 if the corresponding bit in the bitstream is 1.
                     If there is a collision and a bucket is already set to 1, the bucket stays at 1.
                 </t>
                 <t>
                     In the following example the element e0 with M(e0) = {1,3} has been added:
                 </t>
                     <figure anchor="figure_bf_insert_0">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
                 bucket-0     bucket-1       bucket-2      bucket-3
             +-------------+-------------+-------------+-------------+
             |      0      |      1      |      0      |      1      |
             +-------------+-------------+-------------+-------------+
                      ]]></artwork>
                     </figure>
                 <t>
                   It is easy to see that an element e1 with M(e1) = {0,3}
                   could have been added to the BF below, while an element e2
                   with M(e2) = {0,2} cannot be in the set represented by the
                   BF below:
                 </t>
 
                 <figure anchor="figure_bf_contains">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
                 bucket-0     bucket-1       bucket-2      bucket-3
             +-------------+-------------+-------------+-------------+
             |      1      |      0      |      0      |      1      |
             +-------------+-------------+-------------+-------------+
                      ]]></artwork>
                 </figure>
                 <t>
                   The parameters L and k depend on the set size and MUST be
                   chosen carefully to ensure that the BF does not return too
                   many false-positives.
                 </t>
                 <t>
                     It is not possible to remove an element from the BF because buckets can only be set to 1 or 0. Hence it is impossible to
                     differentiate between buckets containing one or more elements. To remove elements from the BF a <xref target="cbf" format="title" />
                     is required.
                 </t>
             </section>
 
             <section anchor="cbf" numbered="true" toc="default">
                 <name>Counting Bloom Filter</name>
                 <t>
                   A Counting Bloom Filter (CBF) is a variation on the idea
                   of a <xref target="bf" format="title" />. With a CBF, buckets are
                   unsigned numbers instead of binary values.
                   This allows the removal of an element from the CBF.
                 </t>
                 <t>
                   Adding an element to the CBF is similar to the adding operation of the BF.
                   However, instead of setting the buckets to 1 the
                   numeric value stored in the bucket is increased by 1.
                   For example, if two colliding elements M(e1) = {1,3} and
                   M(e2) = {0,3} are added to the CBF, bucket 0 and 1 are set
                   to 1 and bucket 3 (the colliding bucket) is set to 2:
                 </t>
                 <figure anchor="figure_cbf_insert_0">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
                 bucket-0     bucket-1       bucket-2      bucket-3
             +-------------+-------------+-------------+-------------+
             |      1      |      1      |      0      |      2      |
             +-------------+-------------+-------------+-------------+
                      ]]></artwork>
                 </figure>
                 <t>
                   The counter stored in the bucket is also called the order of the bucket.
                 </t>
                 <t>
                   To remove an element form the CBF the counters of all buckets
                   the element is mapped to are decreased by 1.
                 </t>
                 <t>
                   For example, removing M(e2) = {1,3} from the CBF above
                   results in:
                 </t>
                 <figure anchor="figure_cbf_remove_0">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
                 bucket-0     bucket-1       bucket-2      bucket-3
             +-------------+-------------+-------------+-------------+
             |      1      |      0      |      0      |      1      |
             +-------------+-------------+-------------+-------------+
                      ]]></artwork>
                 </figure>
                 <t>
                   In practice, the number of bits available for the counters
                   is often finite. For example, given a 4-bit
                   counter, a CBF bucket would overflow 16 elements are mapped
                   to the same bucket. To handle this case, the maximum value
                   (15 in our example) is considered to represent "infinity". Once the
                   order of a bucket reaches "infinity", it is no longer incremented or decremented.
                 </t>
                 <t>
                   The parameters L and k and the number of bits allocated to the counters
                   SHOULD depend on the set size.
                   A CBF will degenerate when subjected to insert and remove iterations of
                   different elements, and eventually all buckets will reach "infinity".
                   The speed of the degradation will depend on the choice of L and k in
                   relation to the number of elements stored in the IBF.
                 </t>
             </section>
         </section>
 
         <section anchor="ibf" numbered="true" toc="default">
         <name>Invertible Bloom Filter</name>
             <t>
                 An Invertible Bloom Filter (IBF) is a further extension of the <xref target="cbf" format="title" />.
                 An IBF extends the <xref target="cbf" format="title" /> with two more operations:
                 decode and set difference. This two extra operations are key to efficiently obtain
                 small differences between large sets.
             </t>
             <section anchor="ibf_structure" numbered="true" toc="default">
                 <name>Structure</name>
                 <t>
                   An IBF consists of an injective mapping function M mapping
                   elements to k out of L buckets. Each of the L buckets stores
                   a signed COUNTER, an IDSUM and an XHASH.
                   An IDSUM is the XOR of various element IDs.
                   An XHASH is the XOR of various hash values.
                   As before, the values used for k, L and the number of bits used
                   for the signed counter and the XHASH depend
                   on the set size and various other trade-offs.
                 </t>
                 <t>
                   If the IBF size is too small or the mapping
                   function does not spread out the elements
                   uniformly, the signed counter can overflow or
                   underflow. As with the CBF, the "maximum" value is
                   thus used to represent "infinite".  As there is no
                   need to distinguish between overflow and
                   underflow, the most canonical representation of
                   "infinite" would be the minimum value of the
                   counter in the canonical 2-complement
                   interpretation.  For example, given a 4-bit
                   counter a value of -8 would be used to represent
                   "infinity".
                 </t>
                     <figure anchor="figure_ibf_structure">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+-------
   count |   COUNTER   |   COUNTER   |   COUNTER   |   COUNTER   |  C...
         +-------------+-------------+-------------+-------------+------
   idSum |    IDSUM    |    IDSUM    |    IDSUM    |     IDSUM   |  I...
         +-------------+-------------+-------------+-------------+------
 hashSum |   HASHSUM   |   HASHSUM   |   HASHSUM   |    HASHSUM  |  H..
         +-------------+-------------+-------------+-------------+-------
                  ]]></artwork>
                     </figure>
 
             </section>
 
             <section anchor="ibf_format_id_generation" numbered="true" toc="default">
               <name>Salted Element ID Calculation</name>
               <t>
                 IBFs are a probabilistic data structure, hence it can be necessary to
                 recompute the IBF in case operations fail, and then try again. The
                 recomputed IBF would ideally be statistically independent of the
                 failed IBF. This is achieved by introducing an IBF-salt. Given that with
                 benign peers failures should be rare, and that we need to be able to
                 "invert" the application of the IBF-salt to the element IDs, we use an
                 unsigned 32 bit non-random IBF-salt value of which the lowest 6 bits will
                 be used to rotate bits in the element ID computation.
               </t>
               <t>
                 64-bit element IDs are generated from a
                 <relref  section="2" target="RFC5869" displayFormat="of">HKDF construction</relref>
                 with HMAC-SHA512 as XTR and HMAC-SHA256 as PRF with a 16-bit KDF-salt set to a
                 unsigned 16-bit representation of zero.
                 The output of the KDF is then truncated to 64-bit.
                 Finally, salting is done by calculating the IBF-salt modulo 64
                 (effectively using only the lowest 6-bits of the IBF-salt)
                 and doing a bitwise right rotation of the output of KDF.  We
                 note that this operation was chosen as it is easily inverted,
                 allowing applications to easily derive element IDs with one
                 IBF-salt value from element IDs generated with a different
                 IBF-salt value.
               </t>
               <t>
                 In case the IBF does not decode, the IBF-salt can be changed to
                 compute different element IDs, which will (likely) be mapped
                 to different buckets, likely allowing the IBF to decode in a
                 subsequent iteration.
               </t>
               <figure anchor="ibf_format_id_generation_pseudo_code">
                 <artwork name="" type="" align="left" alt=""><![CDATA[
 # INPUTS:
 # key: Pre calculated and truncated key from id_calculation function
 # ibf_salt: Salt of the IBF
 # OUTPUT:
 # value: salted key
 FUNCTION salt_key(key,ibf_salt):
   s = (ibf_salt * 7) modulo 64;
   /* rotate key */
   return (key >> s) | (key << (64 - s))
 END FUNCTION
 
 
 # INPUTS:
 # element: element for which we are to calculate the element ID
 # ibf_salt: Salt of the IBF
 # OUTPUT:
 # value: the ID of the element
 FUNCTION id_calculation (element,ibf_salt):
     kdf_salt = 0 // 16 bits
     XTR=HMAC-SHA256
     PRF=HMAC-SHA256
     key = HKDF(XTR, PRF, kdf_salt, element) modulo 2^64
     return salt_key(key, ibf_salt)
 END FUNCTION
                  ]]></artwork>
               </figure>
             </section>
 
             <section anchor="ibf_format_HASH_calculation" numbered="true" toc="default">
               <name>HASH calculation</name>
               <t>
                 The HASH of an element ID is computed by calculating the
                 CRC32 checksum of the 64-bit ID value,
                 which returns a 32-bit value.CRC32 is well-known and described in <relref  section="4.1" target="RFC3385" displayFormat="of">the RFC</relref>.
               </t>
             </section>
 
             <section anchor="ibf_m" numbered="true" toc="default">
               <name>Mapping Function</name>
               <t>
                 The mapping function M decides which buckets a given ID is mapped to.
                 For an IBF, it is beneficial to use an injective mapping function M.
               </t>
               <t>
                 The first index is simply the CRC32 of the ID modulo the IBF size. The second
                 index is calculated by creating a new 64-bit value by shifting the previous 32-bit
                 value left and setting the lower 32 bits to the number of indices already processed.
                 From the resulting 64-bit value, another CRC32 checksum is computed.
                 The subsequent index is the modulo of this CRC32 output.
                 The process is repeated until the desired number of indices is generated.
                 In the case the process computes the same index twice,
                 which would mean this bucket could not get pure again,
                 the second hit is just skipped and the next iteration is used instead,
                 creating an injective mapping function.
               </t>
               <figure anchor="ibf_format_bucket_identification_pseudo_code">
                 <artwork name="" type="" align="left" alt=""><![CDATA[
 # INPUTS:
 # key: the ID of the element calculated
 # k: numbers of buckets per element
 # L: total number of buckets in the IBF
 # OUTPUT:
 # dst: Array with k bucket IDs
 FUNCTION get_bucket_id (key, k, L)
   bucket = CRC32(key)
   i = 0 // unsigned 32-bit index
   filled = 0
   WHILE filled < k DO
 
     element_already_in_bucket = false
     j = 0
     WHILE j < filled DO
       IF dst[j] == bucket modulo L THEN
         element_already_in_bucket = true
       END IF
       j++
     END WHILE
 
     IF !element_already_in_bucket THEN
         dst[filled] = bucket modulo L
         filled = filled + 1
     END IF
 
     x = (bucket << 32) | i // 64 bit result
     bucket = CRC32(x)
     i = i + 1
   END WHILE
   return dst
 END FUNCTION
   ]]></artwork>
               </figure>
             </section>
 
             <section anchor="ibf_operations" numbered="true" toc="default">
               <name>Operations</name>
               <t>
                 When an IBF is created, all counters and IDSUM and HASHSUM values of
                 all buckets are initialized to zero.
               </t>
                 <section anchor="ibv_operations_insert" numbered="true" toc="default">
                     <name>Insert Element</name>
                     <t>
                       To add an element to an IBF, the element is mapped to a subset of k buckets using
                       the injective mapping function M as described in section <xref target="ibf_m" format="title" />. For the buckets selected by the mapping function, the counter is increased by one and the
                       IDSUM field is set to the XOR of the element ID
                       computed as described in section <xref target="ibf_format_id_generation" format="title" />
                       and the previously stored IDSUM. Furthermore,
                       the HASHSUM is set to the XOR of the previously stored HASHSUM
                       and the hash of the element ID computed as described
                       in section <xref target="ibf_format_HASH_calculation" format="title" />.
                     </t>
                     <t>
                       In the following example, the insert operation is illustrated using an element with the
                       ID 0x0102 mapped to {1,3} with a hash of 0x4242, and a second element with the
                       ID 0x0304 mapped to {0,1} and a hash of 0x0101.
                     </t>
                     <t>Empty IBF:</t>
                     <figure anchor="figure_ibf_insert_0">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      0      |      0      |      0      |      0      |
         +-------------+-------------+-------------+-------------+
   idSum |    0x0000   |    0x0000   |    0x0000   |    0x0000   |
         +-------------+-------------+-------------+-------------+
 hashSum |    0x0000   |    0x0000   |    0x0000   |    0x0000   |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                     </figure>
                     <t>Insert first element with ID 0x0102 and hash 0x4242 into {1,3}:</t>
                     <figure anchor="figure_ibf_insert_1">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      0      |      1      |      0      |      1      |
         +-------------+-------------+-------------+-------------+
   idSum |    0x0000   |   0x0102    |    0x0000   |   0x0102    |
         +-------------+-------------+-------------+-------------+
 hashSum |    0x0000   |   0x4242    |    0x0000   |   0x4242    |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                     </figure>
                     <t>Insert second element with ID 0x0304 and hash 0101 into {0,1}:</t>
                     <figure anchor="figure_ibf_insert_2">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      1      |      2      |      0      |      1      |
         +-------------+-------------+-------------+-------------+
   idSum |    0x0304   |   0x0206    |    0x0000   |   0x0102    |
         +-------------+-------------+-------------+-------------+
 hashSum |    0x0101   |   0x4343    |    0x0000   |   0x4242    |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                     </figure>
                 </section>
                 <section anchor="ibf_operations_remove" numbered="true" toc="default">
                     <name>Remove Element</name>
                     <t>
                         To remove an element from the IBF the element is again mapped to a subset of the buckets using M.
                         Then all the counters of the buckets selected by M are reduced by one, the IDSUM is
                         replaced by the XOR of the old IDSUM and the ID of the element being removed, and the
                         HASHSUM is similarly replaced with the XOR of the old HASHSUM and the hash of the ID.
                     </t>
                     <t>
                         In the following example the remove operation is illustrated.
                     </t>
                 <t>IBF with two encoded elements:</t>
                 <figure anchor="figure_ibf_remove_0">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      1      |      2      |      0      |      1      |
         +-------------+-------------+-------------+-------------+
   idSum |    0x0304   |   0x0206    |    0x0000   |   0x0102    |
         +-------------+-------------+-------------+-------------+
 hashSum |   0x0101    |   0x4343    |    0x0000   |   0x4242    |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                 </figure>
                     <t>After removal of element with ID 0x0304 and hash 0x0101 mapped to {0,1} from the IBF:</t>
                     <figure anchor="figure_ibf_remove_1">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      0      |      1      |      0      |      1      |
         +-------------+-------------+-------------+-------------+
   idSum |    0x0000   |   0x0102    |    0x0000   |   0x0102    |
         +-------------+-------------+-------------+-------------+
 hashSum |    0x0000   |   0x4242    |    0x0000   |   0x4242    |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                     </figure>
                     <t>
                       Note that it is possible to "remove" elements from an IBF that were never present
                       in the IBF in the first place. A negative counter value is thus indicative of
                       elements that were removed without having been added.  Note that an IBF bucket
                       counter of zero no longer guarantees that an element mapped to that bucket is not
                       present in the set: a bucket with a counter of zero can be the result of one
                       element being added and a different element (mapped to the same bucket) being removed.
                       To check that an element is not present requires a counter of zero and an
                       IDSUM and HASHSUM of zero --- and some certainty that there was no collision due
                       to the limited number of bits in IDSUM and HASHSUM.  Thus,
                       IBFs are not suitable to replace BFs or IBFs.
                     </t>
                     <t>
                       Buckets in an IBF with a counter of 1 or -1 are crucial for decoding an IBF, as
                       they MIGHT represent only a single element, with the IDSUM being the ID of that element.
                       Following Eppstein <xref target="Eppstein" format="default" />, we will call buckets that only represent a single
                       element <em>pure buckets</em>.
                       Note that due to the possibility of multiple insertion and removal operations
                       affecting the same bucket, not all buckets with a counter of 1 or -1 are
                       actually pure buckets.  Sometimes a counter can be 1 or -1 because N elements
                       mapped to that bucket were added while N-1 or N+1 different elements also
                       mapped to that bucket were removed.
                     </t>
                 </section>
 
                 <section anchor="ibf_operations_decode" numbered="true" toc="default">
                     <name>Extracting elements</name>
                     <t>
                       Extracting elements from an IBF yields IDs of elements from the IBF.
                       Elements are extracted from an IBF by repeatedly performing a
                       decode operation on the IBF.
                     </t>
                     <t>
                       A decode operation requires a pure bucket, that is a bucket to which M
                       only mapped a single element, to succeed.  Thus, if there is no bucket with
                       a counter of 1 or -1, decoding fails. However, as a counter of 1 or -1 is
                       not a guarantee that the bucket is pure, there is also a chance that the
                       decoder returns an IDSUM value that is actually the XOR of several IDSUMs.
                       This is primarily detected by checking that the HASHSUM is the hash of the IDSUM.
                       Only if the HASHSUM also matches, the bucket could be pure.  Additionally,
                       one MUST check that the IDSUM value actually would be mapped by M to
                       the respective bucket. If not, there was a hash collision and the bucket
                       is also not pure.
                     </t>
                     <t>
                       The very rare case that after all these checks a bucket is still
                       falsely identified as pure MUST be detected (say by determining that
                       extracted element IDs do not match any actual elements), and addressed
                       at a higher level in the protocol. As these failures are probabilistic
                       and depend on element IDs and the IBF construction, they can typically
                       be avoided by retrying with different parameters, such as a different
                       way to assign element IDs to elements (by varying the IBF-salt),
                       using a larger value for L, or a different mapping function M.
                       A more common scenario (especially if L was too small) is that
                       IBF decoding fails because there is no pure bucket. In this case, the
                       higher-level protocol generally MUST also retry using different
                       parameters (except if an attack is detected).
                     </t>
                     <t>
                       Suppose the IBF contains a pure bucket. In this case, the IDSUM in the
                       bucket is the ID of an element.  Furthermore, it is then possible
                       to remove that element from the IBF (by inserting it if the counter
                       was negative, and by removing it if the counter was positive). This
                       is likely to cause other buckets to become pure, allowing further
                       elements to be decoded.  Eventually, decoding ought to finish with
                       all counters and IDSUM and HASHSUM values reach zero. However, it is also
                       possible that an IBF only partly decodes and then decoding fails due
                       to the lack of pure buckets after extracting some element IDs.
                     </t>
                     <t>
                       In the following example the successful decoding of an IBF containing
                       the two elements previously added in our running example.
                     </t>
                     <t>
                        We begin with an IBF with two elements added:
                     </t>
                     <figure anchor="figure_ibf_decode_0">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      1      |      2      |      0      |      1      |
         +-------------+-------------+-------------+-------------+
   idSum |   0x0304    |   0x0206    |    0x0000   |   0x0102    |
         +-------------+-------------+-------------+-------------+
 hashSum |   0x0101    |   0x4343    |    0x0000   |   0x4242    |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                     </figure>
                     <t>
                       In the IBF are two pure buckets to decode (buckets 0 and 3) we choose to start with
                       decoding bucket 0. This yields the element with the hash ID 0x0304 and
                       hash 1010. This element ID is mapped to buckets
                       {0,1}.
                       Subtracting this element results in bucket 1 becoming pure:
                     </t>
                     <figure anchor="figure_ibf_decode_1">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      0      |      1      |      0      |      1      |
         +-------------+-------------+-------------+-------------+
   idSum |    0x0000   |   0x0102    |    0x0000   |   0x0102    |
         +-------------+-------------+-------------+-------------+
 hashSum |    0x0000   |   0x4242    |    0x0000   |   0x4242    |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                     </figure>
                     <t>
                         We can now decoding bucket 2 and extract the element
                         with the ID 0x0102 and hash 0x4242. Now the IBF is
                         empty. Extraction completes with the status that
                         the IBF has been successfully decoded.
                     </t>
                     <figure anchor="figure_ibf_decode_2">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      0      |      0      |      0      |      0      |
         +-------------+-------------+-------------+-------------+
   idSum |    0x0000   |    0x0000   |    0x0000   |    0x0000   |
         +-------------+-------------+-------------+-------------+
 hashSum |    0x0000   |    0x0000   |    0x0000   |    0x0000   |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                     </figure>
                 </section>
 
                 <section anchor="ibv_operations_setdiff" numbered="true" toc="default">
                     <name>Set Difference</name>
                     <t>
                         Given addition and removal as defined above, it is possible to define an operation on IBFs
                         that computes an IBF representing the set difference.  Suppose IBF1 represents set A, and
                         IBF2 represents set B.  Then this set difference operation will compute IBF3 which
                         represents the set A - B. Note that this computation can be done only on the IBFs,
                         and does not require access to the elements from set A or B.
 
                         To calculate the IBF representing this set difference, both IBFs MUST have the same
                         length L, the same number of buckets per element k and use the same map M.
                         Naturally, all IDs must have been computed using the same IBF-salt.  Given this,
                         one can compute the IBF representing the set difference by taking the XOR of the IDSUM and HASHSUM values
                         of the respective buckets and subtracting the respective counters.  Care MUST be taken
                         to handle overflows and underflows by setting the counter to "infinity" as necessary.
                         The result is a new IBF with the same number of buckets representing the set difference.
                     </t>
                     <t>
                         This new IBF can be decoded as described in section <xref target="ibf_operations_decode" format="counter" />.
                         The new IBF can have two types of pure buckets with counter set to 1 or -1. If the counter is set to 1
                         the element is missing in the secondary set, and if the counter is set to -1 the element is missing in
                         the primary set.
                     </t>
                     <t>
                         To demonstrate the set difference operation we compare IBF-A with IBF-B and generate as described
                         IBF-AB
                     </t>
                     <t>IBF-A contains the elements with ID 0x0304 and hash 0x0101 mapped to {0,1},
                        and ID 0x0102 and hash 0x4242 mapped to {1,3}:</t>
                     <figure anchor="figure_ibf_setdiff_A">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      1      |      2      |      0      |      1      |
         +-------------+-------------+-------------+-------------+
   idSum |    0x0304   |   0x0206    |    0x0000   |   0x0102    |
         +-------------+-------------+-------------+-------------+
 hashSum |    0x0101   |   0x4343    |    0x0000   |   0x4242    |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                     </figure>
                     <t>IBF-B also contains the element with ID 0x0102 and
                      and another element with ID 0x1345 and hash 0x5050
                      mapped to {1,2}.</t>
                     <figure anchor="figure_ibf_setdiff_B">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      0      |      1      |      1      |      1      |
         +-------------+-------------+-------------+-------------+
   idSum |    0x0000   |    0x1447   |    0x1345   |    0x0102   |
         +-------------+-------------+-------------+-------------+
 hashSum |    0x0000   |    0x9292   |    0x5050   |    0x4242   |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                     </figure>
                 <t>IBF-A minus IBF-B is then:</t>
                 <figure anchor="figure_ibf_setdiff_AB">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
             bucket-0     bucket-1       bucket-2      bucket-3
         +-------------+-------------+-------------+-------------+
   count |      1      |      0      |      -1     |      0      |
         +-------------+-------------+-------------+-------------+
   idSum |    0x0304   |    0x1049   |    0x1345   |    0x0000   |
         +-------------+-------------+-------------+-------------+
 hashSum |    0x0101   |    0x5151   |    0x5050   |    0x0000   |
         +-------------+-------------+-------------+-------------+
                  ]]></artwork>
                 </figure>
                 <t>After calculating and decoding the IBF-AB shows clear that in IBF-A the element with the hash 0x5050
                     is missing (-1 in bucket 2) while in IBF-B the element with the hash 0101 is missing
                     (1 in bucket 0). The element with hash 0x4242 is present in IBF-A and IBF-B and is
                     removed by the set difference operation.  Bucket 2 is not empty.
                 </t>
             </section>
             </section>
 
             <section anchor="ibf_format" numbered="true" toc="default">
                 <name>Wire format</name>
                 <t>
                     For the counter field, we use a variable-size encoding to ensure
                     that even for very large sets the counter should never reach
                     "infinity", while also ensuring that the encoding is compact for
                     small sets.
                     Hence, the counter size transmitted over the wire
                     varies between 1 and 64 bits, depending on the
                     maximum counter in the IBF. A range of 1 to 64 bits
                     should cover most areas of application and can be
                     efficiently implemented on most contemporary CPU
                     architectures and programming languages.
                     The bit length for the transmitted IBF
                     will be communicated in the header of the
                     <em><xref target="messages_ibf" format="title" /></em> message
                     in the "IMCS" field as unsigned 8-bit integer.
                     For implementation details see section <xref target="performance_counter_variable_size" format="title" />.
                 </t>
                 <t>
                     For the "IDSUM", we always use a 64-bit representation.
                     The IDSUM value MUST have sufficient entropy for the
                     mapping function M to yield reasonably random buckets
                     even for very large values of L. With a 32 bit
                     value the chance that multiple elements may be mapped
                     to the same ID would be quite high, even for moderately
                     large sets.  Using more than 64 bits would at best make
                     sense for very large sets, but then it is likely always
                     better to simply afford additional round trips to handle
                     the occasional collision. 64 bits are also a reasonable
                     size for many CPU architectures.
                 </t>
                 <t>
                     For the "HASHSUM", we always use a 32-bit
                     representation.  Here, it is most important to
                     avoid collisions, where different elements are
                     mapped to the same hash, possibly resulting in
                     a bucket being falsely classified as pure.
                     While with 32 bits there remains a non-negligible chance of
                     accidental collisions, our protocol is designed
                     to handle occasional collisions. Hence, at 32 bit the chance is
                     believed to be sufficiently small enough
                     for the protocol to handle those cases efficiently.  Smaller hash
                     values would safe bandwidth, but also substantially
                     increase the chance of collisions. 32 bits are
                     also again a reasonable size for many CPU
                     architectures.
                 </t>
           </section>
         </section>
 
         <section anchor="se" numbered="true" toc="default">
             <name>Strata Estimator</name>
                 <t>
                     Strata Estimators help estimate the size of the set difference between two sets of elements.
                     This is necessary to efficiently determinate the tuning parameters for an IBF, in particular
                     a good value for L.
                 </t>
                 <t>
                     Basically a Strata Estimator (SE) is a series of IBFs (with a rather small value of L=79)
                     in which increasingly large subsets of the full set
                     of elements are added to each IBF.  For the n-th IBF, the function selecting the
                     subset of elements MUST sample to select (probabilistically) 1/(2^n) of all
                     elements.  This can be done by counting the number of trailing bits set to "1"
                     in an element ID, and then inserting the element into the IBF identified by that
                     counter.  As a result, all elements will be mapped to one IBF, with the n-th
                     IBF being statistically expected to contain 1/(2^n) elements.
                 </t>
                 <t>
                     Given two SEs, the set size difference can be estimated by attempting to decode all of the
                     IBFs.  Given that L is set to a fixed and rather small value, IBFs containing large strata
                     will likely fail to decode.  For IBFs that failed to decode, one simply
                     extrapolates the number of elements by scaling the numbers obtained from the
                     other IBFs that did decode.  If none of the IBFs of the SE decoded (which given
                     a reasonable number of IBFs in the SE should be highly unlikely), one can theoretically
                     retry using a different IBF-salt.
                 </t>
                 <t>
                     When decoding the IBFs in the strata estimator, it is possible to determine
                     on which side which part of the difference is. For this purpose, the pure buckets with
                     counter 1 and -1 must be distinguished and assigned to the respective side when decoding
                     the IBFs.
                 </t>
         </section>
 
         <section anchor="modeofoperation" numbered="true" toc="default">
             <name>Mode of Operation</name>
             <t>
               Depending on the state of the two sets the set union
               protocol uses different modes of operation to efficiently
               determinate missing elements between the two sets.
             </t>
             <t>
               The simplest mode is the <em>full synchronisation mode</em>.
               If the difference between the sets of the two
               peers exceeds a certain threshold, the overhead to determine
               which elements are different would outweigh the overhead of
               simply sending the complete set. Hence, the protocol may
               determine that the most efficient method is to exchange the full sets.
             </t>
             <t>
               The second possibility is that the difference between the sets
               is relatively small compared to the set size.
               In this case, the <em>differential synchronisation mode</em> is more efficient.
               Given these two possibilities, the first steps of the protocol are used to
               determine which mode MUST be used.
             </t>
             <t>
               Thus, the set union protocol always begins with the following operation mode independent steps:
             </t>
 
             <t>
               The initiating peer begins in the <strong>Initiating Connection</strong> state and the receiving peer in the <strong>Expecting Connection</strong>
               state. The first step for the initiating peer in the protocol is to send an <em><xref target="messages_operation_request" format="title" /></em> to the receiving peer and
               transition into the <strong>Expect SE</strong> state. After receiving the <em><xref target="messages_operation_request" format="title" /></em> the receiving peer
               transitions to the <strong>Expecting IBF</strong> state and answers with the
               <em><xref target="messages_se" format="title" /></em> message. When the initiating peer receives the <em><xref target="messages_se" format="title" /></em> message,
               it decides with some heuristics which operation mode is likely more suitable for the estimated set difference and the application-provided latency-bandwidth tradeoff.
               The detailed algorithm used to choose between the <xref target="modeofoperation_full-sync" format="title" /> and the <xref target="modeofoperation_individual-elements" format="title" />
               is explained in the section <xref target="modeofoperation_combined-mode" format="title" /> below.
             </t>
             <section anchor="modeofoperation_full-sync" numbered="true" toc="default">
               <name>Full Synchronisation Mode</name>
               <t>
                 When the initiating peer decides to use the full synchronisation mode and it is better that the other peer sends his set first, the initiating
                 peer sends a <em><xref target="messages_request_full" format="title" /></em> message, and transitions from <strong>Expecting SE</strong> to the <strong>Full Receiving</strong> state.
                 If it has been determined that it is better that the initiating peer sends his set first, the initiating peer sends a <em><xref target="messages_send_full" format="title" /></em> message followed by all
                 set elements in <em><xref target="messages_full_element" format="title" /></em> messages to the other peer, followed by the <em><xref target="messages_full_done" format="title" /></em> message, and transitions into the <strong>Full Sending</strong> state.
               </t>
               <t>
                 A state diagram illustrating the state machine used during full synchronization
                 is provided
                 <eref target="https://git.gnunet.org/lsd0003.git/plain/statemachine/state_machine_full.png">here</eref>.
               </t>
               <t><strong>The behavior of the participants the different state is described below:</strong></t>
               <dl>
                 <dt><strong>Expecting IBF:</strong></dt>
                 <dd>
                   If a peer in the <strong>Expecting IBF</strong> state receives a <em><xref target="messages_request_full" format="title" /></em> message from the other peer, the
                   peer sends all the elements of his set followed by a <em><xref target="messages_full_done" format="title" /></em> message to the other peer, and transitions to the
                   <strong>Full Sending</strong> state. If the peer receives an <em><xref target="messages_send_full" format="title" /></em> message followed by
                   <em><xref target="messages_full_element" format="title" /></em> messages, the peer processes the element and transitions to the <strong>Full Receiving</strong> state.
                 </dd>
                 <dt><strong>Full Sending:</strong></dt>
                 <dd>
                   While a peer is in <strong>Full Sending</strong> state the peer expects to continuously receive elements from the other
                   peer. As soon as a the <em><xref target="messages_full_done" format="title" /></em> message is received, the peer transitions into
                   the <strong>Finished</strong> state.
                 </dd>
                 <dt><strong>Full Receiving: </strong></dt>
                 <dd>
                   While a peer is in the <strong>Full Receiving</strong> state, it expects to continuously receive elements from the other
                   peer. As soon as a the <em><xref target="messages_full_done" format="title" /></em> message is received, it sends
                   the remaining elements (those it did not receive) from his set to the other
                   peer, followed by a <em><xref target="messages_full_done" format="title" /></em>.
                   After sending the last message, the peer transitions into the <strong>Finished</strong> state.
                 </dd>
               </dl>
             </section>
             <section anchor="modeofoperation_individual-elements" numbered="true" toc="default">
               <name>Differential Synchronisation Mode</name>
               <t>
                 The message format used by the protocol limits the maximum message size to
                 64 kb. Given that L can be large, an IBF will not always fit within that
                 size limit. To deal with this, larger IBFs are split into multiple messages.
               </t>
               <t>
                 When the initiating peer in the <strong>Expected SE</strong> state decides to use the differential synchronisation mode, it
                 sends an IBF, which may
                 consist of several <em><xref target="messages_ibf" format="title" /></em> messages,
                 to the receiving peer and transitions into the <strong>Passive Decoding</strong> state.
               </t>
               <t>
                 The receiving peer in the <strong>Expecting IBF</strong> state receives the
                 first <em><xref target="messages_ibf" format="title" /></em> message from
                 the initiating peer, and transitions into the <strong>Expecting IBF Last</strong> state
                 if the IBF was split into multiple <em><xref target="messages_ibf" format="title" /></em>
                 messages. If there is just a single <em><xref target="messages_ibf" format="title" /></em>
                 message, the receiving peer
                 transitions directly to the <strong>Active Decoding</strong> state.
               </t>
               <t>
                 The peer that is in the <strong>Active Decoding</strong>, <strong>Finish Closing</strong> or in the <strong>Expecting IBF Last</strong>
                 state is called the active peer, and the peer that is in either the <strong>Passive Decoding</strong> or the <strong>Finish Waiting</strong> state
                 is called the passive peer.
               </t>
               <t>
                 A state diagram illustrating the state machine used during differential synchronization
                 is provided
                 <eref target="https://git.gnunet.org/lsd0003.git/plain/statemachine/differential_state_machine.png">here</eref>.
               </t>
               <t><strong>The behavior of the participants the different states is described below:</strong></t>
               <dl>
                 <dt><strong>Passive Decoding:</strong></dt>
                 <dd>
                   <t>
                     In the <strong>Passive Decoding</strong> state the passive peer reacts to requests from the active peer.
                     The action the passive peer executes depends on the message the passive peer receives in the <strong>Passive Decoding</strong> state from the active peer
                     and is described below on a per message basis.
                   </t>
 
                   <dl>
                     <dt><em><xref target="messages_inquiry" format="title" /></em> message:</dt>
                     <dd>
                       The <em><xref target="messages_inquiry" format="title" /></em> message
                       is received if the active peer requests the SHA-512 hash of one or more elements (by sending the 64 bit element ID)
                       that are missing from the active peer's set.
                       In this case the passive peer answers with <em><xref target="messages_offer" format="title" /></em> messages
                       which contain the SHA-512 hash of the requested element.  If the passive peer does not have an element with
                       a matching element ID, it MUST ignore the inquiry (in this case, a bucket was falsely classified as pure, decoding the IBF will eventually fail, and roles will be swapped).
                       It should be verified that after an falsely classified pure bucket a role change is made.
                       If multiple elements match the 64 bit element ID, the passive
                       peer MUST send offers for all of the matching elements.
                     </dd>
                     <dt><em><xref target="messages_demand" format="title" /></em> message:</dt>
                     <dd>
                       The <em><xref target="messages_demand" format="title" /></em> message
                       is received if the active peer requests a complete element that is missing in the active peers set in response to an offer. If the requested element is known and has not yet been transmitted
                       the passive peer answers with an <em><xref target="messages_elements" format="title" /></em> message which contains the full,
                       application-dependent data of the requested element.  If the passive peer receives a demand for a SHA-512 hash for which
                       it has no corresponding element, a protocol violation is detected and the protocol MUST be aborted.
                       Implementations MUST also abort when facing demands without previous matching offers or for which the passive peer previously transmitted the element to the active peer.
                     </dd>
                     <dt><em><xref target="messages_offer" format="title" /></em> message:</dt>
                     <dd>
                       The <em><xref target="messages_offer" format="title" /></em> message
                       is received if the active peer has decoded an element that is present in the active peers set and is likely be missing in the
                       set of the passive peer. If the SHA-512 hash of the offer is indeed not a hash of any of the elements from the set of
                       the passive peer, the passive peer MUST answer with a <em><xref target="messages_demand" format="title" /></em> message
                       for that SHA-512 hash and remember that it issued this demand. The demand thus needs to be added to a list with unsatisfied demands.
                     </dd>
                     <dt><em><xref target="messages_elements" format="title" /></em> message:</dt>
                     <dd>
                       When a new <em><xref target="messages_elements" format="title" /></em> message has been received the peer checks if a corresponding
                       <em><xref target="messages_demand" format="title" /></em> for the element has been sent
                       and the demand is still unsatisfied.
                       If the element has been demanded the peer checks the element for validity, removes it from the list
                       of pending demands and then saves the element to the set. Otherwise the peer
                       ignores the element.
                     </dd>
                     <dt><em><xref target="messages_ibf" format="title" /></em> message:</dt>
                     <dd>
                       If an <em><xref target="messages_ibf" format="title" /></em> message is received, this
                       indicates that decoding of the IBF on the active site has failed and roles will be swapped.
                       The receiving passive peer transitions into the <strong>Expecting IBF Last</strong> state,
                       and waits for more <em><xref target="messages_ibf" format="title" /></em> messages.
                       There, once the final <em><xref target="messages_ibf_last" format="title" /></em> message has been received, it transitions to <strong>Active Decoding</strong>.
                     </dd>
                     <dt><em><xref target="messages_ibf_last" format="title" /></em> message:</dt>
                     <dd>
                       If an <em><xref target="messages_ibf_last" format="title" /></em> message is received this
                       indicates that there is just one IBF slice left and a direct state and role transition from
                       <strong>Passive Decoding</strong> to <strong>Active Decoding</strong> is initiated.
                     </dd>
                     <dt><em><xref target="messages_done" format="title" /></em> message:</dt>
                     <dd>
                       Receiving the <em><xref target="messages_done" format="title" /></em> message signals
                       the passive peer that all demands of the active peer have been satisfied. Alas, the
                       active peer will continue to process demands from the passive peer.
                       Upon receiving this message, the passive peer transitions into the
                       <strong>Finish Waiting</strong> state.
                     </dd>
                   </dl>
                 </dd>
                 <dt><strong>Active Decoding:</strong></dt>
                 <dd>
                   <t>
                     In the <strong>Active Decoding</strong> state the active peer decodes the IBFs and evaluates the set difference
                     between the active and passive peer. Whenever an element ID is obtained by decoding the IBF, the active peer
                     sends either an offer or an inquiry to the passive peer, depending on which site the decoded element is missing.
                   </t>
                   <t>
                     If the IBF decodes a positive (1) pure bucket, the element is missing on the passive peers site.
                     Thus, the active peer sends an <em><xref target="messages_offer" format="title" /></em> to the passive peer.
                     A negative (-1) pure bucket indicates that an element is missing in the active peers set, so the active peer
                     sends a <em><xref target="messages_inquiry" format="title" /></em> to the passive peer.
                   </t>
                   <t>
                     In case the IBF does not successfully decode anymore, the active peer sends a new IBF computed with a different IBF-salt to the passive peer
                     and changes into <strong>Passive Decoding</strong> state. This initiates a role swap.
                     To reduce overhead and prevent double transmission of offers and elements, the new IBF is created
                     on the local set after updating it with the all of the elements that have been successfully demanded.  Note that the active peer MUST NOT wait for all active demands to be satisfied, as demands can fail if a bucket was falsely classified as pure.
                   </t>
                   <t>
                     As soon as the active peer successfully finished decoding the IBF, the active peer sends a
                     <em><xref target="messages_done" format="title" /></em> message to the passive peer.
                   </t>
                   <t>
                     All other actions taken by the active peer depend on the message the active peer receives from
                     the passive peer. The actions are described below on a per message basis:
                   </t>
                   <dl>
                     <dt><em><xref target="messages_offer" format="title" /></em> message:</dt>
                     <dd>
                       The <em><xref target="messages_offer" format="title" /></em> message indicates that the
                       passive peer received a <em><xref target="messages_inquiry" format="title" /></em> message from
                       the active peer. If a inquiry has been sent and
                       the offered element is missing in the active peers set,
                       the active peer sends a <em><xref target="messages_demand" format="title" /></em> message to the
                       passive peer. The demand needs to be added to a list of unsatisfied demands.
                       In case the received offer is for an element that is already in the set of the peer, the offer MUST BE ignored.
                     </dd>
                     <dt><em><xref target="messages_demand" format="title" /></em> message:</dt>
                     <dd>
                       The <em><xref target="messages_demand" format="title" /></em> message indicates that the
                       passive peer received a <em><xref target="messages_offer" format="title" /></em> from
                       the active peer. The active peer satisfies the demand of the passive peer by sending an
                       <em><xref target="messages_elements" format="title" /></em> message if a offer request
                       for the element was sent earlier. Otherwise, the protocol MUST be aborted, as peers must never send demands for hashes that they have never been offered.
                     </dd>
                     <dt><em><xref target="messages_elements" format="title" /></em> message:</dt>
                     <dd>
                       If element is received that was not demanded or for which
                       the application-specific validation logic fails, the protocol
                       MUST be aborted.  Otherwise, the corresponding demand is marked
                       as satisfied.  Note that this applies only to the differential
                       synchronization mode. In full synchronization, it is perfectly
                       normal to receive
                       <xref target="messages_full_element" format="title" />
                       messages for elements that were not demanded and that might
                       even already be known locally.
                     </dd>
                     <dt><em><xref target="messages_done" format="title" /></em> message:</dt>
                     <dd>
                       Receiving the message <em><xref target="messages_done" format="title" /></em> indicates
                       that all demands of the passive peer have been satisfied. The active peer then changes into the
                       <strong>Finish Closing</strong> state.
                       If the IBF has not finished decoding and the <em><xref target="messages_done" format="title" /></em>
                       is received, the other peer is not in compliance with the protocol and the protocol MUST be aborted.
                     </dd>
                   </dl>
                 </dd>
                 <dt><strong>Expecing IBF Last</strong></dt>
                 <dd>
                   <t>
                     In this state the active peer continuously receives <em><xref target="messages_ibf" format="title" /></em>
                     messages from the passive peer. When the last <em><xref target="messages_ibf_last" format="title" /></em> message is received,
                     the peer changes into the <strong>Active Decoding</strong> state.
                   </t>
                 </dd>
                 <dt><strong>Finish Closing</strong> / <strong>Finish Waiting</strong></dt>
                 <dd>
                   <t>
                     In this states the peers are waiting for all demands to be satisfied and for the synchronisation
                     to be completed. When all demands are satisfied the peer changes into <strong>Finished</strong> state.
                   </t>
                 </dd>
               </dl>
             </section>
             <section anchor="modeofoperation_combined-mode" numbered="true" toc="default">
               <name>Combined Mode</name>
               <t>
                 In the <em>combined mode</em> the protocol decides between
                 <xref target="modeofoperation_full-sync" format="title" /> and
                 the <xref target="modeofoperation_individual-elements" format="title" />
                 to minimize resource consumption.  Typically, the protocol always runs
                 in combined mode, but implementations MAY allow applications to force
                 the use of one of the modes for testing.  In this case, applications MUST
                 ensure that the respective options to force a particular mode are used by
                 both participants.
               </t>
               <t>
                 The <xref target="modeofoperation_individual-elements" format="title" /> is only efficient on small set
                 differences or if the byte-size of the elements is large. If the set difference is estimated to be large
                 the <xref target="modeofoperation_full-sync" format="title" /> is
                 more efficient. The exact heuristics and parameters on which the protocol decides which mode
                 MUST be used are described in the <xref target="performance" format="title" /> section of this document.
               </t>
               <t>
                 There are two main cases when a <xref target="modeofoperation_full-sync" format="title" />
                 is always used.
                 The first case is when one of the peers announces having an empty set. This is announced by setting
                 the SETSIZE field in the <em><xref target="messages_se" format="title" /></em> to 0.
                 <!-- FIXME: why not also do this if sending the elements is about as
                      expensive as sending the SE? Should be a simple calculation. (thesis summermatter: future work) @Christian:
                       As discussed 14.06 we let this comment in here as it is described in the thesis-->
                 The second case is if the application requests full synchronisation explicitly.
                 This is useful for testing and MUST NOT be used in production.
               </t>
               <t>
                 The state diagram illustrating the combined mode can be found
                 <eref target="https://git.gnunet.org/lsd0003.git/plain/statemachine/full_state_machine.png">here</eref>.
               </t>
             </section>
         </section>
 
         <section anchor="messages" numbered="true" toc="default">
             <name>Messages</name>
             <t>
               This section describes the various message formats used by the protocol.
             </t>
             <section anchor="messages_operation_request" numbered="true" toc="default">
                 <name>Operation Request</name>
 
                 <section anchor="messages_operation_request_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         This message is the first message of the protocol and it is sent to signal to the receiving peer
                         that the initiating peer wants to initialize a new connection.
                     </t>
                     <t>
                         This message is sent in the transition between the <strong>Initiating Connection</strong> state and the <strong>Expect SE</strong> state.
                     </t>
                     <t>
                       If a peer receives this message and is willing to run the protocol, it answers by sending back a <em><xref target="messages_se" format="title" /></em> message.
                       Otherwise it simply closes the connection.
                     </t>
                 </section>
                 <section anchor="messages_operation_request_structure" numbered="true" toc="default">
                     <name>Structure</name>
 
                     <figure anchor="figure_operation_request">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |    ELEMENT COUNT      |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |                      APX
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /  APPLICATION DATA                             /
         /                                               /
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                           is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                           is the type of SETU_P2P_OPERATION_REQUEST as registered in <xref target="gana" format="title" />, in network byte order.
                         </dd>
                         <dt>ELEMENT COUNT</dt>
                         <dd>
                           is the number of the elements the requesting party has in its set, as a 32-bit unsigned integer in network byte order.
                         </dd>
                         <dt>APX</dt>
                         <dd>
                           is a SHA-512 hash that identifies the application.
                         </dd>
                         <dt>APPLICATION DATA</dt>
                         <dd>
                           is optional, variable-size application specific data that can be used
                           by the application to decide if it would like to answer the request.
                         </dd>
                     </dl>
                 </section>
             </section>
 
             <section anchor="messages_ibf" numbered="true" toc="default">
                 <name>IBF</name>
 
                 <section anchor="messages_ibf_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The <xref target="messages_ibf" format="title" /> message contains a slice of the IBF.
                     </t>
                     <t>
                         The <em>IBF</em> message is sent at the start of the protocol from the initiating peer in the transaction
                         between <strong>Expect SE</strong> -> <strong>Expecting IBF Last</strong> or when the IBF does not
                         decode and there is a role change in the transition between <strong>Active Decoding</strong> -> <strong>Expecting IBF Last</strong>.
                         This message is only sent if there is more than one IBF slice to be sent. If there is just
                         one slice, then only the <xref target="messages_ibf_last" format="title" /> message is sent.
                     </t>
                 </section>
                 <section anchor="messages_ibf_structure" numbered="true" toc="default">
                     <name>Structure</name>
                     <figure anchor="figure_ibf">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |       IBF SIZE        |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |         OFFSET        |    SALT   |   IMCS    |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |                   IBF-SLICE
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /                                               /
         /                                               /
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte orderwhichdescribes the message size in bytes with the header included.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             the type of SETU_P2P_REQUEST_IBF as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
 
                         <dt>IBF SIZE</dt>
                         <dd>
                             is a 32-bit unsigned integer which signals the total number of buckets in the IBF.  The minimal number of buckets is 37.
                         </dd>
                         <dt>OFFSET</dt>
                         <dd>
                             is a 32-bit unsigned integer which signals the offset of the following IBF slices in the original.
                         </dd>
                         <dt>SALT</dt>
                         <dd>
                             is a 16-bit unsigned integer that contains the IBF-salt which was used to create the
                             IBF.
                         </dd>
                         <dt>IMCS</dt>
                         <dd>
                             is a 16-bit unsigned integer, which describes the number of bits that
                             are required to store a single counter. This is used for the unpacking function as described
                             in the <xref target="performance_counter_variable_size" format="title" /> section.
                         </dd>
                         <dt>IBF-SLICE</dt>
                         <dd>
                           <t>
                             are variable numbers of slices in an array. A single slice contains multiple 64-bit IDSUMS,
                             32-bit HASHSUMS and (1-64)-bit COUNTERS of variable size. All values are in the network byte order. The array of IDSUMS is serialized first, followed
                             by an array of HASHSUMS. Last comes an array of unsigned COUNTERS (details of the COUNTERS encoding are described in section
                             <xref target="performance_counter_variable_size" format="default"/>). The length of the array is
                             defined by MIN( SIZE - OFFSET, MAX_BUCKETS_PER_MESSAGE). MAX_BUCKETS_PER_MESSAGE is defined as
                             32768 divided by the BUCKET_SIZE which ranges between 97-bits when counter uses bit 1 (IMCS=1) and 160-bit when counter size uses 64 bit (IMCS=64).
                           </t>
                           <t>
                             To get the IDSUM field, all IDs (computed with the IBF-salt) hitting a bucket under the map M are added up with a binary XOR operation.
                             See <xref target="ibf_format_id_generation" format="title" /> details about ID generation.
                           </t>
                           <t>
                             The calculation of the HASHSUM field is done accordingly to the calculation of the IDSUM field:
                             all HASHes are added up with a binary XOR operation.
                             The HASH value is calculated as described in detail in section <xref target="ibf_format_HASH_calculation" format="title" />.
                           </t>
                           <t>
                             The algorithm to find the correct bucket in which the ID and the HASH have to be added
                             is described in detail in section <xref target="ibf_m" format="title" />.
                           </t>
                           <t>
                             Test vectors for an implementation can be found in the <xref target="test_vectors" format="title" /> section
                           </t>
                         </dd>
                     </dl>
                     <figure anchor="figure_ibf_slice">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
                              IBF-SLICE
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |                    IDSUMS                     |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |                    IDSUMS                     |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |        HASHSUMS       |        HASHSUMS       |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |        COUNTERS*      |        COUNTERS*      |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /                                               /
         /                                               /
 * Counter size is variable. In this example the IMCS is 32 (4 bytes).
                  ]]></artwork>
                     </figure>
                 </section>
             </section>
 
             <section anchor="messages_ibf_last" numbered="true" toc="default">
               <name>IBF Last</name>
 
               <section anchor="messages_ibf_last_description" numbered="true" toc="default">
                 <name>Description</name>
                 <t>
                   This message indicates to the remote peer that this is the last slice
                   of the Bloom filter. The receiving peer MUST check that the sizes and
                   offsets of all received IBF slices add up to the total IBF SIZE that was
                   given.
                 </t>
                 <t>
                   Receiving this message initiates the state transmissions
                   <strong>Expecting IBF Last</strong> -> <strong>Active Decoding</strong>,
                   <strong>Expecting IBF</strong> -> <strong>Active Decoding</strong> and
                   <strong>Passive Decoding</strong> -> <strong>Active Decoding</strong>. This message
                   can initiate a peer the roll change from <strong>Active Decoding</strong> to
                   <strong>Passive Decoding</strong>.
                 </t>
               </section>
               <section anchor="messages_ibf_last_structure" numbered="true" toc="default">
                 <name>Structure</name>
                 <t>
                   The binary structure is exactly the same as the <xref target="messages_ibf_structure" format="title" /> of
                   the message <xref target="messages_ibf" format="title" /> with a different "MSG TYPE"
                   which is defined in <xref target="gana" format="title" /> "SETU_P2P_IBF_LAST".
                 </t>
               </section>
             </section>
 
             <section anchor="messages_elements" numbered="true" toc="default">
                 <name>Element</name>
 
                 <section anchor="messages_elements_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The <em><xref target="messages_elements" format="title" /></em> message contains an element that is synchronized in the <xref target="modeofoperation_individual-elements" format="title" />
                         and transmits a full element between the peers.
                     </t>
                     <t>
                         This message is sent in the state <strong>Active Decoding</strong> and <strong>Passive Decoding</strong>
                         as answer to a <em><xref target="messages_demand" format="title" /></em> message from the remote peer.
                         The <em><xref target="messages_elements" format="title" /></em> message can also be received in the <strong>Finish Closing</strong> or <strong>Finish Waiting</strong>
                         state after receiving a <em><xref target="messages_done" format="title" /></em> message from the remote peer. In this
                         case the peer changes to the <strong>Finished</strong> state as soon as all demands for elements have been satisfied.
                     </t>
                     <t>
                         This message is exclusively used in the <xref target="modeofoperation_individual-elements" format="title" />.
                     </t>
                 </section>
                 <section anchor="messages_elements_structure" numbered="true" toc="default">
                     <name>Structure</name>
                     <figure anchor="figure_elements">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |   E TYPE  |  PADDING  |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |   E SIZE  |              DATA
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /                                               /
         /                                               /
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             is SETU_P2P_ELEMENTS as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
                         <dt>E TYPE</dt>
                         <dd>
                             is a 16-bit unsigned integer which defines the element type for
                             the application.
                         </dd>
                         <dt>PADDING</dt>
                         <dd>
                             is 16-bit always set to zero.
                         </dd>
                         <dt>E SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer that signals the size of the elements data part.
                         </dd>
                         <dt>DATA</dt>
                         <dd>
                             is a field with variable length that contains the data of the element.
                         </dd>
                     </dl>
                 </section>
             </section>
 
             <section anchor="messages_offer" numbered="true" toc="default">
                 <name>Offer</name>
 
                 <section anchor="messages_offer_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The <em><xref target="messages_offer" format="title" /></em> message is an answer to an <em><xref target="messages_inquiry" format="title" /></em> message
                         and transmits the full hash of an element that has been requested by the other peer.
                         This full hash enables the other peer to check if the element is really missing in his set and
                         eventually sends a <em><xref target="messages_demand" format="title" /></em> message for that element.
                     </t>
                     <t>
                         The offer is sent and received only in the <strong>Active Decoding</strong> and in the <strong>Passive Decoding</strong>
                         state.
                     </t>
                     <t>
                         This message is exclusively sent in the <xref target="modeofoperation_individual-elements" format="title" />.
                     </t>
                 </section>
                 <section anchor="messages_offer_structure" numbered="true" toc="default">
                     <name>Structure</name>
                     <figure anchor="figure_offer">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |         HASH 1
         +-----+-----+-----+-----+-----+-----+-----+-----+
        ...                                             ...
         +-----+-----+-----+-----+-----+-----+-----+-----+
                  HASH 1         |         HASH 2
         +-----+-----+-----+-----+-----+-----+-----+-----+
        ...                                             ...
         +-----+-----+-----+-----+-----+-----+-----+-----+
                  HASH 2         |         HASH n
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /                                               /
         /                                               /
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte order, which describes the message size in bytes header included.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             is SETU_P2P_OFFER as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
                         <dt>HASH n</dt>
                         <dd>
                             contains n (one or more) successive SHA 512-bit hashes of the elements that are being requested with <em><xref target="messages_inquiry" format="title" /></em> messages.
                         </dd>
                     </dl>
                 </section>
             </section>
 
 
             <section anchor="messages_inquiry" numbered="true" toc="default">
                 <name>Inquiry</name>
 
                 <section anchor="messages_inquiry_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The <em><xref target="messages_inquiry" format="title" /></em> message is exclusively sent by the active peer in <strong>Active Decoding</strong> state
                         to request the full hash of an element that is missing in the active peers set. This is normally answered
                         by the passive peer with <em><xref target="messages_offer" format="title" /></em> message.
                     </t>
                     <t>
                         This message is exclusively sent in the <xref target="modeofoperation_individual-elements" format="title" />.
                     </t>
                 </section>
                 <section anchor="messages_inquiry_structure" numbered="true" toc="default">
                     <name>Structure</name>
                     <figure anchor="figure_inquiry">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |          SALT         |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |                   IBF KEY 1                   |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |                   IBF KEY 2                   |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         ...                                            ...
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |                   IBF KEY n                   |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /                                               /
         /                                               /
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             is SETU_P2P_INQUIRY as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
                         <dt>IBF KEY</dt>
                         <dd>
                             contains n (one or more) successive ibf keys (64-bit unsigned integer) for which the inquiry is sent.
                         </dd>
                     </dl>
                 </section>
             </section>
 
             <section anchor="messages_demand" numbered="true" toc="default">
                 <name>Demand</name>
 
                 <section anchor="messages_demand_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The <em><xref target="messages_demand" format="title" /></em> message is sent in the <strong>Active Decoding</strong> and in the <strong>Passive Decoding</strong>
                         state. It is an answer to a received <em><xref target="messages_offer" format="title" /></em> message
                         and is sent if the element described in the <em><xref target="messages_offer" format="title" /></em> message
                         is missing in the peers set. In the normal workflow the answer to the <em><xref target="messages_demand" format="title" /></em> message is an
                         <em><xref target="messages_elements" format="title" /></em> message.
                     </t>
                     <t>
                         This message is exclusively sent in the <xref target="modeofoperation_individual-elements" format="title" />.
                     </t>
                 </section>
                 <section anchor="messages_demand_structure" numbered="true" toc="default">
                     <name>Structure</name>
                     <figure anchor="figure_demand">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |         HASH 1
         +-----+-----+-----+-----+-----+-----+-----+-----+
        ...                                             ...
         +-----+-----+-----+-----+-----+-----+-----+-----+
                  HASH 1         |         HASH 2
         +-----+-----+-----+-----+-----+-----+-----+-----+
        ...                                             ...
         +-----+-----+-----+-----+-----+-----+-----+-----+
                  HASH 2         |         HASH n
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /                                               /
         /                                               /
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte order, which describes the message size in bytes and the header is included.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             the type of SETU_P2P_DEMAND as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
                         <dt>HASH n</dt>
                         <dd>
                             contains n (one or more) successive SHA 512-bit hashes of the elements that are being demanded.
                         </dd>
                     </dl>
                 </section>
             </section>
 
             <section anchor="messages_done" numbered="true" toc="default">
                 <name>Done</name>
 
                 <section anchor="messages_done_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The <em><xref target="messages_done" format="title" /></em> message is sent when all <em><xref target="messages_demand" format="title" /></em> messages
                         have been successfully satisfied and from the perspective of the sender the set is completely synchronized.
                     </t>
                     <t>
                         This message is exclusively sent in the <xref target="modeofoperation_individual-elements" format="title" />.
                     </t>
                 </section>
                 <section anchor="messages_done_structure" numbered="true" toc="default">
                     <name>Structure</name>
                     <figure anchor="figure_done">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |    FINAL CHECKSUM
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /                                               /
         /                                               /
 
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included. The value is always 4 for this message type.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             is SETU_P2P_DONE as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
                         <dt>FINAL CHECKSUM</dt>
                         <dd>
                             a SHA-512 hash XOR sum of the full set after synchronization. This should ensure that the sets are identical in the end!
                         </dd>
                     </dl>
                 </section>
             </section>
 
             <section anchor="messages_full_done" numbered="true" toc="default">
                 <name>Full Done</name>
 
                 <section anchor="messages_full_done_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The <em><xref target="messages_full_done" format="title" /></em> message is sent in the <xref target="modeofoperation_full-sync" format="title" />
                         to signal that all remaining elements of the set have been sent. The message is received and sent in the
                         <strong>Full Sending</strong> and in the <strong>Full Receiving</strong> state. When the <em><xref target="messages_full_done" format="title" /></em> message is received
                         in <strong>Full Sending</strong> state the peer changes directly into <strong>Finished</strong> state. In
                         <strong>Full Receiving</strong> state receiving a <em><xref target="messages_full_done" format="title" /></em> message initiates the sending of
                         the remaining elements that are missing in the set of the other peer.
                     </t>
                 </section>
                 <section anchor="messages_full_done_structure" numbered="true" toc="default">
                     <name>Structure</name>
                     <figure anchor="figure_full_done">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |     FINAL CHECKSUM
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /                                               /
         /                                               /
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included. The value is always 4 for this message type.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             the type of SETU_P2P_FULL_DONE as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
                         <dt> FINAL CHECKSUM</dt>
                         <dd>
                             a SHA-512 hash XOR sum of the full set after synchronization. This should ensure that the sets are identical in the end!
                         </dd>
                     </dl>
                 </section>
             </section>
 
             <section anchor="messages_request_full" numbered="true" toc="default">
                 <name>Request Full</name>
 
                 <section anchor="messages_request_full_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The <em><xref target="messages_request_full" format="title" /></em> message is sent by the initiating peer in <strong>Expect SE</strong> state to the receiving peer, if
                         the operation mode "<xref target="modeofoperation_full-sync" format="title" />" is
                         determined to be the superior <xref target="modeofoperation" format="title" /> and that it is the better choice that
                        the other peer sends his elements first. The initiating peer changes after sending the <em><xref target="messages_request_full" format="title" /></em> message into
                         <strong>Full Receiving</strong> state.
                     </t>
                     <t>
                         The receiving peer receives the <em><xref target="messages_request_full" format="title" /></em> message in the <strong>Expecting IBF</strong>, afterwards the receiving peer
                         starts sending his complete set in <xref target="messages_full_element" format="title" /> messages to the initiating peer.
                     </t>
                 </section>
                 <section anchor="messages_request_full_structure" numbered="true" toc="default">
                     <name>Structure</name>
                     <figure anchor="figure_request_full">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |    REMOTE SET DIFF    |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |   REMOTE SET SIZE     |    LOCAL SET DIFF     |
         +-----+-----+-----+-----+-----+-----+-----+-----+
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included. The value is always 16 for this message type.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             is SETU_P2P_REQUEST_FULL as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
                         <dt>REMOTE SET DIFF</dt>
                         <dd>
                             is a 32-bit unsigned integer in network byte order, which represents the remote (from the perspective of the
                             sending peer) set difference calculated with strata estimator.
                         </dd>
                         <dt>REMOTE SET SIZE</dt>
                         <dd>
                             is a 32-bit unsigned integer in network byte order, which represents the total remote
                             (from the perspective of the sending peer) set size.
                         </dd>
                         <dt>LOCAL SET DIFF</dt>
                         <dd>
                             is a 32-bit unsigned integer in network byte order, which represents the local
                             (from the perspective of the sending peer) set difference calculated with strata estimator.
                         </dd>
                     </dl>
                 </section>
             </section>
 
 
             <section anchor="messages_send_full" numbered="true" toc="default">
                 <name>Send Full</name>
 
                 <section anchor="messages_send_full_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The <em><xref target="messages_send_full" format="title" /></em> message is sent by the initiating peer in <strong>Expect SE</strong> state to the receiving peer if
                         the operation mode "<xref target="modeofoperation_full-sync" format="title" />" is
                         determined as superior <xref target="modeofoperation" format="title" /> and that it is the better choice that the
                         peer sends his elements first. The initiating peer changes after sending the <em><xref target="messages_request_full" format="title" /></em> message into
                         <strong>Full Sending</strong> state.
                     </t>
                     <t>
                         The receiving peer receives the <em><xref target="messages_send_full" format="title" /></em> message in the <strong>Expecting IBF</strong> state, afterwards the receiving peer
                         changes into <strong>Full Receiving</strong> state and expects to receive the set of the remote peer.
                     </t>
                 </section>
                 <section anchor="messages_send_full_structure" numbered="true" toc="default">
                     <name>Structure</name>
                     <figure anchor="figure_send_full">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |    REMOTE SET DIFF    |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |   REMOTE SET SIZE     |    LOCAL SET DIFF     |
         +-----+-----+-----+-----+-----+-----+-----+-----+
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.  The value is always 16 for this message type.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             is SETU_P2P_REQUEST_FULL as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
                         <dt>REMOTE SET DIFF</dt>
                         <dd>
                             is a 32-bit unsigned integer in network byte order, which represents the remote (from the perspective of the sending peer)
                             set difference calculated with strata estimator.
                         </dd>
                         <dt>REMOTE SET SIZE</dt>
                         <dd>
                             is a 32-bit unsigned integer in network byte order, which represents the total remote (from the perspective
                             of the sending peer) set size.
                         </dd>
                         <dt>LOCAL SET DIFF</dt>
                         <dd>
                             is a 32-bit unsigned integer in network byte order, which represents the local (from the perspective of the sending peer)
                             set difference calculated with strata estimator.
                         </dd>
                     </dl>
                 </section>
             </section>
 
 
             <section anchor="messages_se" numbered="true" toc="default">
                 <name>Strata Estimator</name>
 
                 <section anchor="messages_se_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The strata estimator is sent by the receiving peer at the start of the protocol, right after the
                         <xref target="messages_operation_request" format="title" /> message has been received.
                     </t>
                     <t>
                         The strata estimator is used to estimate the difference between the two sets as described in section <xref target="se" format="title" />.
                     </t>
                     <t>
                         When the initiating peer receives the strata estimator, the peer decides which <xref target="modeofoperation" format="title" /> to use
                         for the synchronisation. Depending on the size of the set difference and the <xref target="modeofoperation" format="title" /> the initiating peer
                         changes into <strong>Full Sending</strong>, <strong>Full Receiving</strong> or <strong>Passive Decoding</strong> state.
                     </t>
                     <t>
                         The <em><xref target="messages_se" format="title" /></em> message can contain one, two, four or eight strata estimators with different salts, depending on the initial size of the sets.
                         More details can be found in section <xref target="performance_multi_se" format="title" />.
                     </t>
                     <t>
                         The IBFs in a strata estimator always have 79 buckets. The reason why can be found in <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/> in section 3.4.2.
                     </t>
                     <!-- Give a more precise reference into the thesis for this, do not cite the whole thesis! -->
                 </section>
                 <section anchor="messages_se_structure" numbered="true" toc="default">
                     <name>Structure</name>
                     <figure anchor="figure_se">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE | SEC |     SETSIZE
         +-----+-----+-----+-----+-----+-----+-----+-----+
               SETSIZE                 |     SE-SLICES
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /                                               /
         /                                               /
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             is SETU_P2P_SE as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
                         <dt>SEC</dt>
                         <dd>
                             is a 8-bit unsigned integer in network byte order, which indicates how many strata estimators
                             with different salts are attached to the message. Valid values are 1,2,4 or 8, more details can be found
                             in the section <xref target="performance_multi_se" format="title" />.
                         </dd>
                         <dt>SETSIZE</dt>
                         <dd>
                             is a 64-bit unsigned integer that is defined by the size of the set the SE is
                         </dd>
                         <dt>SE-SLICES</dt>
                         <dd>
                             <t>
                                 are variable numbers of slices in an array. A slice can contain one or more Strata Estimators which
                                 contain multiple IBFs as described in IBF-SLICES in <xref target="messages_ibf_structure" format="default"/>.
                                 A SE slice can contain one to eight Strata Estimators which contain 32 (Defined as Constant SE_STRATA_COUNT) IBFs. Every IBF in
                                 a SE contains 79 Buckets.
                             </t>
                             <t>
                                 The different SEs are built as in detail described in <xref target="performance_multi_se" format="default"/>.
                                 Simply put, the IBFs in each SE are serialized as described in <xref target="messages_ibf_structure" format="default"/> starting with the highest stratum.
                                 Then the created SEs are appended one after the other starting with the SE that was created with a salt of zero.
 
                             </t>
                         </dd>
                     </dl>
                     <figure anchor="figure_se_slice">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
                              SE-SLICE
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |                  SE_1 -> IBF_1
         +-----+-----+-----+-----+-----+-----+-----+-----+
        ...                                             ...
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |                  SE_1 -> IBF_30
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |                  SE_2 -> IBF_1
         +-----+-----+-----+-----+-----+-----+-----+-----+
        ...                                             ...
         /                                               /
         /                                               /
                  ]]></artwork>
                     </figure>
                 </section>
             </section>
 
             <section anchor="messages_sec" numbered="true" toc="default">
               <name>Strata Estimator Compressed</name>
 
               <section anchor="messages_sec_description" numbered="true" toc="default">
                 <name>Description</name>
                 <t>
                   The Strata Estimator can be compressed with gzip as
                   described in <xref target="RFC1951"/> to improve performance. This can be recognized
                   by the different message type number from <xref target="gana" format="title" />.
                 </t>
                 <section anchor="messages_sec_structure" numbered="true" toc="default">
                       <name>Structure</name>
                         <t>
                             The key difference between the compressed and the uncompressed Strata Estimator is that the
                             SE slices are compressed with gzip (<xref target="RFC1951"/>) in the compressed SE.
                             But the header remains uncompressed with both.
                         </t>
                         <t>
                             Since the content of the message is the same as the uncompressed Strata Estimator, the details
                             are not repeated here. For details see section <xref target="messages_se" format="counter" />.
                         </t>
                 </section>
               </section>
             </section>
 
             <section anchor="messages_full_element" numbered="true" toc="default">
                 <name>Full Element</name>
 
                 <section anchor="messages_full_element_description" numbered="true" toc="default">
                     <name>Description</name>
                     <t>
                         The <em><xref target="messages_full_element" format="title" /></em> message is the equivalent of the <xref target="messages_elements" format="title" /> message in
                         the <xref target="modeofoperation_full-sync" format="title" />. It contains a complete element that is missing
                         in the set of the peer that receives this message.
                     </t>
                     <t>
                         The <em><xref target="messages_full_element" format="title" /></em> message is exclusively sent in the transitions <strong>Expecting IBF</strong> -> <strong>Full Receiving</strong> and
                         <strong>Full Receiving</strong> -> <strong>Finished</strong>. The message is only received in the <strong> Full Sending</strong> and
                         <strong>Full Receiving</strong> state.
                     </t>
                     <t>
                         After the last <em><xref target="messages_full_element" format="title" /></em> message has been sent, the <em><xref target="messages_full_done" format="title" /></em> message
                         is sent to conclude the full synchronisation of the element sending peer.
                     </t>
                 </section>
                 <section anchor="messages_full_element_structure" numbered="true" toc="default">
                   <name>Structure</name>
                   <!-- MAYBE just refer to the "ELEMENT" section on structure and only
                        note the different MSG TYPE here? -->
                     <figure anchor="figure_full_element">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
         0     8     16    24    32    40    48    56
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |  MSG SIZE |  MSG TYPE |   E TYPE  |  PADDING  |
         +-----+-----+-----+-----+-----+-----+-----+-----+
         |    SIZE   |   AE TYPE |          DATA
         +-----+-----+-----+-----+-----+-----+-----+-----+
         /                                               /
         /                                               /
                  ]]></artwork>
                     </figure>
                     <t>where:</t>
                     <dl>
                         <dt>MSG SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.
                         </dd>
                         <dt>MSG TYPE</dt>
                         <dd>
                             is SETU_P2P_REQUEST_FULL_ELEMENT as registered in <xref target="gana" format="title" /> in network byte order.
                         </dd>
                         <dt>E TYPE</dt>
                         <dd>
                             is a 16-bit unsigned integer which defines the element type for
                             the application.
                         </dd>
                         <dt>PADDING</dt>
                         <dd>
                             is 16-bit always set to zero
                         </dd>
                         <dt>E SIZE</dt>
                         <dd>
                             is a 16-bit unsigned integer that signals the size of the elements data part.
                         </dd>
                         <dt>AE TYPE</dt>
                         <dd>
                             is a 16-bit unsigned integer that is needed to identify
                             the type of element that is in the data field
                         </dd>
                         <dt>DATA</dt>
                         <dd>
                             is a field with variable length that contains the data of the element.
                         </dd>
                     </dl>
                 </section>
             </section>
         </section>
 
         <section anchor="performance" numbered="true" toc="default">
             <name>Performance Considerations</name>
             <section anchor="performance_formulas" numbered="true" toc="default">
                 <name>Formulas</name>
                 <section anchor="performance_formulas_operationmode" numbered="true" toc="default">
                     <name>Operation Mode</name>
                     <t>
                         The decision which <xref target="modeofoperation" format="title"/> is used is described by the following code.
                         More detailed explanations motivating the design can be found in the accompanying thesis in section 4.5.3.<xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/>
                     </t>
                     <t>
                         The function takes as input the average element size, the local set size, the remote set size, the set differences as estimated from the strata estimator for both the local and remote sets,
                         and the bandwidth/roundtrip tradeoff.
                         The function returns the exact <xref target="modeofoperation" format="title"/> that is predicted to be best as output: FULL_SYNC_REMOTE_SENDING_FIRST
                         if it is likely cheapest that the other peer transmits his elements first, FULL_SYNC_LOCAL_SENDING_FIRST
                         if it is likely cheapest that the elements are transmitted to the other peer directly, and
                         DIFFERENTIAL_SYNC if the differential synchronisation is likely cheapest.
                     </t>
                     <t>
                       The constant IBF_BUCKET_NUMBER_FACTOR is always 2 and IBF_MIN_SIZE is 37.
                       The method for deriving
                         this can be found in the IBF parameter study in <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/> in section 4.5.2.
                     </t>
                     <figure anchor="performance_formulas_operationmode_code">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
 # CONSTANTS:
 # IBF_BUCKET_NUMBER_FACTOR = 2: The amount the IBF gets increased if
 #                               decoding fails
 # RTT_MIN_FULL = 2: Minimal round trips used for full Synchronisation
 # IBF_MIN_SIZE = 37: The minimal size of an IBF
 # MAX_BUCKETS_PER_MESSAGE: Custom value depending on the underlying
 #                          protocol
 # INPUTS:
 # avg_es: The average element size
 # lss: The initial local set size
 # rss: The remote set size
 # lsd: the estimated local set difference calculated by the SE
 # rsd: the estimated remote set difference calculated by the SE
 # rtt: the tradeoff between round trips and bandwidth
 # OUTPUT:
 # FULL_SYNC_REMOTE_SENDING_FIRST, FULL_SYNC_LOCAL_SENDING_FIRST or
 # DIFFERENTIAL_SYNC
 
 FUNCTION decide_operation_mode(avg_es,
                                lss,
                                rss,
                                lsd
                                rsd,
                                rtt)
 
     # If a set size is zero always do full sync
     IF 0 == rss THEN
         RETURN FULL_SYNC_LOCAL_SENDING_FIRST
     END IF
     IF 0 == lss THEN
         RETURN FULL_SYNC_REMOTE_SENDING_FIRST
     END IF
 
     # Estimate required transferred bytes when doing a full
     # synchronisation and transmitting local set first.
     semh = sizeof(ELEMENT_MSG_HEADER)
     estimated_total_diff = rsd + lsd
     total_elements_local_send = rsd + lss
     cost_local_full_sync = avg_es * total_elements_local_send
                            + total_elements_local_send * semh
                            + sizeof(FULL_DONE_MSG_HEADER) * 2
                            + RTT_MIN_FULL * rtt
 
     # Estimate required transferred bytes when doing a full
     # synchronisation and transmitting remote set first.
     total_elements_remote_send = lsd + rss
     cost_remote_full_sync = avg_es * total_elements_remote_send
                             + total_elements_remote_send * semh
                             + sizeof(FULL_DONE_MSG_HEADER) * 2
                             + (RTT_MIN_FULL + 0.5) * rtt
                             + sizeof(REQUEST_FULL_MSG)
 
     # Estimate required transferred bytes when doing a differential
     #  synchronisation
 
     # Estimate messages required to transfer IBF
     ibf_bucket_count = estimated_total_diff * IBF_BUCKET_NUMBER_FACTOR
     IF ibf_bucket_count <= IBF_MIN_SIZE THEN
         ibf_bucket_count = IBF_MIN_SIZE
     END IF
     ibf_message_count = ceil (ibf_bucket_count / MAX_BUCKETS_PER_MESSAGE)
 
     # Estimate average counter length with variable counter
     estimated_counter_bits = MIN (2 * LOG2(lss / ibf_bucket_count),
                                   LOG2(lss))
     estimated_counter_bytes = estimated_counter_bits / 8
 
     # Sum up all messages required to do differential synchronisation
     ibf_bytes = sizeof(IBF_MESSAGE) * ibf_message_count
               + ibf_bucket_count * sizeof(IBF_KEY)
               + ibf_bucket_count * sizeof(IBF_KEYHASH)
               + ibf_bucket_count * estimated_counter_bytes
     # Add 20% overhead to cover IBF retries due to decoding failures
     total_ibf_bytes = ibf_bytes * 1.2
 
     # Estimate other message sizes to be transfered in diff sync
     # Note that we simplify by adding the header each time;
     # if the implementation combines multiple INQUIRY/DEMAND/OFFER
     # requests in one message, the bandwidth would be lower.
     done_size = sizeof(DONE_HEADER)
     element_size = (avg_es + sizeof(ELEMENT_MSG_HEADER))
                  * estimated_total_diff
     inquery_size = (sizeof(IBF_KEY) + sizeof(INQUERY_MSG_HEADER))
                  * estimated_total_diff
     demand_size  = (sizeof(HASHCODE) + sizeof(DEMAND_MSG_HEADER))
                  * estimated_total_diff
     offer_size   = (sizeof(HASHCODE) + sizeof(OFFER_MSG_HEADER))
                  * estimated_total_diff
 
     # Estimate total cost
     diff_cost = element_size + done_size + inquery_size
               + demand_size + offer_size + total_ibf_bytes
               + DIFFERENTIAL_RTT_MEAN * rtt
 
     # Decide for a optimal mode of operation
     full_cost_min = MIN (cost_local_full_sync,
                          cost_remote_full_sync)
     IF full_cost_min < diff_cost THEN
         IF cost_remote_full_sync > cost_local_full_sync THEN
             RETURN FULL_SYNC_LOCAL_SENDING_FIRST
         ELSE
             RETURN FULL_SYNC_REMOTE_SENDING_FIRST
         END IF
     ELSE
         RETURN DIFFERENTIAL_SYNC
     END IF
 END FUNCTION
     ]]></artwork>
                     </figure>
                 </section>
                 <section anchor="performance_formula_ibf_parameters" numbered="true" toc="default">
                     <name>IBF Size</name>
                     <t>
                         The functions, described in this section, calculate a good initial size (initial_ibf_size)
                         and in case of decoding failure, a good next IBF size (get_next_ibf_size).
                     </t>
                     <t>
                         These algorithms are described and justified in more details in
                         <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/> in the parameter study in
                         section 3.5.2, the max IBF counter in section 3.10 and the Improved IBF size in section 3.11.
                     </t>
                     <figure anchor="performance_formula_ibf_parameters_code">
                         <artwork name="" type="" align="left" alt=""><![CDATA[
 # CONSTANTS:
 # IBF_BUCKET_NUMBER_FACTOR = 2: The amount the IBF gets increased
                                 if decoding fails
 # Inputs:
 # sd: Estimated set difference
 # Output:
 # next_size: Size of the initial IBF
 
 FUNCTION initial_ibf_size(sd)
     # We do not go below 37, as 37 buckets should
     # basically always be below one MTU, so there is
     # little to be gained, while a smaller IBF would
     # increase the chance of a decoding failure.
     RETURN MAX(37, IBF_BUCKET_NUMBER_FACTOR * sd);
 END FUNCTION
 
 # CONSTANTS:
 # IBF_BUCKET_NUMBER_FACTOR = 2: The amount the IBF gets increased if
 #                               decoding fails
 # Inputs:
 # de: Number of elements that have been successfully decoded
 # lis: The number of buckets of the last IBF
 # Output:
 # number of buckets for the next IBF
 
 FUNCTION get_next_ibf_size(de, lis)
     next_size = IBF_BUCKET_NUMBER_FACTOR * (lis - de)
     # The MAX operation here also ensures that the
     # result is positive.
     RETURN MAX(37, next_size);
 END FUNCTION
                         ]]></artwork>
                     </figure>
                 </section>
                 <section anchor="performance_num_buck_hash" numbered="true" toc="default">
                     <name>Number of Buckets an Element is Hashed into</name>
                     <t>
                         The number of buckets an element is hashed to is hardcoded to 3. Reasoning and
                         justification can be found in
                         <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/> in the
                         IBF parameter performance study in section 4.5.2.
                     </t>
                 </section>
             </section>
             <section anchor="performance_counter_variable_size" numbered="true" toc="default">
                 <name>Variable Counter Size</name>
                 <t>
                   The number of bits required to represent the counters of an IBF varies
                   due to different parameters as described in section 3.2 of
                   <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/>.
                   Therefore, a packing algorithm has been implemented.
                   This algorithm encodes the IBF counters in their optimal bit-width
                   and thus minimizes the bandwidth needed to transmit the IBF.
                 </t>
                 <t>
                   A simple algorithm is used for the packing.
                   In a first step it is determined, which is the largest counter.
                   The the base 2 logarithm then determines how many bits are needed to store it.
                   In a second step for every counter of every bucket, the counter
                   is stored using this many bits. The resulting bit sequence is then simply concatenated.
                 </t>
                 <t>
                   Three individual functions are used for this purpose.
                   The first one is a function that iterates over each bucket of
                   the IBF to get the maximum counter in the IBF. The second function
                   packs the counters of the IBF, and the third function that unpacks the counters.
                 </t>
                 <t>
                     As a plausibly check to prevent the byzantine upper bound
                     checks in <xref target="security_generic_functions_check_byzantine_boundaries" format="default"/>
                     to fail, implementations must ensure that the
                     estimates of the set size difference added together
                     never exceed the set byzantine upper bound. This
                     could for example happen in case the strata estimator
                     overestimates the set difference.
                 </t>
                 <figure anchor="performance_counter_variable_size_code">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 
 # INPUTS:
 # ibf: The IBF
 # OUTPUTS:
 # returns: Minimal amount of bits required to store the counter
 
 FUNCTION ibf_get_max_counter(ibf)
     max_counter=1 # convince static analysis that we never take log2(0)
     FOR bucket IN ibf DO
         IF bucket.counter > max_counter THEN
             max_counter = bucket.counter
         END IF
     END FOR
     # next bigger discrete number of the binary logarithm of the
     # max counter
     RETURN CEILING( LOG2( max_counter ) )
 END FUNCTION
 
 # INPUTS:
 # ibf: The IBF
 # offset: The offset which defines the starting point from which bucket
 #         the pack operation starts
 # count: The number of buckets in the array that will be packed
 # OUTPUTS:
 # returns: A byte array of packed counters to send over the network
 
 # INPUTS:
 # ibf: The IBF
 # offset: The offset which defines the starting point from which bucket
 #         the pack operation starts
 # count: The number of buckets in the array that will be packed
 # OUTPUTS:
 # returns: A byte array of packed counters to send over the network
 
 FUNCTION pack_counter(ibf, offset, count)
     counter_bytes = ibf_get_max_counter(ibf)
     store_bits = 0
     store = 0
     byte_ctr = 0
     buf=[]
 
     FOR bucket IN ibf[offset] TO ibf[count] DO
         counter = bucket.counter
         byte_len = counter_bytes
 
         WHILE byte_len + store_bits < 8 DO
             bit_to_shift = 0
 
             IF store_bits > 0 OR byte_len > 8 THEN
                 bit_free = 8 - store_bits
                 bit_to_shift = byte_len - bit_free
                 store = store << bit_free
             END IF
             buf[byte_ctr] = (( counter >> bit_to_shift) | store) & 0xFF
             byte_ctr = byte_ctr + 1
             byte_len -= 8 - store_bits
             counter = counter & ((1 << byte_len) - 1)
             store = 0
             store_bits = 0
         END WHILE
         store = (store << byte_len) | counter
         store_bits = store_bits + byte_len
         byte_len = 0
     END FOR
 
     # Write the last partial packed byte to the buffer
     IF store_bits > 0 THEN
         buf[byte_ctr] = store << (8 - store_bits)
         byte_ctr = byte_ctr + 1
     END IF
 
     RETURN buf
 FUNCTION END
 
 # INPUTS:
 # ibf: The IBF
 # offset: The offset which defines the starting point from which bucket
           the packed operation starts
 # count: The number of buckets in the array that will be packed
 # cbl: The bit length of the counter can be found in the
        ibf message in the ibf_counter_bit_length field
 # pd: A byte array which contains the data packed with the pack_counter
       function
 # OUTPUTS:
 # returns: Nothing because the unpacked counter is saved directly
            into the IBF
 
 FUNCTION unpack_counter(ibf, offset, count, cbl, pd)
     ibf_bucket_ctr = 0
     store = 0
     store_bits = 0
     byte_ctr = 0
 
     WHILE TRUE
         byte_read = pd[byte_ctr]
         bit_to_pack_left = 8
         byte_ctr++
 
         WHILE bit_to_pack_left >= 0 DO
 
             # Prevent packet from reading more than required
             IF ibf_bucket_ctr > (count - 1) THEN
                 RETURN
             END IF
 
             IF store_bits + bit_to_pack_left >= cbl THEN
                 bit_use = cbl - store_bits
 
                 IF store_bits > 0 THEN
                     store = store << bit_use
                 END IF
                 bytes_to_shift = bit_to_pack_left - bit_use
                 counter_partial = byte_read >> bytes_to_shift
                 store = store | counter_partial
                 ibf.counter[ibf_bucket_ctr + offset] = store
                 byte_read = byte_read & (( 1 << bytes_to_shift ) - 1)
 
                 bit_to_pack_left -= bit_use
                 ibf_bucket_ctr++
                 store = 0
                 store_bits = 0
             ELSE
                 store_bits = store_bits + bit_to_pack_left
 
                 IF 0 == store_bits THEN
                     store = byte_read
                 ELSE
                     store = store << bit_to_pack_left
                     store = store | byte_read
                 END IF
                 BREAK
             END IF
         END WHILE
     END WHILE
 END FUNCTION
                                     ]]></artwork>
                 </figure>
             </section>
             <section anchor="performance_multi_se" numbered="true" toc="default">
                 <name>Multi Strata Estimators</name>
                 <t>
                     In order to improve the precision of the estimates not only one strata estimator
                     is transmitted for larger sets. One, two, four or eight strata estimators can be
                     transferred. Transmitting multiple strata estimators has the disadvantage that
                     additional bandwidth will be used, so despite the higher precision, it is not
                     always optimal to transmit eight strata estimators. Therefore, the following
                     rules are used, which are based on the average element size multiplied by the number
                     of elements in the set. This value is denoted as "b" in the table:
                 </t>
                 <dl>
                     <dt>SEs</dt>
                     <dd>Rule</dd>
                     <dt>1</dt>
                     <dd>b &lt; 68kb</dd>
                     <dt>2</dt>
                     <dd>b &gt; 68kb</dd>
                     <dt>4</dt>
                     <dd>b &gt; 269kb</dd>
                     <dt>8</dt>
                     <dd>b &gt; 1'077kb</dd>
                 </dl>
                 <t>
                     When creating multiple strata estimators, it is important to salt the keys for the IBFs in the strata
                     estimators differently, using the following bit rotation based salting method:
                 </t>
                 <figure anchor="performance_multi_se_salting_code">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 
 # Inputs:
 # value: Input value to salt (needs to be 64 bit unsigned)
 # salt: Salt to salt value with; Should always be ascending and start
 #       at zero
     i.e. SE1 = Salt 0; SE2 = Salt 1 etc.
 # Output:
 # Returns: Salted value
 
 FUNCTION se_key_salting(value, salt)
     s = (salt * 7) modulo 64
     RETURN (value >> s) | (value << (64 - s))
 END FUNCTION
 
                                     ]]></artwork>
                 </figure>
                 <t>
                     Performance study and details about the reasoning for the used methods can be found in <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/> in section
                     3.4.1 under the title "Added Support for Multiple Strata Estimators".
                     <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/>
                 </t>
             </section>
         </section>
 
 
     <section anchor="security" numbered="true" toc="default">
         <name>Security Considerations</name>
         <t>
             The security considerations in this document focus mainly on the security
             goal of availability. The primary goal of the protocol is to prevent an attacker from
             wasting computing and network resources of the attacked peer.
         </t>
         <t>
             To prevent denial of service attacks, it is vital to check that peers can only
             reconcile a set once in a predefined time span. This is a predefined value and needs
             to be adapted per use basis. To enhance reliability and to allow
             for legitimate failures, say due to network connectivity issues,
             applications SHOULD define a threshold for
             the maximum number of failed reconciliation attempts in a given time period.
         </t>
         <t>
             It is important to close and purge connections after a given timeout
             to prevent draining attacks.
         </t>
         <section anchor="security_generic_functions" numbered="true" toc="default">
             <name>General Security Checks</name>
             <t>
                In this section general checks are described which should be applied to multiple states.
             </t>
 
             <section anchor="security_generic_input_validation" numbered="true" toc="default">
               <name>Input validation</name>
               <t>
                 The format of all received messages needs to be properly validated. This is important to prevent many
                 attacks on the code. The application data MUST be validated by the application using
                 the protocol not by the implementation of the protocol.
                 In case the format validation fails the set operation MUST be terminated.
               </t>
             </section>
             <section anchor="security_generic_functions_check_byzantine_boundaries" numbered="true" toc="default">
                 <name>Byzantine Boundaries</name>
                 <t>
                     To restrict an attacker there should be an upper and lower bound defined and checked
                     at the beginning of the protocol, based
                     on prior knowledge, for the number of elements.
                     The lower byzantine bound can be, for example, the number of elements the
                     other peer had in his set at the last contact.
                     The upper byzantine bound can be a practical maximum e.g. the number
                     of e-voting votes, in Switzerland.
                 </t>
                 <figure anchor="security_generic_functions_missing_message_code">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 # Input:
 # rec: Number of elements in remote set
 # rsd: Number of elements differ in remote set
 # lec: Number of elements in local set
 # lsd: Number of elements differ in local set
 # UPPER_BOUND: Given byzantine upper bound
 # LOWER_BOUND: Given byzantine lower bound
 # Output:
 # returns TRUE if parameters in byzantine bounds otherwise returns FALSE
 FUNCTION check_byzantine_bounds (rec,rsd,lec,lsd)
     IF rec + rsd > UPPER_BOUND THEN
         RETURN FALSE
     END IF
     IF lec + lsd > UPPER_BOUND THEN
         RETURN FALSE
     END IF
     IF rec < LOWER_BOUND THEN
         RETURN FALSE
     END IF
     RETURN TRUE
 END FUNCTION
                     ]]></artwork>
                 </figure>
             </section>
 
             <section anchor="security_generic_functions_check_valid_state" numbered="true" toc="default">
                 <name>Valid State</name>
                 <t>
                     To harden the protocol against attacks, controls were introduced in the improved
                     implementation that check for each message whether the message was received
                     in the correct state. This is central so that an attacker finds as little attack
                     surface as possible and makes it more difficult for the attacker to send the
                     protocol into an endless loop, for example.
                 </t>
             </section>
             <section anchor="security_generic_functions_mfc" numbered="true" toc="default">
                 <name>Message Flow Control</name>
                 <t>
                     For most messages received and sent there needs to be a check in place that checks
                     that a message is not received multiple times. This is solved with a global store (message)
                     and the following code
                 </t>
                 <t>
                     The sequence in which messages are received and sent is arranged in a chain.
                     The messages are dependent on each other. There are dependencies that are
                     mandatory, e.g. for a sent "Demand" message, an "Element" message must
                     always be received. But there are also messages for which a response is
                     not mandatory, e.g. the <em><xref target="messages_inquiry" format="title" /></em> message is only followed by an
                     "Offer" message, if the corresponding element is in the set. Due to this
                     fact, checks can be installed to verify compliance with the following chain.
                 </t>
                 <figure anchor="security_generic_functions_mfc_chain">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 
 
 Chain for
 elements    +---------+    +---------+    +---------+    +---------+
 NOT in IBF  | INQUIRY |--->|  OFFER  |===>|  DEMAND |===>| ELEMENT |
 decoding    +---------+    +---------+    +---------+    +---------+
 peers set
 
 Chain for
 elements    +---------+    +---------+    +---------+
 in IBF      |  OFFER  |--->| DEMAND  |===>| ELEMENT |
 decoding    +---------+    +---------+    +---------+
 peers set
 
             --->: Answer not mandatory
             ===>: Always answer needed.
                     ]]></artwork>
                 </figure>
                 <t>
                     In the message control flow its important to ensure that no duplicated messages are received
                     (Except inquiries where collisions are possible) and only messages are received which are compliant
                     with the flow in <xref target="security_generic_functions_mfc_chain" format="default" />.
                     To link messages the SHA-512 element hashes, that are part of all messages, except in the
                     <em><xref target="messages_inquiry" format="title" /></em> messages, can be used.
                     To link an <em><xref target="messages_inquiry" format="title" /></em> message to an
                     <em><xref target="messages_offer" format="title" /></em> message
                     the SHA-512 hash from the offer has to be salted and converted to the IBF-Key (as described in
                     <xref target="ibf_format_id_generation_pseudo_code" format="default" />). The IBF-Key can
                     be matched with the received <em><xref target="messages_inquiry" format="title" /></em> message.
                 </t>
                 <t>
                     At the end of the set reconciliation operation after receiving and sending the
                     <em><xref target="messages_done" format="title" /></em> message, it should be checked
                     that all demands have been satisfied and all elements have been received.
                 </t>
                 <t>
                     This is based on <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/>, section 5.3 (Message Control Flow).
                 </t>
             </section>
 
             <section anchor="security_generic_functions_active_passive_switches" numbered="true" toc="default">
                 <name>Limit Active/Passive Decoding changes</name>
                 <t>
                   To prevent an attacker from sending a peer into an endless loop between active and passive decoding, a
                   limitation for active/passive roll switches is required.
                   Otherwise, an attacker could
                   force the victim to waste unlimited amount of resources by just transmitting
                   IBFs that do not decode.
                   This can be implemented by
                   a simple counter which terminates the operation after a predefined number of switches.
                   The maximum number of switches needs to be defined in such a way that it is
                   very improbable that more switches are required in a legitimate interaction,
                   and hence the malicious behavior of the other peer is assured.
                 </t>
                 <t>
                   The question after how many active/passive switches it can be assumed that the other peer is not honest,
                   depends on the various tuning parameters of the algorithm.
                   Section 5.4 of <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/>
                   demonstrates that the probability of decoding failure is less than
                   15% for each round. The probability that there will be n legitimate
                   active/passive changes is thus less than 0.15^{round number}.
                   Which means that after about 30 active/passive switches it can be said with a certainty of 2^80 that one of the peers
                   is not following the protocol.
                   Hence, participants MUST impose a maximum of 30 active/passive changes.
                 </t>
             </section>
 
             <section anchor="security_generic_functions_full_plausibility_check" numbered="true" toc="default">
                 <name>Full Synchronisation Plausibility Check</name>
                 <t>
                     An attacker can try to use up a peer's bandwidth by pretending that the peer
                     needs full synchronisation, even if the set difference is very small and the attacker
                     only has a few (or even zero) elements that are not already synchronised.
                     In such a case, it would be ideal if the plausibility could already be checked
                     during full synchronisation as to whether the other peer was honest or not with
                     regard to the estimation of the set size difference and thus the choice of mode
                     of operation.
                 </t>
                 <t>
                     In order to calculate this plausibility, section 5.5 of <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/> describes a formula, which depicts the probability with which one
                     can calculate the corresponding plausibility based on the number of
                     new and repeated elements after each received element.
                 </t>
                 <t>
                     Besides this approach from probability theory, there is an additional check
                     that can be made. After the entire set has been transferred to the other peer,
                     no known elements may be returned by the second peer, since the second peer
                     should only return the elements that are missing from the initial peer's set.
                 </t>
                 <t>
                     This two approaches are implemented in the following pseudocode:
                 </t>
 
                 <figure anchor="security_generic_functions_full_plausibility_check_code">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 # Input:
 # SECURITY_LEVEL: The security level used e.g. 2^80
 # state: The statemachine state
 # rs: Estimated remote set difference
 # lis: Number of elements in set
 # rd: Number of duplicated elements received
 # rf: Number of fresh elements received
 # Output:
 # Returns TRUE if full synchronisation is plausible and FALSE otherwise
 
 FUNCTION full_sync_plausibility_check (state,rs,lis,rd,rf)
     security_level_lb = -1 * SECURITY_LEVEL
 
     # Make sure that no element is received double when
     # all elements already are transmitted to the oder side.
     IF FULL_SENDING == state AND rd > 0 THEN
         RETURN FALSE
     END IF
 
     # Probabilistic algorithm to check for plausible
     # element distribution
     IF FULL_RECEIVING == state THEN
 
         # Prevent division by 0
         IF  0 <= rs THEN
             rs = 1
         END IF
 
         # Formula to verify plausibility
         base = 1 - (rs / (lis + rs))
         exponent = rd - rf * lis / rs
         value = exponent * (LOG2(base)/LOG2(2))
         IF value < security_level_lb OR value > SECURITY_LEVEL THEN
             RETURN FALSE
         END IF
     END IF
     RETURN TRUE
 END FUNCTION
                     ]]></artwork>
                 </figure>
             </section>
     </section>
 
         <section anchor="security_states" numbered="true" toc="default">
             <name>States</name>
 
             <t>
                 In this section the security considerations for each valid message
                 in all states is described, if any other message
                 is received the peer MUST terminate the operation.
             </t>
 
             <section anchor="security_states_expecting_ibf" numbered="true" toc="default">
                 <name>Expecting IBF</name>
                 <t>Security considerations for received messages:</t>
                 <dl>
                     <dt><xref target="messages_request_full" format="title" /></dt>
                     <dd>
                         <t>
                         It needs to be checked that the full synchronisation mode with receiving peer
                         sending first is plausible according to the algorithm deciding which operation mode
                         is applicable as described in <xref target="performance_formulas_operationmode" format="default"/>.
                         </t>
 
                     </dd>
                     <dt><xref target="messages_ibf" format="title" /></dt>
                     <dd>
                         <t>
                             It needs to be checked that the differential synchronisation mode is plausible according
                             to the algorithm deciding which operation mode
                             is applicable as described in <xref target="performance_formulas_operationmode" format="default"/>.
                         </t>
 
                     </dd>
                     <dt><xref target="messages_send_full" format="title" /></dt>
                     <dd>
                         <t>
                             It needs to be checked that the full synchronisation mode with initiating peer
                             sending first is plausible according to the algorithm deciding which operation mode
                             is applicable as described in <xref target="performance_formulas_operationmode" format="default"/>.
                         </t>
                     </dd>
                 </dl>
             </section>
 
             <section anchor="security_states_full_sending" numbered="true" toc="default">
                 <name>Full Sending</name>
                 <t>Security considerations for received messages:</t>
                 <dl>
                     <dt><xref target="messages_full_element" format="title" /></dt>
                     <dd>
                       <t>
                         When receiving full elements there needs to be checked, that every
                         element is a valid element, that no element has been received more than once, and
                         that not more elements have been received than the other peer has committed
                         to at the beginning of the operation. The plausibility should also be checked
                         with an algorithm as described in <xref target="security_generic_functions_full_plausibility_check" format="default"/>.
                       </t>
                     </dd>
                     <dt><xref target="messages_full_done" format="title" /></dt>
                     <dd>
                       <t>
                         When receiving the <em><xref target="messages_full_done" format="title" /></em>
                         message, it is important to check that
                         not fewer elements have been received than the other peer has committed to
                         send at the beginning of the operation.
                         If the sets differ (the FINAL CHECKSUM field in the <xref target="messages_full_done" format="title" />
                         message does not match to the SHA-512 hash XOR sum of the local set), the operation has failed and the
                         reconciliation MUST be aborted. It is a strong indicator
                         that something went wrong (eg. some hardware bug). This should never occur!
                       </t>
                     </dd>
                 </dl>
             </section>
 
             <section anchor="security_states_expecting_ibf_last" numbered="true" toc="default">
                 <name>Expecting IBF Last</name>
                 <t>Security considerations for received messages:</t>
                 <dl>
                     <dt><xref target="messages_ibf" format="title" /></dt>
                     <dd>
                         <t>
                           The application should check that the overall size of the IBF
                           that is being transmitted is within its resource bounds, and
                           abort the protocol if its resource limits are likely to be
                           exceeded, or if the size is implausible for the given operation.
                         </t>
                         <t>
                           It needs to be checked that the offset (message field "OFFSET")
                           for every received <em><xref target="messages_ibf" format="title" /></em> message
                           is strictly monotonic increasing and is a multiple of the MAX_BUCKETS_PER_MESSAGE
                           defined in the <xref target="constants" format="title" /> section, otherwise the
                           connection MUST be aborted.
                         </t>
                         <t>
                           Another sanity check is to ensure that the "OFFSET" message field never
                           is higher than the "IBF SIZE" field in the <em><xref target="messages_ibf" format="title" /></em>
                           message.
                         </t>
                     </dd>
                     <dt><xref target="messages_ibf_last" format="title" /></dt>
                     <dd>
                         <t>
                             When all <em><xref target="messages_ibf" format="title" /></em> messages have
                             been received an <em><xref target="messages_ibf_last" format="title" /></em> message
                             should conclude the transmission of the IBF and a change to the <strong>Active Decoding</strong>
                             phase should be ensured.
                         </t>
                         <t>
                             To verify that all IBFs have been received, a simple validation can be made.
                             The number of buckets in the <em><xref target="messages_ibf_last" format="title" /></em> message
                             added to the value in the message OFFSET field should always be equal to the "IBF SIZE".
                         </t>
                         <t>
                             Further plausibility checks can be made. One is to ensure that after each active/passive
                             switch the IBF can never be more than double in size. Another plausibility check is
                             that an IBF probably never will be larger than the byzantine upperbound multiplied by two.
                             The third plausibility check is to take successfully decoded IBF keys (received offers and demands)
                             into account and to validate the size of the received IBF with the in <xref target="performance_formula_ibf_parameters_code" format="default" />
                             described function get_next_ibf_size(). If any of these three checks fail the operation
                             must be aborted.
                         </t>
                     </dd>
                 </dl>
             </section>
 
             <section anchor="security_states_active_decoding" numbered="true" toc="default">
                 <name>Active Decoding</name>
                 <t>
                     In the <strong>Active Decoding</strong> state it is important to prevent an attacker from
                     generating and transmitting an unlimited number of IBFs that all do not decode, or
                     to generate an IBF constructed to send the peers in an endless loop.
                     To prevent an endless loop in decoding, loop detection MUST be implemented.
                     A solution to prevent endless loop is to limit the number of elements decoded from an IBF.
                     This limit is defined by the number of buckets in the IBF. It is not possible that more elements are decoded
                     from an IBF than an IBF has buckets. If more elements than buckets are in an IBF it is not possible to
                     get pure buckets.
                     An additional check that should be implemented, is to store all element IDs
                     that were prior decoded. When a new element ID is decoded from the IBF it should
                     always be checked that no element ID is repeated.
                     If the same element ID is decoded more than once, this is a strong indication
                     for an invalid IBF and the operation MUST be aborted. Notice that the decoded
                     element IDs are salted as described in <xref target="ibf_format_id_generation_pseudo_code" format="default" />
                     so the described bit rotation needs to be reverted before the decoded element ID
                     is stored and compared to the previous decoded element IDs.
                 </t>
                 <t>
                     If the IBF decodes more elements than are plausible, the
                     operation MUST be terminated. Furthermore, if the IBF
                     decoding successfully terminates and fewer elements were
                     decoded than plausible, the operation MUST also be terminated.
                     The upper thresholds for decoded elements from the IBF is the
                     remote set size the other peer has committed too (Case if the complete remote set is
                     new). The lower threshold for decoding element is the absolute value of the difference
                     between the local and remote set size (Case the set difference is only in the set of
                     a single peer). The other peer's committed set sizes
                     is transmitted in the the <strong>Expecting IBF</strong> state.
                 </t>
 
                 <t>Security considerations for received messages:</t>
                 <dl>
                     <dt><xref target="messages_offer" format="title" /></dt>
                     <dd>
                         <t>
                             If an offer for an element, that never has been requested by
                             an inquiry or if an offer is received twice, the operation MUST be terminated.
                             This requirement can be fulfilled by saving lists that keep track of the state of
                             all sent inquiries and offers. When answering offers these lists MUST be checked.
                             The sending and receiving of <xref target="messages_offer" format="title" /> messages should
                             always be protected with an <xref target="security_generic_functions_mfc" format="title" />
                             to secure the protocol against missing, duplicated, out-of-order or unexpected messages.
                         </t>
                     </dd>
                     <dt><xref target="messages_elements" format="title" /></dt>
                     <dd>
                         <t>
                             If an element that never has been requested by
                             a demand or is received twice, the operation MUST be terminated.
                             The sending and receiving of <xref target="messages_elements" format="title" /> messages should
                             always be protected with an <xref target="security_generic_functions_mfc" format="title" />
                             to secure the protocol against missing, duplicated, out-of-order or unexpected messages.
                         </t>
                     </dd>
                     <dt><xref target="messages_demand" format="title" /></dt>
                     <dd>
                         <t>
                         For every received demand an offer has to be sent in advance. If a demand
                         for an element is received, that never has been offered or the offer already has
                         been answered with a demand, the operation MUST be terminated. It is required to implement
                         a list which keeps track of the state of all sent offers and received demands.
                         The sending and receiving of <em><xref target="messages_demand" format="title" /></em> messages should
                         always be protected with an <xref target="security_generic_functions_mfc" format="title" />
                         to secure the protocol against missing, duplicated, out-of-order or unexpected messages.
                         </t>
                     </dd>
                     <dt><xref target="messages_done" format="title" /></dt>
                     <dd>
                         <t>
                             The <em><xref target="messages_done" format="title" /></em> message is only received if the IBF has finished
                             decoding and all offers have been sent. If the <em><xref target="messages_done" format="title" /></em> message is received before
                             the decoding of the IBF is finished or all open demands
                             have been answered, the operation MUST be terminated.
                             If the sets differ (the FINAL CHECKSUM field in the <xref target="messages_done" format="title" />
                             message does not match to the SHA-512 hash XOR sum of the local set), the operation has failed and the
                             reconciliation MUST be aborted. It is a strong indicator
                             that something went wrong (eg. some hardware bug). This should never occur!
                         </t>
                         <t>
                             When a <em><xref target="messages_done" format="title" /></em> message is received the
                             "check_if_synchronisation_is_complete()" function from the <xref target="security_generic_functions_mfc" format="title" />
                             is required to ensure that all demands have been satisfied successfully.
                         </t>
                     </dd>
                 </dl>
             </section>
             <section anchor="security_states_finish_closing" numbered="true" toc="default">
                 <name>Finish Closing</name>
                 <t>
                     In the  <strong>Finish Closing</strong> state the protocol waits for
                     all sent demands to be fulfilled.
                 </t>
                 <t>
                     In case not all sent demands have been answered in time,
                     the operation has failed and MUST be terminated.
                 </t>
                 <t>Security considerations for received messages:</t>
                 <dl>
                     <dt><xref target="messages_elements" format="title" /></dt>
                     <dd>
                         When receiving <xref target="messages_elements" format="title" /> messages it is important
                         to always check the <xref target="security_generic_functions_mfc" format="title" />
                         to secure the protocol against missing, duplicated, out-of-order or unexpected messages.
                     </dd>
                 </dl>
             </section>
             <section anchor="security_states_finished" numbered="true" toc="default">
                 <name>Finished</name>
                 <t>
                     In this state the connection is terminated, so no security considerations are needed.
                 </t>
             </section>
             <section anchor="security_states_expect_se" numbered="true" toc="default">
                 <name>Expect SE</name>
                 <t>Security considerations for received messages:</t>
                 <dl>
                     <dt><xref target="messages_se" format="title" /></dt>
                     <dd>
                         <t>
                             In case the strata estimator does not decode, the
                             operation MUST be terminated to prevent to get to an unresolvable state.
                             The set difference calculated from the strata estimator needs to be plausible,
                             which means within the byzantine boundaries described in section <xref target="security_generic_functions_check_byzantine_boundaries" format="title" />.
                         </t>
                     </dd>
                 </dl>
             </section>
             <section anchor="security_states_full_receiving" numbered="true" toc="default">
                 <name>Full Receiving</name>
                 <t>Security considerations for received messages:</t>
                 <dl>
                     <dt><xref target="messages_full_element" format="title" /></dt>
                     <dd>
                         <t>
                             When receiving full elements there needs to be checked, that every
                             element is a valid element, no element has been received more than once and
                             not more elements are received than the other peer committed
                             to sending at the beginning of the operation. The plausibility should also be checked
                             with an algorithm as described in <xref target="security_generic_functions_full_plausibility_check" format="default"/>.
                         </t>
                     </dd>
                     <dt><xref target="messages_full_done" format="title" /></dt>
                     <dd>
                         <t>
                             When the <em><xref target="messages_full_done" format="title" /></em> message is received from the remote peer, it should be checked that the number of
                             elements received matches the number that the remote peer
                             originally committed to transmitting,
                             otherwise the operation MUST be terminated.
                             If the sets differ (the FINAL CHECKSUM field in the <xref target="messages_full_done" format="title" />
                             message does not match to the SHA-512 hash XOR sum of the local set), the operation has failed and the
                             reconciliation MUST be aborted. It is a strong indicator
                             that something went wrong (eg. some hardware bug). This should never occur!
                         </t>
                     </dd>
                 </dl>
             </section>
             <section anchor="security_states_passive_decoding" numbered="true" toc="default">
                 <name>Passive Decoding</name>
                 <t>Security considerations for received messages:</t>
                 <dl>
                     <dt><xref target="messages_ibf" format="title" /></dt>
                     <dd>
                         <t>
                             In case an <xref target="messages_ibf" format="title" /> message is received by the peer a active/passive role switch
                             is initiated by the active decoding remote peer.
                             A switch into active decoding mode MUST only be permitted for
                             a predefined number of times as described in <xref target="security_generic_functions_active_passive_switches" format="default"/>
                         </t>
                     </dd>
                     <dt><xref target="messages_inquiry" format="title" /></dt>
                     <dd>
                         <t>
                         A check needs to be in place that prevents receiving an inquiry
                         for an element multiple times or more inquiries than are plausible.
                         The upper thresholds for sent/received inquiries is the
                         remote set size the other peer has committed too (Case if the complete remote set is
                         new). The lower threshold for for sent/received inquiries is the absolute value of the
                         set difference between the local and remote set size
                         (Case the set difference is only in the set of a single peer).
                         The other peer's committed set sizes is transmitted in the the <strong>Expecting IBF</strong> state.
                         Beware that it is possible to get key collisions and an inquiry for the same key
                         can be transmitted multiple times, so the threshold should take this into account.
                         The sending and receiving of <em><xref target="messages_inquiry" format="title" /></em> messages should
                         always be protected with an <xref target="security_generic_functions_mfc" format="title" />
                         to secure the protocol against missing, duplicated, out-of-order or unexpected messages.
                         </t>
                     </dd>
                     <dt><xref target="messages_demand" format="title" /></dt>
                     <dd>
                         Same action as described for <em><xref target="messages_demand" format="title" /></em> message in section
                         <xref target="security_states_active_decoding" format="title"/>.
                     </dd>
                     <dt><xref target="messages_offer" format="title" /></dt>
                     <dd>
                         Same action as described for <em><xref target="messages_offer" format="title" /></em> message in section
                         <xref target="security_states_active_decoding" format="title"/>.
                     </dd>
                     <dt><xref target="messages_done" format="title" /></dt>
                     <dd>
                         Same action as described for <em><xref target="messages_done" format="title" /></em> message in section
                         <xref target="security_states_active_decoding" format="title"/>.
                     </dd>
                     <dt><xref target="messages_elements" format="title" /></dt>
                     <dd>
                         Same action as described for <em><xref target="messages_elements" format="title" /></em> message in section
                         <xref target="security_states_active_decoding" format="title"/>.
                     </dd>
 
                 </dl>
             </section>
             <section anchor="security_states_finish_waiting" numbered="true" toc="default">
                 <name>Finish Waiting</name>
                 <t>
                     In the  <strong>Finish Waiting</strong> state the protocol waits for
                     all transmitted demands to be fulfilled.
                 </t>
                 <t>
                     In case not all transmitted demands have been answered at this time, the operation
                     has failed and the protocol MUST be terminated with an error.
                 </t>
                 <t>Security considerations for received messages:</t>
                 <dl>
                     <dt><xref target="messages_elements" format="title" /></dt>
                     <dd>
                         When receiving <xref target="messages_elements" format="title" /> messages it is important
                         to always check the <xref target="security_generic_functions_mfc" format="title" />
                         to secure the protocol against missing, duplicated, out-of-order or unexpected messages.
                     </dd>
                 </dl>
             </section>
         </section>
     </section>
     <section anchor="constants" numbered="true" toc="default">
         <name>Constants</name>
         <t>
             The following table contains constants used by the protocol. The constants marked with a * are
             validated through experiments in <xref target="byzantine_fault_tolerant_set_reconciliation" format="default"/>.
         </t>
         <figure anchor="figure_constants">
             <artwork name="" type="" align="left" alt=""><![CDATA[
 Name                        | Value      | Description
 ----------------------------+------------+-------------------------------
 SE_STRATA_COUNT             | 32         | Number of IBFs in a strata
                                            estimator.
 IBF_HASH_NUM*               | 3          | Number of times an element is
                                            hashed to an IBF.
                                            (from section 4.5.2)
 IBF_FACTOR*                 | 2          | The factor by which the size
                                            of the IBF is increased in
                                            case of decoding failure or
                                            initially from the set
                                            difference.
                                            (from section 4.5.2)
 MAX_BUCKETS_PER_MESSAGE     | 1120       | Maximum bucket of an IBF
                                            that are transmitted in
                                            single message.
 IBF_MIN_SIZE*               | 37         | Minimal number of buckets
                                            in an IBF. (from section 3.8)
 DIFFERENTIAL_RTT_MEAN*      | 3.65145    | The average RTT that is
                                            needed for a differential
                                            synchronisation.
 SECURITY_LEVEL*             | 2^80       | Security level for
                                            probabilistic security
                                            algorithms. (from section 5.8)
 PROBABILITY_FOR_NEW_ROUND*  | 0.15       | The probability for a IBF
                                            decoding failure in the
                                            differential synchronisation
                                            mode. (from section 5.4)
 DIFFERENTIAL_RTT_MEAN*      | 3.65145    | The average RTT that is needed
                                            for a differential
                                            synchronisation.
                                            (from section 4.5.3)
 MAX_IBF_SIZE                | 1048576    | Maximal number of buckets in
                                            an IBF.
 AVG_BYTE_SIZE_SE*           | 4221       | Average byte size of a single
                                            strata estimator.
                                            (from section 3.4.3)
 VALID_NUMBER_SE*            | [1,2,4,8]  | Valid number of SE's
                                            (from section 3.4)
             ]]></artwork>
         </figure>
 
      </section>
         <section anchor="gana" numbered="true" toc="default">
             <name>GANA Considerations</name>
             <t>
                 GANA is requested to amend the "GNUnet Message Type" <xref target="GANA" format="default"/> registry
                 as follows:
             </t>
             <figure anchor="figure_purposenums">
                 <artwork name="" type="" align="left" alt=""><![CDATA[
 Type    | Name                       | References | Description
 --------+----------------------------+------------+----------------------
  559    | SETU_P2P_REQUEST_FULL      | [This.I-D] | Request the full set
                                                     of the other peer.
  710    | SETU_P2P_SEND_FULL         | [This.I-D] | Signals to send the
                                                     full set to the other
                                                     peer.
  560    | SETU_P2P_DEMAND            | [This.I-D] | Demand the whole
                                                     element from the
                                                     otherpeer, given
                                                     only the hash code.
  561    | SETU_P2P_INQUIRY           | [This.I-D] | Tell the other peer
                                                     to send a list of
                                                     hashes that match
                                                     an IBF key.
  562    | SETU_P2P_OFFER             | [This.I-D] | Tell the other peer
                                                     which hashes match
                                                     a given IBF key.
  563    | SETU_P2P_OPERATION_REQUEST | [This.I-D] | Request a set union
                                                     operation from a
                                                     remote peer.
  564    | SETU_P2P_SE                | [This.I-D] | Strata Estimator
                                                     uncompressed.
  565    | SETU_P2P_IBF               | [This.I-D] | Invertible Bloom
                                                     Filter slices.
  566    | SETU_P2P_ELEMENTS          | [This.I-D] | Actual set elements.
  567    | SETU_P2P_IBF_LAST          | [This.I-D] | Invertible Bloom
                                                     Filter Last Slices.
  568    | SETU_P2P_DONE              | [This.I-D] | Set operation is
                                                     done.
  569    | SETU_P2P_SEC               | [This.I-D] | Strata Estimator
                                                     compressed.
  570    | SETU_P2P_FULL_DONE         | [This.I-D] | All elements in
                                                     full synchronisation
                                                     mode have been sent
                                                     is done.
  571    | SETU_P2P_FULL_ELEMENT      | [This.I-D] | Send an actual
                                                     element in full
                                                     synchronisation mode.
 
            ]]></artwork>
             </figure>
         </section>
         <!-- gana -->
         <section anchor="contributors" numbered="true" toc="default">
             <name>Contributors</name>
             <t>
 	        The GNUnet implementation of the byzantine fault tolerant set reconciliation
 	        protocol was originally implemented by Florian Dold.
             </t>
         </section>
     </middle>
     <back>
         <references>
             <name>Normative References</name>
             &RFC5869;
             &RFC2119;
             &RFC3385;
             &RFC1951;
 
             <reference anchor="byzantine_fault_tolerant_set_reconciliation" target="https://summermatter.net/byzantine-fault-tolerant-set-reconciliation-summermatter.pdf">
                 <front>
                     <title>Byzantine Fault Tolerant Set Reconciliation</title>
                     <author initials="E." surname="Summermatter" fullname="Elias Summermatter">
                         <organization>University of Applied Sciences Bern</organization>
                     </author>
                     <date year="2021"/>
                 </front>
             </reference>
 
             <reference anchor="GANA" target="https://gana.gnunet.org/">
                 <front>
                     <title>GNUnet Assigned Numbers Authority (GANA)</title>
                     <author>
                         <organization>GNUnet e.V.</organization>
                     </author>
                     <date month="April" year="2020"/>
                 </front>
             </reference>
 
             <reference anchor="CryptographicallySecureVoting" target="https://git.gnunet.org/bibliography.git/plain/docs/ba_dold_voting_24aug2014.pdf">
                 <front>
                     <title>Cryptographically Secure, Distributed Electronic Voting</title>
                     <author initials="F." surname="Dold" fullname="Florian Dold">
                         <organization>Technische Universität München</organization>
                     </author>
                 </front>
             </reference>
 
 
             <reference anchor="ByzantineSetUnionConsensusUsingEfficientSetReconciliation" target="https://doi.org/10.1186/s13635-017-0066-3">
                 <front>
                     <title>Byzantine set-union consensus using efficient set reconciliation</title>
                     <author initials="F." surname="Dold" fullname="Florian Dold">
                         <organization>Technische Universität München</organization>
                     </author>
                     <author initials="C." surname="Grothoff" fullname="Christian Grothoff">
                         <organization>Inria, Domaine de Voluceau Rocquencourt</organization>
                     </author>
                 </front>
             </reference>
             <reference anchor="Eppstein" target="https://doi.org/10.1145/2018436.2018462">
                 <front>
                     <title>What’s the Difference? Efficient Set Reconciliation without Prior Context</title>
                     <author initials="D." surname="Eppstein" fullname="David Eppstein">
                         <organization>U.C. Irvine</organization>
                     </author>
                     <author initials="M." surname="Goodrich" fullname="Michael T. Goodrich">
                         <organization>U.C. Irvine</organization>
                     </author>
                     <author initials="F." surname="Uyeda" fullname="Frank Uyeda">
                         <organization>U.C. San Diego</organization>
                     </author>
                     <author initials="G." surname="Varghese" fullname="George Varghese">
                         <organization>U.C. San Diego</organization>
                     </author>
                 </front>
             </reference>
 
             <reference anchor="GNS" target="https://doi.org/10.1007/978-3-319-12280-9_9">
                 <front>
                     <title>A Censorship-Resistant, Privacy-Enhancing and Fully Decentralized Name System</title>
                     <author initials="M." surname="Wachs" fullname="Matthias Wachs">
                         <organization>Technische Universitaet Muenchen</organization>
                     </author>
 
                     <author initials="M." surname="Schanzenbach" fullname="Martin Schanzenbach">
                         <organization>Technische Universitaet Muenchen</organization>
                     </author>
 
                     <author initials="C." surname="Grothoff"
                             fullname="Christian Grothoff">
                         <organization>Technische Universitaet Muenchen</organization>
                     </author>
                     <date year="2014"/>
                 </front>
             </reference>
 
         </references>
         <section anchor="test_vectors" numbered="true" toc="default">
             <name>Test Vectors</name>
             <section anchor="test_vectors_map_function" numbered="true" toc="default">
                 <name>Map Function</name>
                 <t>
                     INPUTS:
                 </t>
                 <figure anchor="test_vectors_map_function_inputs">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 k: 3
 ibf_size: 300
 
 key1: 0xFFFFFFFFFFFFFFFF (64-bit)
 key2: 0x0000000000000000 (64-bit)
 key3: 0x00000000FFFFFFFF (64-bit)
 key4: 0xC662B6298512A22D (64-bit)
 key5: 0xF20fA7C0AA0585BE (64-bit)
            ]]></artwork>
                 </figure>
                 <t>
                     OUTPUT:
                 </t>
                 <figure anchor="test_vectors_map_function_outputs">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 key1: ["122","157","192"]
 key2: ["85","243","126"]
 key3: ["208","101","222"]
 key4: ["239","269","56"]
 key5: ["150","104","33"]
            ]]></artwork>
                 </figure>
             </section>
             <section anchor="test_vectors_id_function" numbered="true" toc="default">
                 <name>ID Calculation Function</name>
                 <t>
                     INPUTS:
                 </t>
                 <figure anchor="test_vectors_id_function_inputs">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 element 1: 0xFFFFFFFFFFFFFFFF (64-bit)
 element 2: 0x0000000000000000 (64-bit)
 element 3: 0x00000000FFFFFFFF (64-bit)
 element 4: 0xC662B6298512A22D (64-bit)
 element 5: 0xF20fA7C0AA0585BE (64-bit)
            ]]></artwork>
                 </figure>
                 <t>
                     OUTPUT:
                 </t>
                 <figure anchor="test_vectors_id_function_outputs">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 element 1: 0x5AFB177B
 element 2: 0x64AB557C
 element 3: 0xCB5DB740
 element 4: 0x8C6A2BB2
 element 5: 0x7EC42981
            ]]></artwork>
                 </figure>
             </section>
             <section anchor="test_counter_compression_function" numbered="true" toc="default">
                 <name>Counter Compression Function</name>
                 <t>
                     INPUTS:
                 </t>
                 <figure anchor="test_counter_compression_function_inputs">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 counter serie 1: [1,8,10,6,2] (min bytes 4)
 counter serie 2: [26,17,19,15,2,8] (min bytes 5)
 counter serie 3: [4,2,0,1,3] (min bytes 3)
            ]]></artwork>
                 </figure>
                 <t>
                     OUTPUT:
                 </t>
                 <figure anchor="test_counter_compression_function_outputs">
                     <artwork name="" type="" align="left" alt=""><![CDATA[
 counter serie 1: 0x18A62
 counter serie 2: 0x3519BC48
 counter serie 3: 0x440B
            ]]></artwork>
                 </figure>
             </section>
         </section>
     </back>
 </rfc>