]> Seccom GmbH

Brunnmattstrasse 44 Bern 3007 CH elias.summermatter@seccom.ch

Berner Fachhochschule

Hoeheweg 80 Biel/Bienne 2501 CH grothoff@gnunet.org

General Independent Stream name systems This document contains a protocol specification for Byzantine fault-tolerant Set Reconciliation.

Introduction This document describes a byzantine fault tolerant set reconciliation protocol used to efficient and securely compute the union of two sets across a network. 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) . 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 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" . 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. 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. 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. 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. 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. 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. 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 .

Background

Bloom Filter 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". 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:

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} ]]>

A typical mapping function is constructed by hashing the element, for example using the well-known HKDF construction. 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. 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. In the following example the element e0 with M(e0) = {1,3} has been added:

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:

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. 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 is required.

Counting Bloom Filter A Counting Bloom Filter (CBF) is a variation on the idea of a . With a CBF, buckets are unsigned numbers instead of binary values. This allows the removal of an element from the CBF. 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:

The counter stored in the bucket is also called the order of the bucket. To remove an element form the CBF the counters of all buckets the element is mapped to are decreased by 1. For example, removing M(e2) = {1,3} from the CBF above results in:

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. 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.

Invertible Bloom Filter An Invertible Bloom Filter (IBF) is a further extension of the . An IBF extends the with two more operations: decode and set difference. This two extra operations are key to efficiently obtain small differences between large sets.

Structure 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. 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".

Salted Element ID Calculation 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. 64-bit element IDs are generated from a HKDF construction 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. 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.

> s) | (key << (64 - s)) # 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) ]]>

HASH calculation 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 the RFC.

Mapping Function 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. 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.

Operations When an IBF is created, all counters and IDSUM and HASHSUM values of all buckets are initialized to zero.

Insert Element 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 . 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 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 . 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. Empty IBF:

Insert first element with ID 0x0102 and hash 0x4242 into {1,3}:

Insert second element with ID 0x0304 and hash 0101 into {0,1}:

Remove Element 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. In the following example the remove operation is illustrated. IBF with two encoded elements:

After removal of element with ID 0x0304 and hash 0x0101 mapped to {0,1} from the IBF:

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. 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 , we will call buckets that only represent a single element pure buckets. 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.

Extracting elements 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. 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. 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). 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. In the following example the successful decoding of an IBF containing the two elements previously added in our running example. We begin with an IBF with two elements added:

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:

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.

Set Difference 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. This new IBF can be decoded as described in section . 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. To demonstrate the set difference operation we compare IBF-A with IBF-B and generate as described IBF-AB 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}:

IBF-B also contains the element with ID 0x0102 and and another element with ID 0x1345 and hash 0x5050 mapped to {1,2}.

IBF-A minus IBF-B is then:

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.

Wire format 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 message in the "IMCS" field as unsigned 8-bit integer. For implementation details see section . 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. 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.

Strata Estimator 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. 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. 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. 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.

Mode of Operation 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. The simplest mode is the full synchronisation mode. 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. The second possibility is that the difference between the sets is relatively small compared to the set size. In this case, the differential synchronisation mode is more efficient. Given these two possibilities, the first steps of the protocol are used to determine which mode MUST be used. Thus, the set union protocol always begins with the following operation mode independent steps: The initiating peer begins in the Initiating Connection state and the receiving peer in the Expecting Connection state. The first step for the initiating peer in the protocol is to send an to the receiving peer and transition into the Expect SE state. After receiving the the receiving peer transitions to the Expecting IBF state and answers with the message. When the initiating peer receives the 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 and the is explained in the section below.

Full Synchronisation Mode 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 message, and transitions from Expecting SE to the Full Receiving state. If it has been determined that it is better that the initiating peer sends his set first, the initiating peer sends a message followed by all set elements in messages to the other peer, followed by the message, and transitions into the Full Sending state. A state diagram illustrating the state machine used during full synchronization is provided here. The behavior of the participants the different state is described below:

Expecting IBF:

If a peer in the Expecting IBF state receives a message from the other peer, the peer sends all the elements of his set followed by a message to the other peer, and transitions to the Full Sending state. If the peer receives an message followed by messages, the peer processes the element and transitions to the Full Receiving state.

Full Sending:

While a peer is in Full Sending state the peer expects to continuously receive elements from the other peer. As soon as a the message is received, the peer transitions into the Finished state.

Full Receiving:

While a peer is in the Full Receiving state, it expects to continuously receive elements from the other peer. As soon as a the message is received, it sends the remaining elements (those it did not receive) from his set to the other peer, followed by a . After sending the last message, the peer transitions into the Finished state.

Differential Synchronisation Mode 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. When the initiating peer in the Expected SE state decides to use the differential synchronisation mode, it sends an IBF, which may consist of several messages, to the receiving peer and transitions into the Passive Decoding state. The receiving peer in the Expecting IBF state receives the first message from the initiating peer, and transitions into the Expecting IBF Last state if the IBF was split into multiple messages. If there is just a single message, the receiving peer transitions directly to the Active Decoding state. The peer that is in the Active Decoding, Finish Closing or in the Expecting IBF Last state is called the active peer, and the peer that is in either the Passive Decoding or the Finish Waiting state is called the passive peer. A state diagram illustrating the state machine used during differential synchronization is provided here. The behavior of the participants the different states is described below:

Passive Decoding:

In the Passive Decoding 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 Passive Decoding state from the active peer and is described below on a per message basis.

message:

The 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 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.

message:

The 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 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.

message:

The 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 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.

message:

When a new message has been received the peer checks if a corresponding 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.

message:

If an 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 Expecting IBF Last state, and waits for more messages. There, once the final message has been received, it transitions to Active Decoding.

message:

If an message is received this indicates that there is just one IBF slice left and a direct state and role transition from Passive Decoding to Active Decoding is initiated.

message:

Receiving the 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 Finish Waiting state.

Active Decoding:

In the Active Decoding 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. If the IBF decodes a positive (1) pure bucket, the element is missing on the passive peers site. Thus, the active peer sends an 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 to the passive peer. 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 Passive Decoding 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. As soon as the active peer successfully finished decoding the IBF, the active peer sends a message to the passive peer. 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:

message:

The message indicates that the passive peer received a 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 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.

message:

The message indicates that the passive peer received a from the active peer. The active peer satisfies the demand of the passive peer by sending an 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.

message:

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 messages for elements that were not demanded and that might even already be known locally.

message:

Receiving the message indicates that all demands of the passive peer have been satisfied. The active peer then changes into the Finish Closing state. If the IBF has not finished decoding and the is received, the other peer is not in compliance with the protocol and the protocol MUST be aborted.

Expecing IBF Last

In this state the active peer continuously receives messages from the passive peer. When the last message is received, the peer changes into the Active Decoding state.

Finish Closing / Finish Waiting

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 Finished state.

Combined Mode In the combined mode the protocol decides between and the 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. The 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 is more efficient. The exact heuristics and parameters on which the protocol decides which mode MUST be used are described in the section of this document. There are two main cases when a 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 to 0. The second case is if the application requests full synchronisation explicitly. This is useful for testing and MUST NOT be used in production. The state diagram illustrating the combined mode can be found here.

Messages This section describes the various message formats used by the protocol.

Operation Request

Description 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. This message is sent in the transition between the Initiating Connection state and the Expect SE state. If a peer receives this message and is willing to run the protocol, it answers by sending back a message. Otherwise it simply closes the connection.

Structure

where:

MSG SIZE

is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.

MSG TYPE

is the type of SETU_P2P_OPERATION_REQUEST as registered in , in network byte order.

ELEMENT COUNT

is the number of the elements the requesting party has in its set, as a 32-bit unsigned integer in network byte order.

APX

is a SHA-512 hash that identifies the application.

APPLICATION DATA

is optional, variable-size application specific data that can be used by the application to decide if it would like to answer the request.

IBF

Description The message contains a slice of the IBF. The IBF message is sent at the start of the protocol from the initiating peer in the transaction between Expect SE -> Expecting IBF Last or when the IBF does not decode and there is a role change in the transition between Active Decoding -> Expecting IBF Last. 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 message is sent.

Structure

where:

MSG SIZE

is a 16-bit unsigned integer in network byte orderwhichdescribes the message size in bytes with the header included.

MSG TYPE

the type of SETU_P2P_REQUEST_IBF as registered in in network byte order.

IBF SIZE

is a 32-bit unsigned integer which signals the total number of buckets in the IBF. The minimal number of buckets is 37.

OFFSET

is a 32-bit unsigned integer which signals the offset of the following IBF slices in the original.

SALT

is a 16-bit unsigned integer that contains the IBF-salt which was used to create the IBF.

IMCS

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 section.

IBF-SLICE

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 ). 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). 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 details about ID generation. 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 . The algorithm to find the correct bucket in which the ID and the HASH have to be added is described in detail in section . Test vectors for an implementation can be found in the section

IBF Last

Description 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. Receiving this message initiates the state transmissions Expecting IBF Last -> Active Decoding, Expecting IBF -> Active Decoding and Passive Decoding -> Active Decoding. This message can initiate a peer the roll change from Active Decoding to Passive Decoding.

Structure The binary structure is exactly the same as the of the message with a different "MSG TYPE" which is defined in "SETU_P2P_IBF_LAST".

Element

Description The message contains an element that is synchronized in the and transmits a full element between the peers. This message is sent in the state Active Decoding and Passive Decoding as answer to a message from the remote peer. The message can also be received in the Finish Closing or Finish Waiting state after receiving a message from the remote peer. In this case the peer changes to the Finished state as soon as all demands for elements have been satisfied. This message is exclusively used in the .

Structure

where:

MSG SIZE

is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.

MSG TYPE

is SETU_P2P_ELEMENTS as registered in in network byte order.

E TYPE

is a 16-bit unsigned integer which defines the element type for the application.

PADDING

is 16-bit always set to zero.

E SIZE

is a 16-bit unsigned integer that signals the size of the elements data part.

DATA

is a field with variable length that contains the data of the element.

Offer

Description The message is an answer to an 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 message for that element. The offer is sent and received only in the Active Decoding and in the Passive Decoding state. This message is exclusively sent in the .

Structure

where:

MSG SIZE

is a 16-bit unsigned integer in network byte order, which describes the message size in bytes header included.

MSG TYPE

is SETU_P2P_OFFER as registered in in network byte order.

HASHES

contains one or more successive SHA 512-bit hashes of the elements that are being requested with messages.

Inquiry

Description The message is exclusively sent by the active peer in Active Decoding 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 message. This message is exclusively sent in the .

Structure

where:

MSG SIZE

is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.

MSG TYPE

is SETU_P2P_INQUIRY as registered in in network byte order.

IBF KEY

contains one or more successive ibf keys (64-bit unsigned integer) for which the inquiry is sent.

Demand

Description The message is sent in the Active Decoding and in the Passive Decoding state. It is an answer to a received message and is sent if the element described in the message is missing in the peers set. In the normal workflow the answer to the message is an message. This message is exclusively sent in the .

Structure

where:

MSG SIZE

is a 16-bit unsigned integer in network byte order, which describes the message size in bytes and the header is included.

MSG TYPE

the type of SETU_P2P_DEMAND as registered in in network byte order.

HASH

contains one or more successive SHA 512-bit hashes of the elements that are being demanded.

Done

Description The message is sent when all messages have been successfully satisfied and from the perspective of the sender the set is completely synchronized. This message is exclusively sent in the .

Structure

where:

MSG SIZE

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.

MSG TYPE

is SETU_P2P_DONE as registered in in network byte order.

Full Done

Description The message is sent in the to signal that all remaining elements of the set have been sent. The message is received and sent in the Full Sending and in the Full Receiving state. When the message is received in Full Sending state the peer changes directly into Finished state. In Full Receiving state receiving a message initiates the sending of the remaining elements that are missing in the set of the other peer.

Structure

where:

MSG SIZE

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.

MSG TYPE

the type of SETU_P2P_FULL_DONE as registered in in network byte order.

Request Full

Description The message is sent by the initiating peer in Expect SE state to the receiving peer, if the operation mode "" is determined to be the superior and that it is the better choice that the other peer sends his elements first. The initiating peer changes after sending the message into Full Receiving state. The receiving peer receives the message in the Expecting IBF, afterwards the receiving peer starts sending his complete set in messages to the initiating peer.

Structure

where:

MSG SIZE

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.

MSG TYPE

is SETU_P2P_REQUEST_FULL as registered in in network byte order.

REMOTE SET DIFF

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.

REMOTE SET SIZE

is a 32-bit unsigned integer in network byte order, which represents the total remote (from the perspective of the sending peer) set size.

LOCAL SET DIFF

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.

Send Full

Description The message is sent by the initiating peer in Expect SE state to the receiving peer if the operation mode "" is determined as superior and that it is the better choice that the peer sends his elements first. The initiating peer changes after sending the message into Full Sending state. The receiving peer receives the message in the Expecting IBF state, afterwards the receiving peer changes into Full Receiving state and expects to receive the set of the remote peer.

Structure

where:

MSG SIZE

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.

MSG TYPE

is SETU_P2P_REQUEST_FULL as registered in in network byte order.

REMOTE SET DIFF

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.

REMOTE SET SIZE

is a 32-bit unsigned integer in network byte order, which represents the total remote (from the perspective of the sending peer) set size.

LOCAL SET DIFF

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.

Strata Estimator

Description The strata estimator is sent by the receiving peer at the start of the protocol, right after the message has been received. The strata estimator is used to estimate the difference between the two sets as described in section . When the initiating peer receives the strata estimator, the peer decides which to use for the synchronisation. Depending on the size of the set difference and the the initiating peer changes into Full Sending, Full Receiving or Passive Decoding state. The 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 . The IBFs in a strata estimator always have 79 buckets. The reason why can be found in in section 3.4.2.

Structure

where:

MSG SIZE

is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.

MSG TYPE

is SETU_P2P_SE as registered in in network byte order.

SEC

is a 8-bit unsigned integer in networkf 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 .

SETSIZE

is a 64-bit unsigned integer that is defined by the size of the set the SE is

SE-SLICES

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 . 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. The different SEs are built as in detail described in . Simply put, the IBFs in each SE are serialized as described in 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.

Strata Estimator Compressed

Description The Strata Estimator can be compressed with gzip as described in to improve performance. This can be recognized by the different message type number from .

Structure The key difference between the compressed and the uncompressed Strata Estimator is that the SE slices are compressed with gzip () in the compressed SE. But the header remains uncompressed with both. Since the content of the message is the same as the uncompressed Strata Estimator, the details are not repeated here. For details see section .

Full Element

Description The message is the equivalent of the message in the . It contains a complete element that is missing in the set of the peer that receives this message. The message is exclusively sent in the transitions Expecting IBF -> Full Receiving and Full Receiving -> Finished. The message is only received in the Full Sending and Full Receiving state. After the last message has been sent, the message is sent to conclude the full synchronisation of the element sending peer.

Structure

where:

MSG SIZE

is a 16-bit unsigned integer in network byte order, which describes the message size in bytes with the header included.

MSG TYPE

is SETU_P2P_REQUEST_FULL_ELEMENT as registered in in network byte order.

E TYPE

is a 16-bit unsigned integer which defines the element type for the application.

PADDING

is 16-bit always set to zero

E SIZE

is a 16-bit unsigned integer that signals the size of the elements data part.

AE TYPE

is a 16-bit unsigned integer that is needed to identify the type of element that is in the data field

DATA

is a field with variable length that contains the data of the element.

Performance Considerations

Formulas

Operation Mode The decision which 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. 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 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. 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 in section 4.5.2.

cost_local_full_sync) RETURN FULL_SYNC_LOCAL_SENDING_FIRST ELSE RETURN FULL_SYNC_REMOTE_SENDING_FIRST END IF ELSE RETURN DIFFERENTIAL_SYNC END IF ]]>

IBF Size 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). These algorithms are described and justified in more details in 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.

Number of Buckets an Element is Hashed into The number of buckets an element is hashed to is hardcoded to 3. Reasoning and justification can be found in in the IBF parameter performance study in section 4.5.2.

Variable Counter Size The number of bits required to represent the counters of an IBF varies due to different parameters as described in section 3.2 of . 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. 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. 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.

max_counter 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 ) ) # 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 buffer=[] FOR bucket IN ibf[offset] to ibf[count] counter = bucket.counter byte_len = counter_bytes WHILE byte_len + store_bits < 8 bit_to_shift = 0 IF store_bits > 0 OR byte_len > 8 bit_free = 8 - store_bits bit_to_shift = byte_len - bit_free store = store << bit_free END IF buffer[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 buffer[byte_ctr] = store << (8 - store_bits) byte_ctr = byte_ctr + 1 END IF RETURN buffer 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_to_read = pd[byte_ctr] bit_to_pack_left = 8 byte_ctr++ WHILE bit_to_pack_left >= 0 # Prevent packet from reading more than required IF ibf_bucket_ctr > (count - 1) RETURN END IF IF store_bits + bit_to_pack_left >= cbl bit_use = cbl - store_bits IF store_bits > 0 store = store << bit_use END IF bytes_to_shift = bit_to_pack_left - bit_use counter_partial = byte_to_read >> bytes_to_shift store = store | counter_partial ibf.counter[ibf_bucket_ctr + offset] = store byte_to_read = byte_to_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 store = byte_to_read ELSE store = store << bit_to_pack_left store = store | byte_to_read END IF BREAK END IF END WHILE END WHILE END FUNCTION ]]>

Multi Strata Estimators 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:

SEs

Rule

1

b < 68kb

2

b > 68kb

4

b > 269kb

8

b > 1'077kb

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:

> s) | (value << (64 - s)) END FUNCTION ]]>

Performance study and details about the reasoning for the used methods can be found in in section 3.4.1 under the title "Added Support for Multiple Strata Estimators".

Security Considerations 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. 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. It is important to close and purge connections after a given timeout to prevent draining attacks.

General Security Checks In this section general checks are described which should be applied to multiple states.

Input validation 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.

Byzantine Boundaries 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.

UPPER_BOUND RETURN FALSE IF END IF lec + lsd > UPPER_BOUND RETURN FALSE IF END IF rec < LOWER_BOUND RETURN FALSE IF END RETURN TRUE FUNCTION END ]]>

For the byzantine upper bound checks to function correctly, 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 if the strata estimator overestimates the set difference.

Valid State 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.

Message Flow Control 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 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 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.

| OFFER | ===> | DEMAND | ===> | ELEMENT | peers set +---------+ +---------+ +---------+ +---------+ Chain for elements +---------+ +---------+ +---------+ in IBF decoding | OFFER | ---> | DEMAND | ===> | ELEMENT | peers set +---------+ +---------+ +---------+ --->: Answer not mandatory ===>: Always answer needed. ]]>

A possible implementation of the check in pseudocode could look as follows:

This is based on , section 5.3 (Message Control Flow).

Limit Active/Passive Decoding changes 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. 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 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.

Full Synchronisation Plausibility Check 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. In order to calculate this plausibility, section 5.5 of 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. 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. This two approaches are implemented in the following pseudocode:

0 RETURN FALSE END IF # Probabilistic algorithm to check for plausible # element distribution IF FULL_RECEIVING == state # Prevent division by 0 IF 0 <= rs 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 RETURN FALSE END IF END IF RETURN TRUE END FUNCTION ]]>

States 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.

Expecting IBF Security considerations for received messages:

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 .

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 .

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 .

Full Sending Security considerations for received messages:

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 .

When receiving the 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, a resynchronisation is required. The number of possible resynchronisation MUST be limited, to prevent resource exhaustion attacks.

Expecting IBF Last Security considerations for received messages:

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.

Active Decoding In the Active Decoding 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. The simplest solution is to prevent decoding of more than a given number of elements. A more robust solution is to implement a algorithm that detects a loop by analyzing past partially decoded IBFs. This can be achieved by saving the hash of all prior partly decoded IBFs hashes in a hashmap and check for every inserted hash, if it is already in the hashmap. 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 and lower thresholds for the number of decoded elements can be calculated with the peers set sizes and the other peer's committed set sizes from the Expecting IBF state. Security considerations for received messages:

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 messages should always be protected with an to secure the protocol against missing, duplicated, out-of-order or unexpected messages.

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 messages should always be protected with an to secure the protocol against missing, duplicated, out-of-order or unexpected messages.

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 messages should always be protected with an to secure the protocol against missing, duplicated, out-of-order or unexpected messages.

The message is only received if the IBF has finished decoding and all offers have been sent. If the message is received before the decoding of the IBF is finished or all open offers and demands have been answered, the operation MUST be terminated. If the sets differ, a resynchronisation is required. The number of possible resynchronisation MUST be limited to prevent resource exhaustion attacks. When a message is received the "check_if_synchronisation_is_complete()" function from the is required to ensure that all demands have been satisfied successfully.

Finish Closing In the Finish Closing state the protocol waits for all sent demands to be fulfilled. In case not all sent demands have been answered in time, the operation has failed and MUST be terminated. Security considerations for received messages:

When receiving messages it is important to always check the to secure the protocol against missing, duplicated, out-of-order or unexpected messages.

Finished In this state the connection is terminated, so no security considerations are needed.

Expect SE Security considerations for received messages:

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 . In case of compressed strata estimators the unpacking algorithm needs to be protected against unpacking memory corruption (memory overflow).

Full Receiving Security considerations for received messages:

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 .

When the 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. After receiving the message, future elements MUST NOT be accepted. If the sets differ, a resynchronisation is required. The number of possible resynchronisation MUST be limited to prevent resource exhaustion attacks.

Passive Decoding Security considerations for received messages:

In case an message is received by the peer a active/passive role switch is initiated by the active decoding remote peer. In this moment the peer MUST wait for all open offers and demands to be fulfilled, to prevent retransmission before switching into active decoding operation mode. A switch into active decoding mode MUST only be permitted for a predefined number of times as described in

A check needs to be in place that prevents receiving an inquiry for an element multiple times or more inquiries than are plausible. The sending and receiving of messages should always be protected with an to secure the protocol against missing, duplicated, out-of-order or unexpected messages.

Same action as described for message in section .

Same action as described for message in section .

Same action as described for message in section .

Same action as described for message in section .

Finish Waiting In the Finish Waiting state the protocol waits for all transmitted demands to be fulfilled. 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. Security considerations for received messages:

When receiving messages it is important to always check the to secure the protocol against missing, duplicated, out-of-order or unexpected messages.

Constants The following table contains constants used by the protocol. The constants marked with a * are validated through experiments in .

GANA Considerations GANA is requested to amend the "GNUnet Message Type" registry as follows:

Contributors The GNUnet implementation of the byzantine fault tolerant set reconciliation protocol was originally implemented by Florian Dold.

Normative References &RFC5869; &RFC2119; &RFC3385; &RFC1951; University of Applied Sciences Bern GNUnet e.V. Technische Universität München Technische Universität München Inria, Domaine de Voluceau Rocquencourt U.C. Irvine U.C. Irvine U.C. San Diego U.C. San Diego Technische Universitaet Muenchen Technische Universitaet Muenchen Technische Universitaet Muenchen

Test Vectors

Map Function INPUTS:

OUTPUT:

ID Calculation Function INPUTS:

OUTPUT:

Counter Compression Function INPUTS:

OUTPUT: