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BonehGolle-psp2002.pdf (390319B)

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    409 %%EndFont 
    410 %%BeginFont: CMMI5
    411 %!PS-AdobeFont-1.1: CMMI5 1.100
    412 %%CreationDate: 1996 Aug 02 08:21:10
    413 
    414 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
    415 
    416 11 dict begin
    417 /FontInfo 7 dict dup begin
    418 /version (1.100) readonly def
    419 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
    420 /FullName (CMMI5) readonly def
    421 /FamilyName (Computer Modern) readonly def
    422 /Weight (Medium) readonly def
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    424 /isFixedPitch false def
    425 end readonly def
    426 /FontName /CMMI5 def
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    513 %%EndFont 
    514 %%BeginFont: CMEX10
    515 %!PS-AdobeFont-1.1: CMEX10 1.00
    516 %%CreationDate: 1992 Jul 23 21:22:48
    517 
    518 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
    519 
    520 11 dict begin
    521 /FontInfo 7 dict dup begin
    522 /version (1.00) readonly def
    523 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
    524 /FullName (CMEX10) readonly def
    525 /FamilyName (Computer Modern) readonly def
    526 /Weight (Medium) readonly def
    527 /ItalicAngle 0 def
    528 /isFixedPitch false def
    529 end readonly def
    530 /FontName /CMEX10 def
    531 /PaintType 0 def
    532 /FontType 1 def
    533 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
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    548 currentdict end
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    633 %%EndFont 
    634 %%BeginFont: CMR6
    635 %!PS-AdobeFont-1.1: CMR6 1.0
    636 %%CreationDate: 1991 Aug 20 16:39:02
    637 
    638 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
    639 
    640 11 dict begin
    641 /FontInfo 7 dict dup begin
    642 /version (1.0) readonly def
    643 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
    644 /FullName (CMR6) readonly def
    645 /FamilyName (Computer Modern) readonly def
    646 /Weight (Medium) readonly def
    647 /ItalicAngle 0 def
    648 /isFixedPitch false def
    649 end readonly def
    650 /FontName /CMR6 def
    651 /PaintType 0 def
    652 /FontType 1 def
    653 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
    654 /Encoding 256 array
    655 0 1 255 {1 index exch /.notdef put} for
    656 dup 161 /Gamma put
    657 dup 162 /Delta put
    658 dup 164 /Lambda put
    659 dup 181 /grave put
    660 dup 182 /acute put
    661 dup 40 /parenleft put
    662 dup 41 /parenright put
    663 dup 43 /plus put
    664 dup 48 /zero put
    665 dup 49 /one put
    666 dup 50 /two put
    667 dup 51 /three put
    668 dup 52 /four put
    669 dup 53 /five put
    670 dup 54 /six put
    671 dup 56 /eight put
    672 dup 61 /equal put
    673 readonly def
    674 /FontBBox{-20 -250 1193 750}readonly def
    675 /UniqueXX 5000789 def
    676 currentdict end
    677 currentfile eexec
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    789 cleartomark
    790 
    791 %%EndFont 
    792 %%BeginFont: MSBM10
    793 %!PS-AdobeFont-1.1: MSBM10 2.1
    794 %%CreationDate: 1993 Sep 17 11:10:37
    795 
    796 % Math Symbol fonts were designed by the American Mathematical Society.
    797 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
    798 
    799 11 dict begin
    800 /FontInfo 7 dict dup begin
    801 /version (2.1) readonly def
    802 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
    803 /FullName (MSBM10) readonly def
    804 /FamilyName (Euler) readonly def
    805 /Weight (Medium) readonly def
    806 /ItalicAngle 0 def
    807 /isFixedPitch false def
    808 end readonly def
    809 /FontName /MSBM10 def
    810 /PaintType 0 def
    811 /FontType 1 def
    812 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
    813 /Encoding 256 array
    814 0 1 255 {1 index exch /.notdef put} for
    815 dup 161 /lessornotequal put
    816 dup 162 /greaterornotequal put
    817 dup 164 /notgreaterequal put
    818 dup 181 /lessornotsimilar put
    819 dup 182 /greaterornotsimilar put
    820 dup 70 /F put
    821 dup 71 /G put
    822 dup 83 /S put
    823 dup 90 /Z put
    824 readonly def
    825 /FontBBox{-55 -420 2343 920}readonly def
    826 /UniqueXX 5031982 def
    827 currentdict end
    828 currentfile eexec
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    954 %%CreationDate: 1991 Aug 18 17:46:49
    955 
    956 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
    957 
    958 11 dict begin
    959 /FontInfo 7 dict dup begin
    960 /version (1.0) readonly def
    961 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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    963 /FamilyName (Computer Modern) readonly def
    964 /Weight (Medium) readonly def
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    966 /isFixedPitch false def
    967 end readonly def
    968 /FontName /CMCSC10 def
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   1135 cleartomark
   1136 
   1137 %%EndFont 
   1138 %%BeginFont: CMSY9
   1139 %!PS-AdobeFont-1.1: CMSY9 1.0
   1140 %%CreationDate: 1991 Aug 15 07:22:27
   1141 
   1142 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
   1143 
   1144 11 dict begin
   1145 /FontInfo 7 dict dup begin
   1146 /version (1.0) readonly def
   1147 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
   1148 /FullName (CMSY9) readonly def
   1149 /FamilyName (Computer Modern) readonly def
   1150 /Weight (Medium) readonly def
   1151 /ItalicAngle -14.035 def
   1152 /isFixedPitch false def
   1153 end readonly def
   1154 /FontName /CMSY9 def
   1155 /PaintType 0 def
   1156 /FontType 1 def
   1157 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
   1158 /Encoding 256 array
   1159 0 1 255 {1 index exch /.notdef put} for
   1160 dup 161 /minus put
   1161 dup 162 /periodcentered put
   1162 dup 164 /asteriskmath put
   1163 dup 169 /circleplus put
   1164 dup 178 /bullet put
   1165 dup 181 /reflexsubset put
   1166 dup 182 /reflexsuperset put
   1167 dup 183 /lessequal put
   1168 dup 184 /greaterequal put
   1169 dup 188 /approxequal put
   1170 dup 189 /propersubset put
   1171 dup 33 /arrowright put
   1172 dup 50 /element put
   1173 dup 54 /negationslash put
   1174 dup 65 /A put
   1175 dup 91 /union put
   1176 dup 92 /intersection put
   1177 dup 102 /braceleft put
   1178 dup 103 /braceright put
   1179 dup 106 /bar put
   1180 readonly def
   1181 /FontBBox{-30 -958 1146 777}readonly def
   1182 /UniqueXX 5000819 def
   1183 currentdict end
   1184 currentfile eexec
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   1291 e55f0c0805dab1179634
   1292 0000000000000000000000000000000000000000000000000000000000000000
   1293 0000000000000000000000000000000000000000000000000000000000000000
   1294 0000000000000000000000000000000000000000000000000000000000000000
   1295 0000000000000000000000000000000000000000000000000000000000000000
   1296 0000000000000000000000000000000000000000000000000000000000000000
   1297 0000000000000000000000000000000000000000000000000000000000000000
   1298 0000000000000000000000000000000000000000000000000000000000000000
   1299 0000000000000000000000000000000000000000000000000000000000000000
   1300 cleartomark
   1301 
   1302 %%EndFont 
   1303 %%BeginFont: CMSY5
   1304 %!PS-AdobeFont-1.1: CMSY5 1.0
   1305 %%CreationDate: 1991 Aug 15 07:21:16
   1306 
   1307 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
   1308 
   1309 11 dict begin
   1310 /FontInfo 7 dict dup begin
   1311 /version (1.0) readonly def
   1312 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
   1313 /FullName (CMSY5) readonly def
   1314 /FamilyName (Computer Modern) readonly def
   1315 /Weight (Medium) readonly def
   1316 /ItalicAngle -14.035 def
   1317 /isFixedPitch false def
   1318 end readonly def
   1319 /FontName /CMSY5 def
   1320 /PaintType 0 def
   1321 /FontType 1 def
   1322 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
   1323 /Encoding 256 array
   1324 0 1 255 {1 index exch /.notdef put} for
   1325 dup 161 /minus put
   1326 dup 162 /periodcentered put
   1327 dup 164 /asteriskmath put
   1328 dup 169 /circleplus put
   1329 dup 178 /bullet put
   1330 dup 181 /reflexsubset put
   1331 dup 182 /reflexsuperset put
   1332 dup 183 /lessequal put
   1333 dup 184 /greaterequal put
   1334 dup 188 /approxequal put
   1335 dup 189 /propersubset put
   1336 dup 48 /prime put
   1337 readonly def
   1338 /FontBBox{21 -944 1448 791}readonly def
   1339 /UniqueXX 5000815 def
   1340 currentdict end
   1341 currentfile eexec
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   1421 
   1422 %%EndFont 
   1423 %%BeginFont: CMMI6
   1424 %!PS-AdobeFont-1.1: CMMI6 1.100
   1425 %%CreationDate: 1996 Jul 23 07:53:52
   1426 
   1427 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
   1428 
   1429 11 dict begin
   1430 /FontInfo 7 dict dup begin
   1431 /version (1.100) readonly def
   1432 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
   1433 /FullName (CMMI6) readonly def
   1434 /FamilyName (Computer Modern) readonly def
   1435 /Weight (Medium) readonly def
   1436 /ItalicAngle -14.04 def
   1437 /isFixedPitch false def
   1438 end readonly def
   1439 /FontName /CMMI6 def
   1440 /PaintType 0 def
   1441 /FontType 1 def
   1442 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
   1443 /Encoding 256 array
   1444 0 1 255 {1 index exch /.notdef put} for
   1445 dup 161 /Gamma put
   1446 dup 162 /Delta put
   1447 dup 164 /Lambda put
   1448 dup 169 /Phi put
   1449 dup 174 /alpha put
   1450 dup 178 /epsilon1 put
   1451 dup 181 /theta put
   1452 dup 182 /iota put
   1453 dup 183 /kappa put
   1454 dup 184 /lambda put
   1455 dup 188 /pi put
   1456 dup 189 /rho put
   1457 dup 39 /phi1 put
   1458 dup 59 /comma put
   1459 dup 61 /slash put
   1460 dup 70 /F put
   1461 dup 83 /S put
   1462 dup 96 /lscript put
   1463 dup 97 /a put
   1464 dup 98 /b put
   1465 dup 99 /c put
   1466 dup 101 /e put
   1467 dup 103 /g put
   1468 dup 104 /h put
   1469 dup 105 /i put
   1470 dup 106 /j put
   1471 dup 107 /k put
   1472 dup 110 /n put
   1473 dup 112 /p put
   1474 dup 113 /q put
   1475 dup 114 /r put
   1476 dup 115 /s put
   1477 dup 118 /v put
   1478 dup 119 /w put
   1479 dup 120 /x put
   1480 dup 121 /y put
   1481 dup 122 /z put
   1482 readonly def
   1483 /FontBBox{11 -250 1241 750}readonly def
   1484 /UniqueXX 5087381 def
   1485 currentdict end
   1486 currentfile eexec
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   1753 cleartomark
   1754 
   1755 %%EndFont 
   1756 %%BeginFont: CMMI9
   1757 %!PS-AdobeFont-1.1: CMMI9 1.100
   1758 %%CreationDate: 1996 Jul 23 07:53:55
   1759 
   1760 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
   1761 
   1762 11 dict begin
   1763 /FontInfo 7 dict dup begin
   1764 /version (1.100) readonly def
   1765 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
   1766 /FullName (CMMI9) readonly def
   1767 /FamilyName (Computer Modern) readonly def
   1768 /Weight (Medium) readonly def
   1769 /ItalicAngle -14.04 def
   1770 /isFixedPitch false def
   1771 end readonly def
   1772 /FontName /CMMI9 def
   1773 /PaintType 0 def
   1774 /FontType 1 def
   1775 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
   1776 /Encoding 256 array
   1777 0 1 255 {1 index exch /.notdef put} for
   1778 dup 161 /Gamma put
   1779 dup 162 /Delta put
   1780 dup 164 /Lambda put
   1781 dup 169 /Phi put
   1782 dup 174 /alpha put
   1783 dup 178 /epsilon1 put
   1784 dup 181 /theta put
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   1791 dup 39 /phi1 put
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   1793 dup 59 /comma put
   1794 dup 60 /less put
   1795 dup 61 /slash put
   1796 dup 62 /greater put
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   1799 dup 67 /C put
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   1801 dup 71 /G put
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   1803 dup 77 /M put
   1804 dup 79 /O put
   1805 dup 80 /P put
   1806 dup 82 /R put
   1807 dup 83 /S put
   1808 dup 84 /T put
   1809 dup 85 /U put
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   1838 readonly def
   1839 /FontBBox{-29 -250 1075 750}readonly def
   1840 /UniqueXX 5087384 def
   1841 currentdict end
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   2255 cleartomark
   2256 
   2257 %%EndFont 
   2258 %%BeginFont: CMTI9
   2259 %!PS-AdobeFont-1.1: CMTI9 1.0
   2260 %%CreationDate: 1991 Aug 18 21:08:07
   2261 
   2262 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
   2263 
   2264 11 dict begin
   2265 /FontInfo 7 dict dup begin
   2266 /version (1.0) readonly def
   2267 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
   2268 /FullName (CMTI9) readonly def
   2269 /FamilyName (Computer Modern) readonly def
   2270 /Weight (Medium) readonly def
   2271 /ItalicAngle -14.04 def
   2272 /isFixedPitch false def
   2273 end readonly def
   2274 /FontName /CMTI9 def
   2275 /PaintType 0 def
   2276 /FontType 1 def
   2277 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
   2278 /Encoding 256 array
   2279 0 1 255 {1 index exch /.notdef put} for
   2280 dup 161 /Gamma put
   2281 dup 162 /Delta put
   2282 dup 164 /Lambda put
   2283 dup 169 /Phi put
   2284 dup 174 /ff put
   2285 dup 175 /fi put
   2286 dup 178 /ffl put
   2287 dup 181 /grave put
   2288 dup 182 /acute put
   2289 dup 183 /caron put
   2290 dup 184 /breve put
   2291 dup 188 /germandbls put
   2292 dup 189 /ae put
   2293 dup 194 /Oslash put
   2294 dup 39 /quoteright put
   2295 dup 40 /parenleft put
   2296 dup 41 /parenright put
   2297 dup 44 /comma put
   2298 dup 45 /hyphen put
   2299 dup 46 /period put
   2300 dup 48 /zero put
   2301 dup 49 /one put
   2302 dup 50 /two put
   2303 dup 51 /three put
   2304 dup 52 /four put
   2305 dup 53 /five put
   2306 dup 54 /six put
   2307 dup 55 /seven put
   2308 dup 56 /eight put
   2309 dup 57 /nine put
   2310 dup 58 /colon put
   2311 dup 65 /A put
   2312 dup 67 /C put
   2313 dup 68 /D put
   2314 dup 69 /E put
   2315 dup 71 /G put
   2316 dup 73 /I put
   2317 dup 75 /K put
   2318 dup 76 /L put
   2319 dup 77 /M put
   2320 dup 78 /N put
   2321 dup 79 /O put
   2322 dup 80 /P put
   2323 dup 82 /R put
   2324 dup 83 /S put
   2325 dup 84 /T put
   2326 dup 85 /U put
   2327 dup 87 /W put
   2328 dup 88 /X put
   2329 dup 91 /bracketleft put
   2330 dup 93 /bracketright put
   2331 dup 97 /a put
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   2334 dup 100 /d put
   2335 dup 101 /e put
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   2338 dup 104 /h put
   2339 dup 105 /i put
   2340 dup 106 /j put
   2341 dup 107 /k put
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   2343 dup 109 /m put
   2344 dup 110 /n put
   2345 dup 111 /o put
   2346 dup 112 /p put
   2347 dup 113 /q put
   2348 dup 114 /r put
   2349 dup 115 /s put
   2350 dup 116 /t put
   2351 dup 117 /u put
   2352 dup 118 /v put
   2353 dup 119 /w put
   2354 dup 120 /x put
   2355 dup 121 /y put
   2356 dup 122 /z put
   2357 readonly def
   2358 /FontBBox{-35 -250 1148 750}readonly def
   2359 /UniqueXX 5000827 def
   2360 currentdict end
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   2876 cleartomark
   2877 
   2878 %%EndFont 
   2879 %%BeginFont: CMBX9
   2880 %!PS-AdobeFont-1.1: CMBX9 1.0
   2881 %%CreationDate: 1991 Aug 20 16:36:25
   2882 
   2883 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
   2884 
   2885 11 dict begin
   2886 /FontInfo 7 dict dup begin
   2887 /version (1.0) readonly def
   2888 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
   2889 /FullName (CMBX9) readonly def
   2890 /FamilyName (Computer Modern) readonly def
   2891 /Weight (Bold) readonly def
   2892 /ItalicAngle 0 def
   2893 /isFixedPitch false def
   2894 end readonly def
   2895 /FontName /CMBX9 def
   2896 /PaintType 0 def
   2897 /FontType 1 def
   2898 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
   2899 /Encoding 256 array
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   2903 dup 164 /Lambda put
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   2956 dup 116 /t put
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   2958 dup 118 /v put
   2959 dup 120 /x put
   2960 dup 121 /y put
   2961 dup 122 /z put
   2962 readonly def
   2963 /FontBBox{-58 -250 1195 750}readonly def
   2964 /UniqueXX 5000767 def
   2965 currentdict end
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   3302 
   3303 %%EndFont 
   3304 %%BeginFont: CMR9
   3305 %!PS-AdobeFont-1.1: CMR9 1.0
   3306 %%CreationDate: 1991 Aug 20 16:39:59
   3307 
   3308 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
   3309 
   3310 11 dict begin
   3311 /FontInfo 7 dict dup begin
   3312 /version (1.0) readonly def
   3313 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
   3314 /FullName (CMR9) readonly def
   3315 /FamilyName (Computer Modern) readonly def
   3316 /Weight (Medium) readonly def
   3317 /ItalicAngle 0 def
   3318 /isFixedPitch false def
   3319 end readonly def
   3320 /FontName /CMR9 def
   3321 /PaintType 0 def
   3322 /FontType 1 def
   3323 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
   3324 /Encoding 256 array
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   3422 readonly def
   3423 /FontBBox{-39 -250 1036 750}readonly def
   3424 /UniqueXX 5000792 def
   3425 currentdict end
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   3939 
   3940 %%EndFont 
   3941 %%BeginFont: CMSY6
   3942 %!PS-AdobeFont-1.1: CMSY6 1.0
   3943 %%CreationDate: 1991 Aug 15 07:21:34
   3944 
   3945 % Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
   3946 
   3947 11 dict begin
   3948 /FontInfo 7 dict dup begin
   3949 /version (1.0) readonly def
   3950 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
   3951 /FullName (CMSY6) readonly def
   3952 /FamilyName (Computer Modern) readonly def
   3953 /Weight (Medium) readonly def
   3954 /ItalicAngle -14.035 def
   3955 /isFixedPitch false def
   3956 end readonly def
   3957 /FontName /CMSY6 def
   3958 /PaintType 0 def
   3959 /FontType 1 def
   3960 /FontMatrix [0.001 0 0 0.001 0 0] readonly def
   3961 /Encoding 256 array
   3962 0 1 255 {1 index exch /.notdef put} for
   3963 dup 161 /minus put
   3964 dup 162 /periodcentered put
   3965 dup 164 /asteriskmath put
   3966 dup 169 /circleplus put
   3967 dup 174 /circledivide put
   3968 dup 175 /circledot put
   3969 dup 176 /circlecopyrt put
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   3978 dup 188 /approxequal put
   3979 dup 189 /propersubset put
   3980 dup 194 /follows put
   3981 dup 48 /prime put
   3982 dup 50 /element put
   3983 dup 121 /dagger put
   3984 dup 196 /spade put
   3985 readonly def
   3986 /FontBBox{-4 -948 1329 786}readonly def
   3987 /UniqueXX 5000816 def
   3988 currentdict end
   3989 currentfile eexec
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   4324 b(correctness)h(in)27201 42754 y(real)348 b(time)h(when)e(the)h(n)-28
   4325 b(um)g(b)28 b(er)346 b(of)h(inputs)h(is)g(large.)473
   4326 b(F)-85 b(or)347 b(instance,)27201 43916 y(w)-28 b(e)357
   4327 b(estimate)h(that)f(to)g(pro)-28 b(v)g(e)357 b(that)g(one)g(million)h
   4328 (v)-28 b(otes)358 b(ha)-28 b(v)g(e)356 b(b)28 b(een)27201
   4329 45079 y(mixed)268 b(correctly)e(w)-28 b(ould)265 b(require)h(some)h(20)
   4330 e(hours)g(on)h(a)g(1GHz)g(PC.)27201 46241 y(In)280 b(e\256ect,)294
   4331 b(existing)281 b(mix)h(serv)-28 b(ers)279 b(are)h(to)28
   4332 b(o)281 b(slo)-28 b(w)279 b(to)h(mix)h(non-trivial)27201
   4333 47403 y(n)-28 b(um)g(b)28 b(ers)341 b(of)g(inputs)g(in)h(real)g(time.)
   4334 28198 48566 y(In)421 b(this)h(pap)28 b(er,)443 b(w)-28
   4335 b(e)421 b(prop)28 b(ose)422 b(an)g(exceptionally)i(e\261cien)-28
   4336 b(t)424 b(new)27201 49728 y(metho)28 b(d)314 b(for)d(pro)-28
   4337 b(ving)312 b(that)g(the)h(output)f(pro)28 b(duced)312
   4338 b(b)-28 b(y)312 b(a)g(mix)i(net-)27201 50890 y(w)-28
   4339 b(ork)356 b(is)h(almost)g(en)-28 b(tirely)358 b(correct.)501
   4340 b(The)356 b(real)h(cost)f(of)h(generating)27201 52053
   4341 y(a)382 b(pro)28 b(of)382 b(of)f(almost)i(en)-28 b(tirely)383
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   4350 b(cor-)27201 56702 y(rect)403 b(mixing.)639 b(In)401
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   4356 60189 y(w)-28 b(ork)334 b(pro)-28 b(v)-57 b(ably)334
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   4359 b(probabilit)-28 b(y)-85 b(.)643 b(F)-85 b(or)403 b(example,)421
   4360 b(a)404 b(t)-28 b(ypical)404 b(ap-)27201 62513 y(plication)360
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   4364 b(v)g(olving)420 b(one)f(million)i(ballots)f(is)27201
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   4375 b(tirely)243 b(correct.)421 b(This)241 b(new)g(tec)-28
   4376 b(hnique)242 b(consists)p eop
   4377 %%Page: 2 2
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   4382 b(the)g(subset)g Fj(S)394 b Fo(to)-2022 1409 y(the)g(mix)h(serv)-28
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   4384 b(a)g(subset)g Fj(S)23028 986 y Fr(0)23720 1409 y Fo(of)-2022
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   4386 2726 y Fi(S)8794 2535 y Fh(0)9433 2571 y Fo(=)285 b Fj(\274)11100
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   4392 b(hes)251 b(mix)h(inputs)f(to)g(mix)h(outputs.)425 b(On)251
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   4394 b(w)313 b(in)g(sections)i(6)e(and)h(7)f(that)h(the)g(problem)g(of)g
   4395 (\257nding)-2022 7221 y Fj(S)-1339 6797 y Fr(0)-641 7221
   4396 y Fo(suc)-28 b(h)401 b(that)h Fj(\274)4594 7376 y Fi(S)5132
   4397 7185 y Fh(0)5871 7221 y Fo(=)385 b Fj(\274)7638 7332
   4398 y Fi(S)8633 7221 y Fo(b)28 b(ecomes)402 b(often)g(imp)28
   4399 b(ossible)403 b(if)f(the)g(set)-2022 8383 y(of)318 b(outputs)h(pro)28
   4400 b(duced)318 b(b)-28 b(y)319 b(the)g(mix)h(serv)-28 b(er)319
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   4406 13032 y(where)287 b(it)i(ma)-28 b(y)288 b(b)28 b(e)288
   4407 b(used)g(to)g(guaran)-28 b(tee)288 b(almost)g(instan)-28
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   4410 b(tirely)305 b(correct.)443 b(This)-2022 15357 y(guaran)-28
   4411 b(tee)283 b(will)h(often)f(b)28 b(e)284 b(enough)f(to)h(announce)f(the)
   4412 h(result)f(of)g(the)-2022 16519 y(election)396 b(instan)-28
   4413 b(tly)-85 b(.)615 b(W)-85 b(e)394 b(can)g(use)h(in)g(parallel)g(to)f
   4414 (our)g(pro)28 b(of)394 b(an)-28 b(y)-2022 17681 y(one)280
   4415 b(of)g(a)g(n)-28 b(um)g(b)28 b(er)280 b(of)g(slo)-28
   4416 b(w)g(er)279 b(pro)28 b(ofs)280 b(of)g(p)28 b(erfect-correctness)281
   4417 b(with-)-2022 18844 y(out)354 b(ha)-28 b(ving)354 b(to)h(mix)g(the)g
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   4420 b(e)451 b(in)-28 b(v)g(olv)g(ed)452 b(again\).)784 b(What)451
   4421 b(mak)-28 b(es)451 b(this)-2022 21168 y(p)28 b(ossible)446
   4422 b(is)f(that)g(our)f(pro)28 b(of)444 b(w)-28 b(orks)444
   4423 b(with)g(the)i(fastest)f(and)f(also)-2022 22331 y(most)458
   4424 b(common)h(implemen)-28 b(tation)460 b(of)f(mix)g(net)-28
   4425 b(w)g(orks:)688 b(ElGamal)-2022 23493 y(re-encryption)342
   4426 b(mix)g(nets)g(\(see)g(section)g(2.1\).)-1026 24655 y(The)375
   4427 b(rest)h(of)g(this)g(pap)28 b(er)376 b(is)g(organized)g(as)f(follo)-28
   4428 b(ws.)559 b(In)375 b(section)-2022 25818 y(2,)442 b(w)-28
   4429 b(e)422 b(de\257ne)h(mix)g(net)-28 b(w)g(orks)421 b(and)h(their)g(prop)
   4430 28 b(erties,)443 b(as)422 b(w)-28 b(ell)423 b(as)-2022
   4431 26980 y(re-encryption)505 b(mix)h(nets.)947 b(In)504
   4432 b(section)i(3,)546 b(w)-28 b(e)504 b(surv)-28 b(ey)505
   4433 b(existing)-2022 28142 y(tec)-28 b(hniques)264 b(for)f(pro)-28
   4434 b(ving)263 b(the)h(correct)g(execution)h(of)f(mix)g(nets.)430
   4435 b(W)-85 b(e)-2022 29305 y(in)-28 b(tro)28 b(duce)280
   4436 b(our)g(mec)-28 b(hanism)281 b(for)f(pro)-28 b(ving)280
   4437 b(almost)h(en)-28 b(tirely)282 b(correct)-2022 30467
   4438 y(mixing)459 b(in)f(section)h(4.)805 b(In)457 b(section)i(5,)488
   4439 b(w)-28 b(e)457 b(prop)28 b(ose)458 b(a)g(new)g(mix)-2022
   4440 31629 y(net)-28 b(w)g(ork)296 b(proto)28 b(col)298 b(based)f(on)g(a)h
   4441 (pro)28 b(of)297 b(of)g(almost)h(en)-28 b(tirely)299
   4442 b(correct)-2022 32791 y(mixing)280 b(and)f(examine)i(the)f(prop)28
   4443 b(erties)280 b(of)f(this)h(mix)g(net)g(in)g(section)-2022
   4444 33954 y(6.)446 b(In)313 b(section)h(7,)320 b(w)-28 b(e)313
   4445 b(pro)-28 b(v)g(e)313 b(that)h(our)e(mix)j(net)f(is)g(almost)g(en)-28
   4446 b(tirely)-2022 35116 y(correct.)455 b(W)-85 b(e)342 b(conclude)f(in)h
   4447 (section)g(8.)-2022 37717 y Fp(2.)1328 b(MIX)332 b(DEFINITIONS)f(AND)i
   4448 (PR)-40 b(OPER)-53 b(TIES)-1026 39156 y Fo(A)343 b Fm(mix)368
   4449 b(server)453 b Fo(tak)-28 b(es)343 b(as)h(input)f(a)g(set)h(of)g
   4450 (ciphertexts)g(and)f(out-)-2022 40319 y(puts)354 b(in)g(a)h(random)f
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   4454 y(decrypts)447 b(to)g(the)h(same)f(plain)-28 b(text.)775
   4455 b(W)-85 b(e)447 b(describ)28 b(e)448 b(later)g(in)f(this)-2022
   4456 43806 y(section)366 b(the)g(encryption)g(sc)-28 b(heme)367
   4457 b(used)e(b)-28 b(y)366 b(mix)h(serv)-28 b(ers)365 b(and)g(the)-2022
   4458 44968 y(re-encryption)370 b(pro)28 b(cess.)540 b(The)369
   4459 b(main)h(prop)28 b(ert)-28 b(y)369 b(desired)h(of)f(a)h(mix)-2022
   4460 46130 y(serv)-28 b(er)433 b(is)i(that)f(the)g(p)28 b(erm)-28
   4461 b(utation)435 b(that)f(matc)-28 b(hes)434 b(input)g(cipher-)-2022
   4462 47293 y(texts)412 b(to)g(output)g(re-encryptions)g(should)f(b)28
   4463 b(e)413 b(kno)-28 b(wn)410 b(to)i(no)f(one)-2022 48455
   4464 y(but)341 b(the)h(mix)g(serv)-28 b(er)341 b(itself.)-1026
   4465 49617 y(Mixing)353 b(t)-28 b(ypically)353 b(in)-28 b(v)g(olv)g(es)353
   4466 b(not)e(one)h(but)g(sev)-28 b(eral)352 b(mix)h(serv)-28
   4467 b(ers)-2022 50780 y(that)320 b(op)28 b(erate)321 b(sequen)-28
   4468 b(tially)323 b(on)d(the)g(same)h(data.)449 b(This)320
   4469 b(is)g(called)i(a)-2022 51942 y Fm(mix)415 b(network)503
   4470 b Fo(or)395 b(mix)i(net.)619 b(Consider)395 b(a)g(mix)i(net)f(that)f
   4471 (consists)-2022 53104 y(of)319 b Fj(n)f Fo(serv)-28 b(ers.)448
   4472 b(Users)319 b(submit)h(encrypted)f(inputs)h(to)f(the)h(mix)g(net-)-2022
   4473 54267 y(w)-28 b(ork.)434 b(These)280 b(inputs)h(are)f(mixed)i(in)f(a)f
   4474 (random)g(order)g(b)-28 b(y)280 b(the)h(\257rst)-2022
   4475 55429 y(mix)362 b(serv)-28 b(er.)516 b(The)361 b(output)g(of)g(this)h
   4476 (\257rst)f(mix)h(serv)-28 b(er)361 b(is)h(passed)f(to)-2022
   4477 56591 y(the)309 b(second)g(mix)g(serv)-28 b(er,)316 b(whic)-28
   4478 b(h)308 b(mixes)i(it)f(again)g(and)g(passes)f(it)h(to)-2022
   4479 57753 y(the)303 b(third)g(mix)h(serv)-28 b(er)303 b(whic)-28
   4480 b(h)303 b(mixes)h(it)g(y)-28 b(et)304 b(again,)311 b(and)303
   4481 b(so)g(on)g(un-)-2022 58916 y(til)313 b(the)f(output)g(of)g(the)h
   4482 Fj(n)p Fo(-th)f(serv)-28 b(er)312 b(b)28 b(ecomes)313
   4483 b(the)g(\257nal)f(output)g(of)-2022 60078 y(the)385 b(mix)h(net.)586
   4484 b(The)385 b(relationship)g(b)28 b(et)-28 b(w)g(een)386
   4485 b(inputs)e(and)h(outputs)-2022 61240 y(of)432 b(a)g(mix)i(net)f(is)f
   4486 (not)h(kno)-28 b(wn)431 b(to)i(an)-28 b(y)432 b(single)i(serv)-28
   4487 b(er,)455 b(but)432 b(could)-2022 62403 y(only)342 b(b)28
   4488 b(e)342 b(learn)-28 b(t)341 b(b)-28 b(y)341 b(a)g(coalition)i(of)e(all)
   4489 h(the)g(serv)-28 b(ers.)-1026 63565 y(As)495 b(is)h(customary)-85
   4490 b(,)534 b(w)-28 b(e)495 b(mo)28 b(del)497 b(all)f(comm)-28
   4491 b(unication)496 b(b)28 b(et)-28 b(w)g(een)-2022 64727
   4492 y(users)443 b(and)h(mix)h(serv)-28 b(ers,)469 b(as)444
   4493 b(w)-28 b(ell)444 b(as)g(among)g(mix)h(serv)-28 b(ers)444
   4494 b(with)-2022 65890 y(a)480 b Fm(bul)52 b(letin)495 b(b)-52
   4495 b(o)g(ar)g(d)p Fo(.)873 b(The)480 b(bulletin)i(b)28 b(oard)481
   4496 b(is)g(a)f(publicly)i(shared)-2022 67052 y(piece)374
   4497 b(of)f(memory)i(to)e(whic)-28 b(h)373 b(all)h(participan)-28
   4498 b(ts)373 b(ha)-28 b(v)g(e)373 b(read)g(access)-2022 68214
   4499 y(and)327 b(app)28 b(endiv)-28 b(e,)332 b(sequen)-28
   4500 b(tial)330 b(write)e(access)g(with)g(authen)-28 b(tication.)-2022
   4501 69377 y(W)-85 b(e)356 b(further)f(assume)h(that)g(all)h(participan)-28
   4502 b(ts)356 b(\(users)f(and)h(serv)-28 b(ers\))-2022 70539
   4503 y(ha)g(v)g(e)472 b(p)28 b(olynomially)476 b(b)28 b(ounded)474
   4504 b(computational)g(resources.)850 b(W)-85 b(e)-2022 71701
   4505 y(consider)338 b(an)g(adv)-28 b(ersary)337 b(that)i(ma)-28
   4506 b(y)338 b(statically)i(con)-28 b(trol)338 b(an)-28 b(y)338
   4507 b(n)-28 b(um-)27201 -915 y(b)28 b(er)388 b(of)f(users)g(and)g(up)g(to)g
   4508 (all)i(the)e(mix)i(serv)-28 b(ers)387 b(min)-28 b(us)387
   4509 b(one.)594 b(\(W)-85 b(e)27201 247 y(re\257ne)283 b(and)f(justify)g
   4510 (this)h(adv)-28 b(ersarial)282 b(mo)28 b(del)284 b(in)e(section)h(3\).)
   4511 436 b(Giv)-28 b(en)27201 1409 y(these)299 b(assumptions,)307
   4512 b(w)-28 b(e)297 b(require)h(the)g(follo)-28 b(wing)298
   4513 b(securit)-28 b(y)297 b(prop)28 b(er-)27201 2571 y(ties)343
   4514 b(of)e(a)g(mix)h(net)-28 b(w)g(ork:)28682 4557 y Fg(\262)498
   4515 b Fn(Correctness)p Fo(:)408 b(the)242 b(set)g(of)g(ciphertexts)h
   4516 (output)f(b)-28 b(y)242 b(the)g(mix)29692 5719 y(net)-28
   4517 b(w)g(ork)489 b(m)-28 b(ust)490 b(\\matc)-28 b(h")490
   4518 b(the)h(set)g(of)f(input)h(ciphertexts.)29692 6882 y(This)330
   4519 b(means)h(that)g(ev)-28 b(ery)331 b(output)g(is)f(a)h(re-encryption)g
   4520 (of)g(an)29692 8044 y(input,)549 b(and)506 b(no)h(t)-28
   4521 b(w)g(o)505 b(outputs)i(are)g(re-encryption)g(of)g(the)29692
   4522 9206 y(same)342 b(input.)28682 11219 y Fg(\262)498 b
   4523 Fn(Priv)-65 b(acy)p Fo(:)571 b(no)399 b(adv)-28 b(ersary)398
   4524 b(can)g(matc)-28 b(h)399 b(an)-28 b(y)399 b(output)f(of)h(the)29692
   4525 12381 y(mix)427 b(net)-28 b(w)g(ork)426 b(to)g(the)h(corresp)28
   4526 b(onding)426 b(input)h(with)f(proba-)29692 13543 y(bilit)-28
   4527 b(y)272 b(b)28 b(etter)273 b(than)e(1)p Fj(=n)f Fo(where)h
   4528 Fj(n)g Fo(is)h(the)f(n)-28 b(um)g(b)28 b(er)271 b(of)g(inputs.)29692
   4529 14705 y(Our)297 b(mixnet)j(guaran)-28 b(tees)297 b(a)g(sligh)-28
   4530 b(tly)299 b(w)-28 b(eak)g(er)297 b(notion)h(of)f(pri-)29692
   4531 15868 y(v)-57 b(acy)342 b(as)f(explained)i(in)f(section)g(6.)28682
   4532 17880 y Fg(\262)498 b Fn(Robustness)p Fo(:)415 b(the)258
   4533 b(mix)h(net)-28 b(w)g(ork)256 b(m)-28 b(ust)257 b(pro)28
   4534 b(duce)258 b(a)g(correct)29692 19042 y(output)299 b(irresp)28
   4535 b(ectiv)-28 b(e)302 b(of)d(p)28 b(ossible)301 b(serv)-28
   4536 b(er)299 b(faults)h(or)f(failures.)28682 21054 y Fg(\262)498
   4537 b Fn(Univ)-33 b(ersal)340 b(v)-33 b(eri\257abilit)g(y)p
   4538 Fo(:)436 b(a)293 b(mix)i(net)g(is)f(univ)-28 b(ersally)295
   4539 b(v)-28 b(er-)29692 22217 y(i\257able)263 b(if)f(a)f(coalition)i(of)f
   4540 (all)g(users)g(and)f Fm(al)52 b(l)365 b Fo(mix)262 b(serv)-28
   4541 b(ers)262 b(only)29692 23379 y(ev)-28 b(er)415 b(succeeds)g(in)f(con)
   4542 -28 b(vincing)415 b(an)f(outside)h(v)-28 b(eri\257er)415
   4543 b(of)f(the)29692 24541 y(correct)332 b(execution)i(of)e(a)h(mix)g(net)
   4544 -28 b(w)g(ork)331 b(when)h(the)h(mix)h(net-)29692 25704
   4545 y(w)-28 b(ork)299 b(w)-28 b(as)299 b(indeed)j(executed)g(correctly)-85
   4546 b(.)442 b(Note)301 b(that)f(univ)-28 b(er-)29692 26866
   4547 y(sal)317 b(v)-28 b(eri\257abilit)g(y)319 b(uses)e(a)g(stronger)f(adv)
   4548 -28 b(ersarial)317 b(mo)28 b(del)319 b(than)29692 28028
   4549 y(that)413 b(used)g(to)g(de\257ne)h(priv)-57 b(acy)413
   4550 b(and)g(correctness,)431 b(since)414 b Fm(al)52 b(l)29692
   4551 29190 y Fo(mix)413 b(serv)-28 b(ers)411 b(ma)-28 b(y)412
   4552 b(participate)g(in)g(the)h(coalition)f(to)g(c)-28 b(heat)29692
   4553 30353 y(an)341 b(outside)h(v)-28 b(eri\257er.)28682 32365
   4554 y Fg(\262)498 b Fn(E\261ciency)p Fo(:)431 b(while)292
   4555 b(not)g(a)g(securit)-28 b(y)292 b(prop)28 b(ert)-28 b(y)-85
   4556 b(,)301 b(lo)-28 b(w)291 b(compu-)29692 33527 y(tational)405
   4557 b(o)-28 b(v)g(erhead)403 b(is)h(the)h(holy)f(grail)g(of)g(secure)g(mix)
   4558 h(net-)29692 34689 y(w)-28 b(orks.)27201 36764 y Fp(2.1)1329
   4559 b(Re\255encryption)331 b(Mix)i(Nets)28198 38203 y Fo(In)279
   4560 b(this)h(section,)294 b(w)-28 b(e)279 b(de\257ne)h(re-encryption)h(mix)
   4561 g(net)-28 b(w)g(orks.)434 b(Re-)27201 39365 y(encryption)417
   4562 b(mix)g(nets)f(w)-28 b(ere)416 b(originally)h(prop)28
   4563 b(osed)416 b(in)g([22].)681 b(The)27201 40527 y(particular)383
   4564 b(re-encryption)g(mix)h(of)e([22])h(w)-28 b(as)382 b(brok)-28
   4565 b(en)382 b(in)h([24,)394 b(25],)27201 41690 y(and)263
   4566 b(later)g(\257xed)g(b)-28 b(y)263 b([21].)430 b(A)262
   4567 b(v)-28 b(ery)263 b(large)g(n)-28 b(um)g(b)28 b(er)262
   4568 b(of)h(constructions)27201 42852 y(based)405 b(on)g(re-encryption)h
   4569 (mix)g(nets)f(ha)-28 b(v)g(e)405 b(since)h(b)28 b(een)406
   4570 b(prop)28 b(osed)27201 44014 y(\(w)-28 b(e)341 b(will)h(surv)-28
   4571 b(ey)342 b(them)g(brie\260y)g(in)f(the)h(follo)-28 b(wing)341
   4572 b(section\).)28198 45177 y(Re-encryption)228 b(mix)h(net)-28
   4573 b(w)g(orks)226 b(op)28 b(erate)229 b(in)e(t)-28 b(w)g(o)227
   4574 b(distinct)h(phases.)27201 46339 y(In)347 b(the)h(\257rst)f(phase)h
   4575 (\(the)g(mixing)h(phase\),)g(the)f(input)g(ciphertexts)27201
   4576 47501 y(are)f(sh)-28 b(u\262ed)346 b(and)h(re-encrypted.)472
   4577 b(In)347 b(the)g(second)g(phase)f(\(the)i(de-)27201 48664
   4578 y(cryption)364 b(phase\),)369 b(the)364 b(mixed)g(ciphertexts)g(that)g
   4579 (are)f(the)g(output)27201 49826 y(of)395 b(the)g(\257rst)f(phase)g(are)
   4580 h(decrypted.)615 b(The)395 b(serv)-28 b(ers)394 b(that)g(p)28
   4581 b(erform)27201 50988 y(the)375 b(mixing)g(in)f(the)h(\257rst)e(phase,)
   4582 383 b(and)373 b(the)i(serv)-28 b(ers)373 b(that)h(p)28
   4583 b(erform)27201 52151 y(the)414 b(decryption)g(in)f(the)h(second)f
   4584 (phase)g(need)h(not)f(b)28 b(e)414 b(the)f(same,)27201
   4585 53313 y(although)389 b(they)h Fm(c)-52 b(an)467 b Fo(b)28
   4586 b(e)389 b(the)h(same.)598 b(W)-85 b(e)389 b(treat)g(them)g(separately)
   4587 27201 54475 y(in)420 b(order)f(to)g(de\257ne)h(our)f(adv)-28
   4588 b(ersarial)419 b(mo)28 b(del)421 b(more)e(clearly)-85
   4589 b(.)691 b(W)-85 b(e)27201 55637 y(call)313 b(the)f(\257rst)g(group)f
   4590 Fm(mixing)338 b(servers)396 b Fo(and)311 b(the)h(second)g(group)f
   4591 Fm(de-)27201 56800 y(cryption)365 b(servers)p Fo(.)28308
   4592 59020 y Ff(Definition)554 b Fo(2.1.)498 b Fm(\(A)-52
   4593 b(dversarial)283 b(mo)-52 b(del\))283 b(We)f(c)-52 b(onsider)282
   4594 b(an)g(ad-)27201 60182 y(versary)378 b(that)f(may)f(static)-52
   4595 b(al)52 b(ly)377 b(c)-52 b(ontr)g(ol)377 b(any)f(numb)-52
   4596 b(er)379 b(of)d(users,)381 b(up)27201 61345 y(to)426
   4597 b(al)52 b(l)427 b(the)f(mixing)g(servers)i(minus)f(one,)441
   4598 b(and)425 b(up)h(to)f(a)h(minority)27201 62507 y(of)366
   4599 b(de)-52 b(cryption)365 b(servers.)28198 64727 y Fo(This)256
   4600 b(static)h(adv)-28 b(ersarial)257 b(mo)28 b(del)258 b(is)e(commonly)i
   4601 (considered,)274 b(but)27201 65890 y(it)412 b(will)g(b)28
   4602 b(e)412 b(particularly)f(appropriate)g(for)f(the)i(mix)g(net)f(w)-28
   4603 b(e)411 b(pro-)27201 67052 y(p)28 b(ose.)441 b(Since)297
   4604 b(our)e(mixing)i(is)f(so)g(fast,)305 b(it)297 b(is)f(not)f(practically)
   4605 j(imp)28 b(or-)27201 68214 y(tan)-28 b(t)293 b(to)h(consider)f(dynamic)
   4606 h(adv)-28 b(ersaries.)440 b(The)293 b(same)h(ma)-28 b(y)293
   4607 b(not)h(b)28 b(e)27201 69377 y(true)313 b(for)g(slo)-28
   4608 b(w)g(er)312 b(mixes,)320 b(making)314 b(suc)-28 b(h)312
   4609 b(sc)-28 b(hemes)314 b(p)28 b(ossibly)314 b(w)-28 b(eak)g(er)27201
   4610 70539 y(or)433 b(more)g(complicated.)731 b(As)433 b(w)-28
   4611 b(e)432 b(shall)h(see,)457 b(the)434 b(in)-28 b(v)g(olv)g(emen)g(t)434
   4612 b(of)27201 71701 y(decryption)412 b(serv)-28 b(ers)410
   4613 b(is)h(v)-28 b(ery)411 b(m)-28 b(uc)g(h)410 b(more)h(limited)i(than)e
   4614 (that)g(of)p eop
   4615 %%Page: 3 3
   4616 3 2 bop 948 -1978 49851 45 v 926 -816 45 1163 v 1546
   4617 -1165 a Fo(Sc)-28 b(heme)p 17426 -816 V 21501 w(Computational)342
   4618 b(Cost)p 44387 -816 V 10100 w(T)-85 b(rade-o\256)p 50776
   4619 -816 V 17448 -772 26961 45 v 926 346 45 1163 v 17426
   4620 346 V 18046 -2 a(Re-encryption)p 24925 346 V 6018 w(Pro)28
   4621 b(of)p 38226 346 V 6017 w(Decryption)p 44387 346 V 50776
   4622 346 V 948 391 49851 45 v 926 1553 45 1163 v 1546 1204
   4623 a(Cut)340 b(and)h(Cho)28 b(ose)342 b(ZK)f([26,)h(21])p
   4624 17426 1553 V 6482 w(2)p Fj(n)p 24925 1553 V 5946 w Fo(642)p
   4625 Fj(nk)3389 b Fo(=)285 b Fj(O)28 b Fo(\()p Fj(nk)h Fo(\))p
   4626 38226 1553 V 1554 w(\(2)227 b(+)h(4)p Fj(k)29 b Fo(\))p
   4627 Fj(n)p 44387 1553 V 50776 1553 V 926 2715 V 1546 2367
   4628 a Fo(P)-28 b(airwise)340 b(P)-28 b(erm)g(utations)341
   4629 b([2,)h(16])p 17426 2715 V 5931 w(2)p Fj(n)p 24925 2715
   4630 V 3782 w Fo(7)p Fj(n)170 b Fo(log)185 b Fj(n)p Fo(\(2)p
   4631 Fj(k)257 b Fg(\241)228 b Fo(1\))1195 b(=)285 b Fj(O)28
   4632 b Fo(\()p Fj(nk)h Fo(\))p 38226 2715 V 1554 w(\(2)227
   4633 b(+)h(4)p Fj(k)29 b Fo(\))p Fj(n)p 44387 2715 V 50776
   4634 2715 V 926 3878 V 1546 3529 a Fo(Matrix)342 b(Represen)-28
   4635 b(tation)341 b([9])p 17426 3878 V 7544 w(2)p Fj(n)p 24925
   4636 3878 V 4666 w Fo(18)p Fj(n)p Fo(\(2)p Fj(k)257 b Fg(\241)228
   4637 b Fo(1\))2080 b(=)285 b Fj(O)28 b Fo(\()p Fj(nk)h Fo(\))p
   4638 38226 3878 V 1554 w(\(2)227 b(+)h(4)p Fj(k)29 b Fo(\))p
   4639 Fj(n)p 44387 3878 V 50776 3878 V 926 5040 V 1546 4691
   4640 a Fo(P)-28 b(olynomial)342 b(Sc)-28 b(heme)342 b([20])p
   4641 17426 5040 V 8464 w(2)p Fj(n)p 24925 5040 V 4922 w Fo(8)p
   4642 Fj(n)p Fo(\(2)p Fj(k)257 b Fg(\241)228 b Fo(1\))2336
   4643 b(=)285 b Fj(O)28 b Fo(\()p Fj(nk)h Fo(\))p 38226 5040
   4644 V 1554 w(\(2)227 b(+)h(4)p Fj(k)29 b Fo(\))p Fj(n)p 44387
   4645 5040 V 50776 5040 V 926 6202 V 1546 5854 a Fo(Randomized)341
   4646 b(P)-28 b(artial)341 b(Chec)-28 b(king)342 b([18])p 17426
   4647 6202 V 3783 w(2)p Fj(n)p 24925 6202 V 4666 w(n=)p Fo(2\(2)p
   4648 Fj(k)257 b Fg(\241)228 b Fo(1\))2080 b(=)285 b Fj(O)28
   4649 b Fo(\()p Fj(nk)h Fo(\))p 38226 6202 V 1554 w(\(2)227
   4650 b(+)h(4)p Fj(k)29 b Fo(\))p Fj(n)p 44387 6202 V 2463
   4651 w Fo(Priv)-57 b(acy)p 50776 6202 V 926 7365 V 1546 7016
   4652 a(Optimistic)343 b(Mixing)g([11])p 17426 7365 V 8957
   4653 w(6)p Fj(n)p 24925 7365 V 5629 w Fo(6)228 b(+)g(12)p
   4654 Fj(k)3381 b Fo(=)285 b Fj(O)28 b Fo(\()p Fj(k)h Fo(\))p
   4655 38226 7365 V 1607 w(\(5)227 b(+)h(10)p Fj(k)29 b Fo(\))p
   4656 Fj(n)p 44387 7365 V 50776 7365 V 926 8527 V 1546 8178
   4657 a Fo(Pro)f(of-of-Subpro)g(duct)340 b([this)i(pap)28 b(er])p
   4658 17426 8527 V 4470 w(2)p Fj(n)p 24925 8527 V 4903 w Fo(2)p
   4659 Fj(\256)q Fo(\(2)p Fj(k)257 b Fg(\241)228 b Fo(1\))2625
   4660 b(=)285 b Fj(O)28 b Fo(\()p Fj(k)h Fo(\))p 38226 8527
   4661 V 1863 w(\(2)227 b(+)h(4)p Fj(k)29 b Fo(\))p Fj(n)p 44387
   4662 8527 V 1553 w Fo(Correctness)p 50776 8527 V 948 8571
   4663 49851 45 v -2022 11554 a Fn(Figure)412 b(1:)562 b(Real)413
   4664 b(cost)f(p)33 b(er)412 b(serv)-33 b(er)413 b(\(for)g(a)e(total)j(of)e
   4665 Fj(k)441 b Fn(serv)-33 b(ers\))413 b(of)f(mixing)i Fj(n)d
   4666 Fn(items)i(using)g(di\256eren)-33 b(t)412 b(mixes.)582
   4667 b(The)411 b(cost)i(is)-2022 12717 y(measured)434 b(in)g(terms)h(of)f
   4668 (the)g(n)-33 b(um)g(b)33 b(er)434 b(of)g(exp)33 b(onen)-33
   4669 b(tiations.)650 b(Note)434 b(that)g Fj(\256)h Fn(is)f(the)g(securit)-33
   4670 b(y)435 b(parameter)g(of)g(our)f(sc)-33 b(heme)-2022
   4671 13879 y(\(e.g.)525 b Fo(1)285 b Fg(\267)g Fj(\256)g Fg(\267)g
   4672 Fo(5)p Fn(\))-2022 17228 y Fo(mixing)418 b(serv)-28 b(ers.)683
   4673 b(Since)418 b(this)g(mak)-28 b(es)418 b(decryption)g(serv)-28
   4674 b(ers)417 b(easier)-2022 18390 y(to)482 b(recruit)g(than)g(mixing)i
   4675 (serv)-28 b(ers,)518 b(it)482 b(is)h(appropriate)f(that)g(our)-2022
   4676 19552 y(adv)-28 b(ersarial)381 b(mo)28 b(del)383 b(requires)f(a)f(ma)57
   4677 b(jorit)-28 b(y)382 b(of)f(honest)h(decryption)-2022
   4678 20715 y(serv)-28 b(ers)341 b(but)g(only)h(a)f(single)h(honest)g(mixing)
   4679 h(serv)-28 b(er.)-1026 21877 y(F)-85 b(or)227 b(concreteness,)251
   4680 b(w)-28 b(e)227 b(base)h(our)f(presen)-28 b(tation)228
   4681 b(of)f(re-encryption)-2022 23039 y(mix)513 b(nets)g(on)f(an)g(ElGamal)i
   4682 (implemen)-28 b(tation.)970 b(ElGamal)514 b(is)f(a)-2022
   4683 24201 y(probabilistic)448 b(public-k)-28 b(ey)449 b(cryptosystem.)776
   4684 b(The)448 b(public)g(param-)-2022 25364 y(eters)394 b(are)g(a)h(m)-28
   4685 b(ultiplicativ)g(e)396 b(group)e Fe(G)g Fo(of)g(prime)h(order)e
   4686 Fj(q)430 b Fo(\(either)-2022 26526 y(a)469 b(subgroup)f(of)i
   4687 Fe(Z)5491 26103 y Fr(\244)5491 26701 y Fi(p)6443 26526
   4688 y Fo(or)f(an)g(elliptic)j(curv)-28 b(e)469 b(o)-28 b(v)g(er)469
   4689 b Fe(F)18651 26637 y Fi(p)19133 26526 y Fo(\),)503 b(a)469
   4690 b(genera-)-2022 27688 y(tor)384 b Fj(g)421 b Fo(of)385
   4691 b(the)g(group)f Fe(G)p Fo(,)396 b(and)385 b(an)g(elemen)-28
   4692 b(t)386 b Fj(y)393 b Fo(=)358 b Fj(g)17904 27265 y Fi(x)18423
   4693 27688 y Fo(.)586 b(The)385 b(priv)-57 b(ate)-2022 28851
   4694 y(k)-28 b(ey)353 b(consists)h(of)f(the)g(v)-57 b(alue)354
   4695 b Fj(x)p Fo(.)492 b(ElGamal)354 b(is)f(seman)-28 b(tically)355
   4696 b(secure)-2022 30013 y([27])446 b(under)f(the)h(assumption)g(that)f
   4697 (the)h(Decisional)h(Di\261e)g(Hell-)-2022 31175 y(man)394
   4698 b(\(DDH\))h(problem)f(is)h(hard)e(in)i Fe(G)p Fo(.)614
   4699 b(T)-85 b(o)393 b(encrypt)i(a)f(plain)-28 b(text)-2022
   4700 32338 y Fj(m)368 b Fg(2)h Fe(G)p Fo(,)405 b(a)391 b(user)h(c)-28
   4701 b(ho)28 b(oses)392 b(a)g(random)f Fj(r)397 b Fg(2)369
   4702 b Fe(Z)15877 32449 y Fi(q)16727 32338 y Fo(and)391 b(pro)28
   4703 b(duces)392 b(the)-2022 33500 y(follo)-28 b(wing)354
   4704 b(ciphertext:)483 b(\()p Fj(g)8527 33077 y Fi(r)8989
   4705 33500 y Fj(;)171 b(my)10886 33077 y Fi(r)11348 33500
   4706 y Fo(\).)495 b(Note)355 b(that)f(a)g(ciphertext)i(is)f(a)-2022
   4707 34662 y(pair)425 b(of)h(t)-28 b(w)g(o)425 b(elemen)-28
   4708 b(ts)428 b(of)e Fe(G)p Fo(.)710 b(T)-85 b(o)425 b(decrypt)i(a)f
   4709 (ciphertext)h(\()p Fj(a;)172 b(b)p Fo(\),)-2022 35825
   4710 y(a)488 b(user)h(computes)g Fj(a)6542 35401 y Fi(x)7061
   4711 35825 y Fj(=b)g Fo(whic)-28 b(h)488 b(yields)i(the)f(plain)-28
   4712 b(text)490 b Fj(m)f Fo(since)-2022 36987 y(\()p Fj(g)-1100
   4713 36564 y Fi(r)-638 36987 y Fo(\))-240 36564 y Fi(x)563
   4714 36987 y Fo(=)285 b Fj(y)2184 36564 y Fi(r)2646 36987
   4715 y Fo(.)-1026 38149 y(The)556 b(ElGamal)i(cryptosystem)f(mak)-28
   4716 b(es)557 b(re-encryption)h(of)e(ci-)-2022 39311 y(phertexts)496
   4717 b(p)28 b(ossible)496 b(with)f(kno)-28 b(wledge)496 b(of)f(only)h(the)g
   4718 (public)f(pa-)-2022 40474 y(rameters.)891 b(T)-85 b(o)486
   4719 b(re-encrypt)h(a)g(ciphertext)h Fj(c)527 b Fo(=)f(\()p
   4720 Fj(g)18981 40051 y Fi(r)19444 40474 y Fj(;)171 b(my)21341
   4721 40051 y Fi(r)21803 40474 y Fo(\),)523 b(one)-2022 41636
   4722 y(only)342 b(needs)h(to)f(c)-28 b(ho)28 b(ose)342 b(a)g(new)g(random)f
   4723 (v)-57 b(alue)343 b Fj(r)17374 41213 y Fr(0)17959 41636
   4724 y Fg(2)285 b Fe(Z)19591 41747 y Fi(q)20391 41636 y Fo(and)342
   4725 b(com-)-2022 42996 y(pute)367 b Fj(c)779 42573 y Fr(0)1406
   4726 42996 y Fo(=)328 b(\()p Fj(g)3452 42573 y Fi(r)4159 42996
   4727 y Fg(\242)246 b Fj(g)5213 42573 y Fi(r)5620 42294 y Fh(0)5974
   4728 42996 y Fj(;)171 b(my)7871 42573 y Fi(r)8578 42996 y
   4729 Fg(\242)246 b Fj(y)9648 42573 y Fi(r)10055 42294 y Fh(0)10409
   4730 42996 y Fo(\).)534 b(It)367 b(is)h(easy)g(to)g(con)-28
   4731 b(vince)368 b(oneself)-2022 44158 y(that)357 b(the)i(ciphertexts)g
   4732 Fj(c)f Fo(and)f Fj(c)10531 43735 y Fr(0)11187 44158 y
   4733 Fo(decrypt)h(to)g(the)g(same)g(plain)-28 b(text)-2022
   4734 45321 y Fj(m)p Fo(.)566 b(F)-85 b(urthermore,)387 b(one)379
   4735 b(cannot)f(test)h(if)f Fj(c)14758 44897 y Fr(0)15435
   4736 45321 y Fo(is)h(a)f(re-encryption)h(of)-2022 46483 y
   4737 Fj(c)341 b Fo(if)h(the)g(DDH)f(problem)h(is)f(hard)g(in)g
   4738 Fe(G)p Fo(.)-915 48868 y Ff(Definition)553 b Fo(2.2.)499
   4739 b Fm(R)-52 b(e-encryption)489 b(mix)g(networks)h(\(ElGamal)-2022
   4740 50031 y(implementation\))-542 52127 y Fg(\262)498 b Fn(Key)286
   4741 b(generation)p Fm(:)429 b(al)52 b(l)281 b(de)-52 b(cryption)280
   4742 b(servers)i(jointly)d(gener-)468 53289 y(ate)f(the)g(p)-52
   4743 b(ar)g(ameters)278 b Fo(\()p Fj(q)36 b(;)170 b(g)36 b(;)171
   4744 b(x;)h(y)36 b Fo(\))278 b Fm(of)f(an)g(ElGamal)h(cryptosys-)468
   4745 54451 y(tem)382 b(in)f(a)f(gr)-52 b(oup)381 b Fe(G)g
   4746 Fm(of)g(or)-52 b(der)381 b Fj(q)416 b Fm(gener)-52 b(ate)g(d)382
   4747 b(by)f Fj(g)36 b Fm(,)385 b(using)c(for)468 55613 y(example)412
   4748 b(the)f(thr)-52 b(eshold)411 b(key)h(gener)-52 b(ation)411
   4749 b(pr)-52 b(oto)g(c)g(ol)410 b(of)g(Pe)-52 b(d-)468 56776
   4750 y(ersen)363 b([23,)e(10].)468 b(The)361 b(public)h(p)-52
   4751 b(ar)g(ameters)362 b Fj(q)36 b(;)170 b(g)36 b(;)171 b(y)398
   4752 b Fm(ar)-52 b(e)361 b(made)468 57938 y(public,)343 b(while)336
   4753 b(the)g(private)h(key)g Fj(x)g Fm(is)f(shar)-52 b(e)g(d)336
   4754 b(among)h(the)g(de-)468 59100 y(cryption)365 b(servers)k(in)c(a)h
   4755 Fo(\()p Fj(t;)171 b(k)29 b Fo(\))366 b Fm(se)-52 b(cr)g(et)367
   4756 b(sharing)f(scheme.)-542 61333 y Fg(\262)498 b Fn(Submission)437
   4757 b(of)e(inputs)p Fm(:)538 b(users)400 b(submit)g(to)f(the)g(mix)g(net)
   4758 468 62495 y(ElGamal)310 b(encrypte)-52 b(d)309 b(inputs)f
   4759 Fo(\()p Fj(g)13172 62072 y Fi(r)13634 62495 y Fj(;)171
   4760 b(my)15531 62072 y Fi(r)15993 62495 y Fo(\))309 b Fm(using)g(the)g(p)
   4761 -52 b(ar)g(ame-)468 63657 y(ters)321 b(gener)-52 b(ate)g(d)321
   4762 b(ab)-52 b(ove.)455 b(Users)321 b(ar)-52 b(e)320 b(also)g(r)-52
   4763 b(e)g(quir)g(e)g(d)320 b(to)f(submit)468 64820 y(a)293
   4764 b(pr)-52 b(o)g(of)291 b(of)h(know)52 b(le)-52 b(dge)294
   4765 b(for)e(the)h(c)-52 b(orr)g(esp)g(onding)293 b(plaintext)f
   4766 Fj(m)p Fm(.)468 65982 y(This)363 b(c)-52 b(an)363 b(b)-52
   4767 b(e)364 b(done)f(for)g(example)i(by)e(pr)-52 b(oving)364
   4768 b(know)52 b(le)-52 b(dge)364 b(of)468 67144 y Fj(r)427
   4769 b Fm(with)396 b(r)-52 b(esp)g(e)g(ct)399 b(to)f Fj(g)8760
   4770 66721 y Fi(r)9222 67144 y Fm(,)405 b(using)399 b Fj(my)14108
   4771 66721 y Fi(r)14968 67144 y Fm(as)f(input)f(to)g(a)h(r)-52
   4772 b(andom)468 68307 y(or)g(acle)367 b(\(se)-52 b(e)367
   4773 b([13,)e(27]\).)-542 70539 y Fg(\262)498 b Fn(Mixing)536
   4774 b(phase)p Fm(:)695 b(mixing)479 b(server)h Fj(M)16158
   4775 70650 y Fi(i)16988 70539 y Fm(r)-52 b(e)g(c)g(eives)480
   4776 b(as)f(input)468 71701 y(the)320 b(set)f(of)g(ElGamal)h(ciphertexts)g
   4777 (output)e(by)h(mixing)h(server)29692 17228 y Fj(M)30681
   4778 17339 y Fi(i)p Fr(\241)p Fd(1)32078 17228 y Fm(.)445
   4779 b(Server)292 b Fj(M)36965 17339 y Fi(i)37607 17228 y
   4780 Fm(p)-52 b(ermutes)291 b(and)f(r)-52 b(e-r)g(andomizes)290
   4781 b(\(i.e.)445 b(r)-52 b(e-)29692 18390 y(encrypts\))399
   4782 b(al)52 b(l)398 b(these)h(ciphertexts,)406 b(and)397
   4783 b(outputs)g(a)h(new)g(set)29692 19552 y(of)444 b(ciphertexts,)465
   4784 b(which)445 b(is)f(then)h(p)-52 b(asse)g(d)445 b(to)g
   4785 Fj(M)48520 19663 y Fi(i)p Fd(+1)49898 19552 y Fm(.)707
   4786 b(Server)29692 20715 y Fj(M)30681 20826 y Fi(i)31363
   4787 20715 y Fm(must)331 b(also)f(pr)-52 b(ovide)331 b(a)f(pr)-52
   4788 b(o)g(of)329 b(of)h(c)-52 b(orr)g(e)g(ct)331 b(exe)-52
   4789 b(cution)331 b(\(this)29692 21877 y(wil)52 b(l)366 b(b)-52
   4790 b(e)367 b(discusse)-52 b(d)366 b(in)f(much)i(detail)e(b)-52
   4791 b(elow\).)28682 23637 y Fg(\262)498 b Fn(Decryption)454
   4792 b(phase)p Fm(:)565 b(a)413 b(quorum)h(of)f(de)-52 b(cryption)412
   4793 b(servers)29692 24799 y(jointly)349 b(p)-52 b(erform)350
   4794 b(a)g(thr)-52 b(eshold)350 b(de)-52 b(cryption)349 b(of)h(the)g
   4795 (\257nal)g(out-)29692 25962 y(put,)358 b(and)e(pr)-52
   4796 b(ovide)357 b(a)g(zer)-52 b(o-know)52 b(le)-52 b(dge)358
   4797 b(pr)-52 b(o)g(of)356 b(of)g(c)-52 b(orr)g(e)g(ctness)29692
   4798 27124 y(for)366 b(de)-52 b(cryption.)27201 29581 y Fp(3.)1329
   4799 b(RELA)-126 b(TED)332 b(W)-13 b(ORK)28198 31020 y Fo(The)245
   4800 b(main)i(di\261cult)-28 b(y)247 b(of)f(re-encryption)h(mixnets)g(lies)g
   4801 (in)g(design-)27201 32183 y(ing)395 b(computationally)i(e\261cien)-28
   4802 b(t)396 b(w)-28 b(a)g(ys)393 b(for)h(mix)i(serv)-28 b(ers)394
   4803 b(to)h Fm(pr)-52 b(ove)27201 33345 y Fo(that)389 b(they)h(mixed)g(and)f
   4804 (re-encrypted)h(their)f(inputs)g(correctly)g(in)27201
   4805 34507 y(the)318 b(mixing)g(phase.)448 b(W)-85 b(e)317
   4806 b(surv)-28 b(ey)317 b(here)h(some)f(tec)-28 b(hniques)318
   4807 b(that)f(are)27201 35670 y(represen)-28 b(tativ)g(e)445
   4808 b(of)f(the)h(progress)e(made)i(and)e(compare)i(the)f(e\261-)27201
   4809 36832 y(ciency)283 b(of)f(these)h(tec)-28 b(hniques)283
   4810 b(in)f(Figure)g(1.)436 b(The)281 b(table)i(in)f(Figure)g(1)27201
   4811 37994 y(compares)402 b(the)h(real)g(cost)f(p)28 b(er)402
   4812 b(serv)-28 b(er)402 b(\(for)g(a)g(total)h(of)f Fj(k)432
   4813 b Fo(serv)-28 b(ers\))27201 39156 y(of)481 b(mixing)h
   4814 Fj(n)f Fo(items.)875 b(The)481 b(cost)g(is)g(expressed)h(as)f(the)g(n)
   4815 -28 b(um)g(b)28 b(er)27201 40319 y(of)472 b(exp)28 b(onen)-28
   4816 b(tiations)473 b(required)e(to)h(re-encrypt)g(the)g(inputs,)504
   4817 b(v)-28 b(er-)27201 41481 y(ify)462 b(correctness)e(and)g(decrypt)h
   4818 (the)h(outputs.)813 b(W)-85 b(e)461 b(do)f(not)h(tak)-28
   4819 b(e)27201 42643 y(in)g(to)455 b(accoun)-28 b(t)455 b(the)g(cost)g(of)g
   4820 (op)28 b(erations)455 b(suc)-28 b(h)455 b(as)f(additions)i(and)27201
   4821 43806 y(m)-28 b(ultiplications)362 b(that)f(are)f(m)-28
   4822 b(uc)g(h)359 b(faster)h(to)g(p)28 b(erform)360 b(than)g(exp)28
   4823 b(o-)27201 44968 y(nen)-28 b(tiations.)432 b(Where)269
   4824 b(applicable,)285 b(the)269 b(table)g(also)g(men)-28
   4825 b(tions)269 b(what)27201 46130 y(trade-o\256)342 b(is)g(made)g(for)e
   4826 (e\261ciency)-85 b(.)28198 47293 y(The)365 b(\257rst)h(metho)28
   4827 b(ds)367 b(to)f(pro)-28 b(v)g(e)365 b(the)i(correctness)f(of)g(the)g
   4828 (mixing)27201 48455 y(w)-28 b(ere)266 b(based)h(on)f(cut-and-c)-28
   4829 b(ho)28 b(ose)267 b(zero-kno)-28 b(wledge)268 b(pro)28
   4830 b(ofs)266 b([26,)282 b(21,)27201 49617 y(1].)879 b(Though)481
   4831 b(m)-28 b(uc)g(h)481 b(w)-28 b(ork)481 b(w)-28 b(en)g(t)482
   4832 b(in)-28 b(to)482 b(designing)h(e\261cien)-28 b(t)483
   4833 b(cus-)27201 50780 y(tomized)361 b(zero-kno)-28 b(wledge)359
   4834 b(pro)28 b(ofs,)364 b(these)c(sc)-28 b(hemes)359 b(remain)h(com-)27201
   4835 51942 y(putationally)395 b(exp)28 b(ensiv)-28 b(e.)613
   4836 b(T)-85 b(o)392 b(mak)-28 b(e)394 b(the)g(pro)28 b(ofs)392
   4837 b(more)i(e\261cien)-28 b(t,)27201 53104 y(an)433 b(approac)-28
   4838 b(h)432 b(prop)28 b(osed)434 b(indep)28 b(enden)-28 b(tly)435
   4839 b(b)-28 b(y)433 b(Millimix)k([16])d(and)27201 54267 y(MIP-2)384
   4840 b([2,)394 b(3])384 b(is)f(to)h(decomp)28 b(ose)384 b(a)f(p)28
   4841 b(erm)-28 b(utation)384 b(on)f Fj(n)g Fo(elemen)-28 b(ts)27201
   4842 55429 y(in)g(to)378 b Fj(n)170 b Fo(log)185 b Fj(n)377
   4843 b Fo(pairwise)h(p)28 b(erm)-28 b(utations)378 b(called)g(comparitors.)
   4844 564 b(The)27201 56591 y(mix)321 b(serv)-28 b(er)319 b(then)h(pro)-28
   4845 b(v)g(es)318 b(correct)i(execution)h(of)e(all)h(the)g(compari-)27201
   4846 57753 y(tors)293 b(one)g(b)-28 b(y)293 b(one,)303 b(whic)-28
   4847 b(h)292 b(can)h(b)28 b(e)294 b(done)f(e\261cien)-28 b(tly)295
   4848 b(with)e(a)f(v)-57 b(arian)-28 b(t)27201 58916 y(of)366
   4849 b(the)g(Chaum-P)-28 b(edersen)365 b([6])i(proto)28 b(col)366
   4850 b(that)f(pro)-28 b(v)g(es)365 b(equalit)-28 b(y)368 b(of)27201
   4851 60078 y(discrete)362 b(logarithms.)514 b(The)360 b(sc)-28
   4852 b(hemes)362 b(recen)-28 b(tly)361 b(prop)28 b(osed)361
   4853 b(b)-28 b(y)360 b(F)-85 b(u-)27201 61240 y(ruk)-57 b(a)-28
   4854 b(w)g(a)420 b(and)h(Sak)-28 b(o)420 b([9],)443 b(and)420
   4855 b(Ne\256)i([20])g(o\256er)f(y)-28 b(et)422 b(more)f(e\261cien)-28
   4856 b(t)27201 62403 y(pro)28 b(ofs)341 b(of)g(correct)g(mixing.)28198
   4857 63565 y(W)-85 b(e)229 b(no)-28 b(w)228 b(compare)i(in)f(more)g(detail)i
   4858 (our)d(new)h(mix)i(net)f(to)f(the)h(t)-28 b(w)g(o)27201
   4859 64727 y(sc)g(hemes)389 b(to)g(whic)-28 b(h)388 b(it)h(is)f(closest:)551
   4860 b(Randomized)389 b(P)-28 b(artial)388 b(Chec)-28 b(k-)27201
   4861 65890 y(ing)468 b([18])f(and)g(Optimistic)i(Mixing)g([11].)833
   4862 b(Randomized)468 b(P)-28 b(artial)27201 67052 y(Chec)g(king)290
   4863 b(\(RPC\))f(trades-o\256)h(some)g(priv)-57 b(acy)290
   4864 b(for)f(more)h(e\261ciency:)27201 68214 y(correctness)447
   4865 b(is)h(v)-28 b(eri\257ed)447 b(b)-28 b(y)447 b(asking)h(eac)-28
   4866 b(h)446 b(mix)j(serv)-28 b(er)446 b(to)i(rev)-28 b(eal)27201
   4867 69377 y(a)424 b(randomly)f(selected)i(fraction)e(of)g(its)h
   4868 (input/output)f(relations.)27201 70539 y(This)330 b(guaran)-28
   4869 b(tees)330 b(with)g(high)g(probabilit)-28 b(y)331 b(that)f(all)h(but)f
   4870 (an)g(exp)28 b(o-)27201 71701 y(nen)-28 b(tially)264
   4871 b(small)f(n)-28 b(um)g(b)28 b(er)262 b(of)g(inputs)g(w)-28
   4872 b(ere)262 b(pro)28 b(cessed)263 b(correctly)g(b)-28 b(y)p
   4873 eop
   4874 %%Page: 4 4
   4875 4 3 bop -2022 -915 a Fo(the)279 b(mix)h(serv)-28 b(er.)434
   4876 b(On)280 b(the)f(do)-28 b(wnside,)291 b(RPC)278 b(o\256ers)h(a)g(w)-28
   4877 b(eak)g(er)278 b(guar-)-2022 247 y(an)-28 b(tee)342 b(of)f(priv)-57
   4878 b(acy)-85 b(.)457 b(Since)343 b(eac)-28 b(h)341 b(mix)i(serv)-28
   4879 b(er)341 b(rev)-28 b(eals)343 b(a)e(fraction)h(of)-2022
   4880 1409 y(its)429 b(input/output)f(relations,)451 b(priv)-57
   4881 b(acy)429 b(b)28 b(ecomes)430 b(a)f Fm(glob)-52 b(al)531
   4882 b Fo(prop-)-2022 2571 y(ert)-28 b(y)308 b(of)h(the)g(mix)h(net)-28
   4883 b(w)g(ork:)438 b(a)309 b Fm(majority)396 b Fo(of)308
   4884 b(honest)h(mix)h(serv)-28 b(ers)308 b(is)-2022 3734 y(required)357
   4885 b(to)g(ensure)g(priv)-57 b(acy)-85 b(,)361 b(rather)c(than)f(a)h
   4886 Fm(single)433 b Fo(mix)358 b(serv)-28 b(er.)-2022 4896
   4887 y(T)-85 b(o)276 b(ac)-28 b(hiev)g(e)279 b(the)f(same)g(con\257dence)g
   4888 (in)g(priv)-57 b(acy)-85 b(,)291 b(RPC)277 b(requires)h(the)-2022
   4889 6058 y(in)-28 b(v)g(olv)g(emen)g(t)275 b(of)g(man)-28
   4890 b(y)274 b(more)h(serv)-28 b(ers)274 b(than)h(the)g(sc)-28
   4891 b(hemes)275 b(w)-28 b(e)274 b(ha)-28 b(v)g(e)-2022 7221
   4892 y(previously)342 b(surv)-28 b(ey)g(ed.)-1026 8383 y(While)479
   4893 b(our)e(mix)h(net)g(b)28 b(ears)478 b(some)g(resem)-28
   4894 b(blance)478 b(to)g(RPC,)f(it)-2022 9545 y(exploits)338
   4895 b(mostly)h(a)e(di\256eren)-28 b(t)338 b(trade-o\256.)455
   4896 b(RPC)336 b(trades)h(o\256)g(mostly)-2022 10708 y(priv)-57
   4897 b(acy)228 b(\(and)f(some)h(correctness\))g(for)f(e\261ciency)-85
   4898 b(,)252 b(while)229 b(our)d(sc)-28 b(heme)-2022 11870
   4899 y(trades)403 b(o\256)h(mostly)h(correctness)f(\(and)g(a)g(little)i
   4900 (priv)-57 b(acy\))404 b(for)g(e\261-)-2022 13032 y(ciency)-85
   4901 b(.)548 b(Con)-28 b(trary)371 b(to)h(RPC,)g(a)g(mix)h(serv)-28
   4902 b(er)372 b(in)g(our)f(sc)-28 b(heme)373 b(do)28 b(es)-2022
   4903 14195 y(not)354 b(rev)-28 b(eal)355 b(individual)h(relationships)f(b)28
   4904 b(et)-28 b(w)g(een)355 b(inputs)f(and)g(out-)-2022 15357
   4905 y(puts,)287 b(but)273 b(only)h(the)g(global)g(relationship)h(b)28
   4906 b(et)-28 b(w)g(een)273 b(a)h(large)f(subset)-2022 16519
   4907 y(of)357 b(the)i(inputs)f(and)g(a)g(large)g(subset)g(of)g(the)g
   4908 (outputs.)506 b(Lik)-28 b(e)358 b(RPC,)-2022 17681 y(our)331
   4909 b(mix)j(o\256ers)f(p)28 b(erfect)333 b(priv)-57 b(acy)333
   4910 b(if)g(there)g(is)g(an)f(honest)h(ma)57 b(jorit)-28 b(y)-2022
   4911 18844 y(of)457 b(mix)i(serv)-28 b(ers.)805 b(But)458
   4912 b(unlik)-28 b(e)459 b(RPC,)e(our)g(mix)i(preserv)-28
   4913 b(es)458 b(some)-2022 20006 y(priv)-57 b(acy)361 b(ev)-28
   4914 b(en)361 b(when)g(there)g(is)g(only)g(a)g(single)h(honest)f(mix)g(serv)
   4915 -28 b(er.)-2022 21168 y(In)488 b(the)i(case)f(of)g(a)g(single)h(honest)
   4916 f(mix)h(serv)-28 b(er,)526 b(as)489 b(w)-28 b(e)489 b(sho)-28
   4917 b(w)487 b(in)-2022 22331 y(section)501 b(6,)541 b(ev)-28
   4918 b(ery)502 b(input)f(is)g(hidden)g(among)g Fj(n=)p Fo(2)18454
   4919 21907 y Fi(\256)19544 22331 y Fo(outputs)g(on)-2022 23493
   4920 y(a)-28 b(v)g(erage,)535 b(where)496 b Fj(n)g Fo(is)h(the)g(total)g(n)
   4921 -28 b(um)g(b)28 b(er)495 b(of)i(inputs)f(and)g Fj(\256)h
   4922 Fo(is)-2022 24655 y(the)365 b(securit)-28 b(y)366 b(parameter)g(\(e.g.)
   4923 528 b Fj(\256)325 b Fo(=)g(4\).)528 b(On)365 b(the)h(do)-28
   4924 b(wnside,)371 b(our)-2022 25818 y(mixnet)285 b(do)28
   4925 b(es)284 b(not)f(guaran)-28 b(tee)284 b(p)28 b(erfect)284
   4926 b(correctness)f(but)h(m)-28 b(ust)283 b(rely)-2022 26980
   4927 y(on)323 b(the)h(parallel)h(execution)h(of)d(a)h(slo)-28
   4928 b(w)g(er)323 b(v)-28 b(eri\257cation)324 b(proto)28 b(col)324
   4929 b(to)-2022 28142 y(guaran)-28 b(tee)341 b(p)28 b(erfect)342
   4930 b(correctness.)-1026 29305 y(Finally)-85 b(,)435 b(a)415
   4931 b(pro)28 b(of)414 b(tec)-28 b(hnique)416 b(similar)f(to)g(ours)f(is)h
   4932 (used)g(in)g([11])-2022 30467 y(to)329 b(build)g(a)g(mix)h(net)-28
   4933 b(w)g(ork)328 b(with)h(di\256eren)-28 b(t)329 b(prop)28
   4934 b(erties.)452 b(The)329 b(pro)28 b(of)-2022 31629 y(of)332
   4935 b(correctness)h(in)g([11])g(consists)g(of)g(a)g(pro)28
   4936 b(of)332 b(that)h(the)g(pro)28 b(duct)333 b(of)-2022
   4937 32791 y Fm(al)52 b(l)498 b Fo(the)396 b(inputs)f(equals)i(the)f(pro)28
   4938 b(duct)395 b(of)g Fm(al)52 b(l)499 b Fo(the)396 b(outputs)f(of)g(the)
   4939 -2022 33954 y(mix)373 b(serv)-28 b(er.)548 b(This,)380
   4940 b(com)-28 b(bined)372 b(with)h(redundancy)f(c)-28 b(hec)g(ks)372
   4941 b(in)g(the)-2022 35116 y(inputs,)477 b(guaran)-28 b(tees)450
   4942 b(p)28 b(erfect)451 b(correctness.)781 b(As)450 b(in)g(our)g(sc)-28
   4943 b(heme,)-2022 36278 y(the)308 b(cost)h(of)f(the)g(pro)28
   4944 b(of)308 b(is)h(indep)28 b(enden)-28 b(t)309 b(of)f(the)h(n)-28
   4945 b(um)g(b)28 b(er)308 b(of)g(inputs)-2022 37441 y(mixed.)487
   4946 b(But)351 b(on)h(the)g(do)-28 b(wnside,)353 b(the)f(redundancy)f(c)-28
   4947 b(hec)g(ks)351 b(in)h(the)-2022 38603 y(inputs)300 b(\(whic)-28
   4948 b(h)300 b(guaran)-28 b(tee)301 b(p)28 b(erfect)301 b(correctness\))g
   4949 (result)g(in)g(a)f(cost)-2022 39765 y(of)308 b(mixing)h(and)g
   4950 (decrypting)f(that)h(is)g(more)f(than)h(t)-28 b(wice)308
   4951 b(as)g(high)h(as)-2022 40928 y(in)341 b(all)h(other)g(re-encryption)g
   4952 (mix)g(net)-28 b(w)g(orks.)-1026 42090 y(T)-85 b(o)246
   4953 b(summarize,)268 b(our)246 b(mix)j(has)e(the)h(lo)-28
   4954 b(w)g(est)247 b(total)h(computational)-2022 43252 y(o)-28
   4955 b(v)g(erhead)386 b(for)h(mixing)h Fj(n)f Fo(inputs.)593
   4956 b(In)386 b(particular)h(the)g(n)-28 b(um)g(b)28 b(er)387
   4957 b(of)-2022 44415 y(exp)28 b(onen)-28 b(tiations)418 b(required)f(to)g
   4958 (pro)-28 b(v)g(e)416 b(correct)g(mixing)i(is)f(a)f(con-)-2022
   4959 45577 y(stan)-28 b(t)495 b(indep)28 b(enden)-28 b(t)496
   4960 b(of)g(the)g(n)-28 b(um)g(b)28 b(er)494 b(of)i(inputs.)918
   4961 b(Our)495 b(sc)-28 b(heme)-2022 46739 y(guaran)g(tees)303
   4962 b(only)h(almost)g(en)-28 b(tirely)305 b(correct)e(mixing:)438
   4963 b(an)-28 b(y)303 b(error)f(in)-2022 47901 y(the)449 b(output)g(is)h
   4964 (detected)g(with)f(probabilit)-28 b(y)450 b(99\045)f(whereas)g(p)28
   4965 b(er-)-2022 49064 y(fectly)325 b(correct)f(mixes)i(pro)-28
   4966 b(vide)324 b(standard)g(guaran)-28 b(tees)323 b(of)i(correct-)-2022
   4967 50226 y(ness)386 b(of)h(1)258 b Fg(\241)g Fo(2)3744 49803
   4968 y Fd(80)4611 50226 y Fo(.)591 b(Our)387 b(sc)-28 b(heme)387
   4969 b(also)g(trades)g(o\256)f(a)h(little)i(priv)-57 b(acy)-2022
   4970 51388 y(\(see)342 b(section)g(6\).)-2022 54765 y Fp(4.)1328
   4971 b(PR)-40 b(OOF)456 b(OF)f(ALMOST)h(ENTIREL)-122 b(Y)455
   4972 b(COR\255)302 56314 y(RECT)332 b(MIXING)-1026 57753 y
   4973 Fo(T)-85 b(o)386 b(illustrate)j(the)f(k)-28 b(ey)389
   4974 b(idea)f(of)f(our)g(pro)28 b(of)387 b(of)g(almost)h(en)-28
   4975 b(tirely)-2022 58916 y(correct)294 b(mixing,)306 b(w)-28
   4976 b(e)294 b(in)-28 b(tro)28 b(duce)295 b(\257rst)f(the)h(follo)-28
   4977 b(wing)294 b(simple)i(prob-)-2022 60078 y(lem.)861 b(Consider)476
   4978 b(a)g(pro)-28 b(v)g(er)475 b(who)h(is)h(committed)h(to)e(t)-28
   4979 b(w)g(o)475 b(sets)i(of)-2022 61240 y Fj(n)456 b Fo(elemen)-28
   4980 b(ts:)687 b Fj(m)4770 61351 y Fd(1)5231 61240 y Fj(;)171
   4981 b(:)g(:)g(:)g(;)g(m)8408 61351 y Fi(n)9451 61240 y Fg(2)476
   4982 b Fe(G)456 b Fo(and)g Fj(m)14849 60817 y Fr(0)14849 61416
   4983 y Fd(1)15310 61240 y Fj(;)171 b(:)g(:)g(:)g(;)g(m)18487
   4984 60817 y Fr(0)18487 61416 y Fi(n)19530 61240 y Fg(2)476
   4985 b Fe(G)p Fo(.)800 b(This)-2022 62403 y(pro)-28 b(v)g(er)391
   4986 b(w)-28 b(an)g(ts)390 b(to)i(con)-28 b(vince)393 b(a)f(v)-28
   4987 b(eri\257er)393 b(that)f(there)h(exists)g(a)f(p)28 b(er-)-2022
   4988 63565 y(m)-28 b(utation)397 b Fj(')f Fo(on)g Fj(n)h Fo(elemen)-28
   4989 b(ts)398 b(suc)-28 b(h)396 b(that)g(for)g(all)i Fj(i)p
   4990 Fo(,)411 b Fj(m)19927 63142 y Fr(0)19927 63750 y Fi(i)20655
   4991 63565 y Fo(=)377 b Fj(m)22730 63720 y Fi(')p Fd(\()p
   4992 Fi(i)p Fd(\))24260 63565 y Fo(.)-2022 64727 y(In)417
   4993 b(other)i(w)-28 b(ords,)436 b(the)419 b(pro)-28 b(v)g(er)418
   4994 b(m)-28 b(ust)418 b(con)-28 b(vince)419 b(the)g(v)-28
   4995 b(eri\257er)418 b(that)-2022 65890 y(the)475 b(set)g
   4996 Fg(f)p Fj(m)3021 65466 y Fr(0)3021 66075 y Fi(i)3372
   4997 65890 y Fg(g)3884 65466 y Fi(n)3884 66075 y(i)p Fd(=1)5737
   4998 65890 y Fo(is)g(a)g(p)28 b(erm)-28 b(utation)475 b(of)g(the)g(set)g
   4999 Fg(f)p Fj(m)20225 66001 y Fi(i)20576 65890 y Fg(g)21088
   5000 65466 y Fi(n)21088 66075 y(i)p Fd(=1)22467 65890 y Fo(.)855
   5001 b(In)-2022 67052 y(addition,)417 b(the)402 b(pro)-28
   5002 b(v)g(er)401 b(m)-28 b(ust)401 b(rev)-28 b(eal)402 b(no)f(information)h
   5003 (ab)28 b(out)402 b(the)-2022 68214 y(sets)366 b Fg(f)p
   5004 Fj(m)1419 68325 y Fi(i)1771 68214 y Fg(g)2283 67791 y
   5005 Fi(n)2283 68399 y(i)p Fd(=1)4027 68214 y Fo(and)g Fg(f)p
   5006 Fj(m)7457 67791 y Fr(0)7457 68399 y Fi(i)7808 68214 y
   5007 Fg(g)8320 67791 y Fi(n)8320 68399 y(i)p Fd(=1)10065 68214
   5008 y Fo(and)g(as)g(little)i(information)f(as)f(p)28 b(os-)-2022
   5009 69377 y(sible)356 b(ab)28 b(out)357 b(the)f(p)28 b(erm)-28
   5010 b(utation)356 b Fj(')p Fo(.)499 b(W)-85 b(e)356 b(assume)g(that)g(the)h
   5011 (pro)-28 b(v)g(er)-2022 70539 y(is)341 b(computationally)i(b)28
   5012 b(ounded.)-1026 71701 y(W)-85 b(e)342 b(prop)28 b(ose)341
   5013 b(the)h(follo)-28 b(wing)341 b(approac)-28 b(h:)28397
   5014 -915 y(1.)499 b(The)430 b(v)-28 b(eri\257er)430 b(c)-28
   5015 b(ho)28 b(oses)430 b(a)h(random)e(subset)h(of)g(indices)h
   5016 Fj(S)489 b Fg(\275)29692 247 y(f)p Fo(1)p Fj(;)171 b(:)g(:)g(:)g(;)g(n)
   5017 p Fg(g)280 b Fo(\(note)g(that)g Fg(j)p Fj(S)57 b Fg(j)284
   5018 b(\274)h Fj(n=)p Fo(2)279 b(with)h(high)f(probabilit)-28
   5019 b(y\).)28397 2198 y(2.)499 b(The)443 b(pro)-28 b(v)g(er)442
   5020 b(rev)-28 b(eals)444 b(to)f(the)h(v)-28 b(eri\257er)443
   5021 b(the)h(set)g Fj(S)49619 1774 y Fr(0)50371 2198 y Fo(=)454
   5022 b Fj(')p Fo(\()p Fj(S)57 b Fo(\))29692 3360 y(de\257ned)417
   5023 b(as)g Fj(')p Fo(\()p Fj(S)57 b Fo(\))410 b(=)g Fg(f)p
   5024 Fj(')p Fo(\()p Fj(s)p Fo(\))p Fg(j)p Fj(s)h Fg(2)f Fj(S)57
   5025 b Fg(g)p Fo(.)682 b(The)416 b(v)-28 b(eri\257er)417 b(c)-28
   5026 b(hec)g(ks)29692 4522 y(that)342 b Fg(j)p Fj(S)57 b Fg(j)284
   5027 b Fo(=)h Fg(j)p Fj(S)35494 4099 y Fr(0)35792 4522 y Fg(j)p
   5028 Fo(.)28397 6473 y(3.)499 b(The)369 b(pro)-28 b(v)g(er)368
   5029 b(sho)-28 b(ws)368 b(that)40230 5726 y Fc(Q)41197 6777
   5030 y Fi(S)41961 6473 y Fj(m)42863 6584 y Fi(i)43545 6473
   5031 y Fo(=)44673 5726 y Fc(Q)45640 6777 y Fi(S)46178 6586
   5032 y Fh(0)46703 6473 y Fj(m)47605 6050 y Fr(0)47605 6658
   5033 y Fi(i)47956 6473 y Fo(.)539 b(W)-85 b(e)370 b(assume)29692
   5034 7635 y(that)235 b(this)h(can)f(b)28 b(e)236 b(done)f(without)h(rev)-28
   5035 b(ealing)236 b(an)-28 b(ything)235 b(ab)28 b(out)29692
   5036 8797 y(the)387 b Fj(m)32403 8908 y Fi(i)33141 8797 y
   5037 Fo(or)g Fj(m)35343 8374 y Fr(0)35343 8982 y Fi(i)35694
   5038 8797 y Fo(.)592 b(The)386 b(v)-28 b(eri\257er)387 b(is)g(satis\257ed)h
   5039 (if)f(this)g(pro)28 b(duct)29692 9960 y(equalit)-28 b(y)343
   5040 b(holds.)28198 11902 y(There)466 b(are)g(a)h(few)f(imp)28
   5041 b(ortan)-28 b(t)467 b(observ)-57 b(ations)467 b(to)g(mak)-28
   5042 b(e)467 b(ab)28 b(out)27201 13064 y(this)342 b(approac)-28
   5043 b(h:)28682 15006 y Fg(\262)498 b Fo(A)308 b(pro)-28 b(v)g(er)306
   5044 b(who)h(kno)-28 b(ws)307 b(a)h(p)28 b(erm)-28 b(utation)309
   5045 b Fj(')e Fo(suc)-28 b(h)307 b(that)h(for)g(all)29692
   5046 16168 y Fj(i)p Fo(,)412 b Fj(m)31640 15745 y Fr(0)31640
   5047 16353 y Fi(i)32368 16168 y Fo(=)378 b Fj(m)34444 16323
   5048 y Fi(')p Fd(\()p Fi(i)p Fd(\))36371 16168 y Fo(will)398
   5049 b(alw)-28 b(a)g(ys)396 b(trivially)j(succeed)f(in)f(steps)g(2)29692
   5050 17331 y(and)341 b(3.)28682 19281 y Fg(\262)498 b Fo(Consider)339
   5051 b(a)h(malicious)h(pro)-28 b(v)g(er)338 b(who)h(do)28
   5052 b(es)341 b(not)f(ha)-28 b(v)g(e)339 b(a)h(p)28 b(er-)29692
   5053 20444 y(m)-28 b(utation)517 b Fj(')f Fo(suc)-28 b(h)515
   5054 b(that)i(for)f(all)h Fj(i)p Fo(,)561 b Fj(m)45775 20020
   5055 y Fr(0)45775 20629 y Fi(i)46702 20444 y Fo(=)577 b Fj(m)48977
   5056 20599 y Fi(')p Fd(\()p Fi(i)p Fd(\))50507 20444 y Fo(.)981
   5057 b(This)29692 21606 y(pro)-28 b(v)g(er)492 b(m)-28 b(ust)493
   5058 b(\257nd)f(a)h(set)h Fj(S)41333 21183 y Fr(0)42123 21606
   5059 y Fo(suc)-28 b(h)492 b(that)i Fg(j)p Fj(S)57 b Fg(j)537
   5060 b Fo(=)g Fg(j)p Fj(S)51043 21183 y Fr(0)51341 21606 y
   5061 Fg(j)494 b Fo(and)29692 22021 y Fc(Q)30659 23073 y Fi(S)31423
   5062 22768 y Fj(m)32325 22879 y Fi(i)32961 22768 y Fo(=)34041
   5063 22021 y Fc(Q)35009 23073 y Fi(S)35547 22882 y Fh(0)36072
   5064 22768 y Fj(m)36974 22345 y Fr(0)36974 22953 y Fi(i)37325
   5065 22768 y Fo(.)437 b(W)-85 b(e)286 b(pro)-28 b(v)g(e)284
   5066 b(in)i(section)g(7)f(that)g(if)h(the)g(set)29692 23931
   5067 y Fg(f)p Fj(m)31106 23507 y Fr(0)31106 24116 y Fi(i)31457
   5068 23931 y Fg(g)410 b Fo(is)h(not)f(a)g(p)28 b(erm)-28 b(utation)411
   5069 b(of)f(the)h(set)f Fg(f)p Fj(m)48369 24042 y Fi(i)48721
   5070 23931 y Fg(g)p Fo(,)427 b(then)411 b(the)29692 25093
   5071 y(probabilit)-28 b(y)366 b(that)g(the)g(pro)-28 b(v)g(er)364
   5072 b(can)i(\257nd)f(a)g(set)h Fj(S)49326 24670 y Fr(0)49989
   5073 25093 y Fo(with)g(the)29692 26255 y(desired)386 b(prop)28
   5074 b(erties)386 b(in)f(p)28 b(olynomial)388 b(time)e(is)g(at)f(most)h(5)p
   5075 Fj(=)p Fo(8,)29692 27417 y(or)285 b(else)i(the)g(discrete)g(logarithm)f
   5076 (problem)g(can)g(b)28 b(e)287 b(solv)-28 b(ed)286 b(in)29692
   5077 28580 y(p)28 b(olynomial)343 b(time)g(in)f Fe(G)p Fo(.)28682
   5078 30530 y Fg(\262)498 b Fo(The)430 b(pro)28 b(of)429 b(leaks)j(a)e
   5079 (little)i(bit)e(of)g(information)h(ab)28 b(out)430 b(the)29692
   5080 31693 y(p)28 b(erm)-28 b(utation)411 b Fj(')p Fo(.)662
   5081 b(Giv)-28 b(en)411 b Fj(S)466 b Fo(and)410 b Fj(S)44107
   5082 31270 y Fr(0)44804 31693 y Fo(=)399 b Fj(')p Fo(\()p
   5083 Fj(S)57 b Fo(\),)427 b(the)411 b(v)-28 b(eri\257er)29692
   5084 32855 y(kno)g(ws)371 b(that)i Fj(')p Fo(\()p Fj(i)p Fo(\))336
   5085 b Fg(2)g Fj(S)38903 32432 y Fr(0)39572 32855 y Fo(if)373
   5086 b(and)f(only)h(if)f Fj(i)337 b Fg(2)e Fj(S)57 b Fo(.)548
   5087 b(This)371 b(is)i(ac-)29692 34017 y(ceptable)307 b(for)e(mix)i(net)-28
   5088 b(w)g(ork)305 b(applications,)315 b(considering)306 b(the)29692
   5089 35180 y(n)-28 b(um)g(b)28 b(er)401 b(of)g(inputs)h Fj(m)38804
   5090 35291 y Fi(i)39557 35180 y Fo(is)g(t)-28 b(ypically)404
   5091 b(large,)417 b(so)402 b(that)f(b)28 b(eing)29692 36342
   5092 y(hidden)342 b(among)f(ab)28 b(out)342 b(half)f(the)h(outputs)f(is)g
   5093 (su\261cien)-28 b(t.)27201 38284 y Fn(Applications)395
   5094 b(to)f(ElGamal)g(re-encryption)h(mixes.)27201 39446 y
   5095 Fo(ElGamal)475 b(re-encryption)f(mix)g(net)-28 b(w)g(orks)473
   5096 b(w)-28 b(ere)472 b(de\257ned)i(in)g(sec-)27201 40609
   5097 y(tion)290 b(2.1.)439 b(Recall)290 b(that)f(the)h(inputs)g(to)f(a)g
   5098 (mix)i(serv)-28 b(er)289 b(are)g(ElGamal)27201 41771
   5099 y(ciphertexts.)451 b(The)325 b(mix)g(serv)-28 b(er)324
   5100 b(mixes)i(these)f(inputs,)j(re-encrypts)27201 42933 y(them,)400
   5101 b(and)387 b(outputs)f(a)h(new)g(set)h(of)f(ElGamal)h(ciphertexts.)594
   5102 b(F)-85 b(ol-)27201 44096 y(lo)-28 b(wing)488 b(the)g(example)h(ab)28
   5103 b(o)-28 b(v)g(e,)525 b(w)-28 b(e)487 b(prop)28 b(ose)488
   5104 b(to)g(v)-28 b(erify)488 b(that)g(the)27201 45258 y(inputs)332
   5105 b(w)-28 b(ere)331 b(mixed)h(correctly)g(b)-28 b(y)331
   5106 b(\257rst)g(computing)h(the)g(pro)28 b(duct)27201 46420
   5107 y(of)302 b(a)g(randomly)h(selected)h(subset)e Fj(S)358
   5108 b Fo(of)302 b(the)h(inputs,)311 b(then)302 b(giving)h
   5109 Fj(S)27201 47582 y Fo(to)290 b(the)h(mix)g(serv)-28 b(er)289
   5110 b(and)h(asking)g(it)g(to)g(pro)28 b(duce)290 b(a)g(subset)g
   5111 Fj(S)50644 47159 y Fr(0)51231 47582 y Fo(of)g(the)27201
   5112 48745 y(outputs)249 b(whose)h(pro)28 b(duct)249 b(is)g(the)h(same.)426
   5113 b(T)-85 b(o)248 b(mak)-28 b(e)250 b(this)g(pro)28 b(of)249
   5114 b(tec)-28 b(h-)27201 49907 y(nique)276 b(w)-28 b(ork)275
   5115 b(with)g(ElGamal)h(ciphertexts,)290 b(w)-28 b(e)275 b(need)h(the)f
   5116 (follo)-28 b(wing)27201 51069 y(t)g(w)g(o)340 b(prop)28
   5117 b(ositions:)28308 53228 y Ff(Pr)-22 b(oposition)554 b
   5118 Fo(4.1.)499 b Fm(\(Multiplic)-52 b(ative)462 b(homomorphism)g(of)g(El-)
   5119 27201 54391 y(Gamal\))451 b(L)-52 b(et)451 b Fo(\()p
   5120 Fj(g)33740 54502 y Fd(1)34202 54391 y Fj(;)171 b(m)35559
   5121 54502 y Fd(1)36020 54391 y Fo(\))450 b Fm(and)g Fo(\()p
   5122 Fj(g)39825 54502 y Fd(2)40287 54391 y Fj(;)171 b(m)41644
   5123 54502 y Fd(2)42105 54391 y Fo(\))451 b Fm(b)-52 b(e)451
   5124 b(ElGamal)f(encryptions)27201 55553 y(of)369 b(plaintexts)g
   5125 Fj(P)33666 55664 y Fd(1)34496 55553 y Fm(and)g Fj(P)37140
   5126 55664 y Fd(2)37601 55553 y Fm(.)480 b(Then)369 b Fo(\()p
   5127 Fj(g)41950 55664 y Fd(1)42412 55553 y Fj(g)42900 55664
   5128 y Fd(2)43362 55553 y Fj(;)171 b(m)44719 55664 y Fd(1)45180
   5129 55553 y Fj(m)46082 55664 y Fd(2)46543 55553 y Fo(\))370
   5130 b Fm(is)f(an)g(ElGamal)27201 56715 y(encryption)d(of)f(the)h(pr)-52
   5131 b(o)g(duct)365 b Fj(P)39460 56826 y Fd(1)39921 56715
   5132 y Fj(P)40575 56826 y Fd(2)41036 56715 y Fm(.)28308 58874
   5133 y Ff(Pr)-22 b(oposition)554 b Fo(4.2.)499 b Fm(\(Chaum-Pe)-52
   5134 b(dersen)547 b(pr)-52 b(oto)g(c)g(ol)545 b([6]\))g(Con-)27201
   5135 60037 y(sider)424 b(a)f(pr)-52 b(over)424 b(who)e(knows)h(values)h
   5136 Fo(\()p Fj(g)36 b(;)171 b(x;)h(h;)f(y)36 b Fo(\))423
   5137 b Fm(in)g Fe(Z)49094 60148 y Fi(p)50000 60037 y Fm(and)f(also)27201
   5138 61199 y(knows)403 b Fo(log)31593 61448 y Fi(g)32246 61199
   5139 y Fj(x)h Fm(and)e Fo(log)36576 61448 y Fi(h)37284 61199
   5140 y Fj(y)36 b Fm(.)581 b(The)403 b(Chaum-Pe)-52 b(dersen)403
   5141 b(pr)-52 b(oto)g(c)g(ol)402 b(al-)27201 62361 y(lows)438
   5142 b(this)g(pr)-52 b(over)438 b(to)g(c)-52 b(onvinc)g(e)438
   5143 b(a)g(veri\257er)i(that)d Fo(log)48337 62610 y Fi(g)48991
   5144 62361 y Fj(x)415 b Fo(=)h(log)52520 62610 y Fi(h)53228
   5145 62361 y Fj(y)27201 63524 y Fm(without)456 b(r)-52 b(eve)g(aling)458
   5146 b(anything)e(ab)-52 b(out)457 b(these)h(discr)-52 b(ete)458
   5147 b(lo)-52 b(garithms.)27201 64686 y(A)449 b(trivial)g(applic)-52
   5148 b(ation)447 b(of)h(the)g(Chaum-Pe)-52 b(dersen)450 b(pr)-52
   5149 b(oto)g(c)g(ol)447 b(is)i(to)27201 65848 y(pr)-52 b(ove)351
   5150 b(that)d(two)h(ElGamal)i(ciphertexts)f(ar)-52 b(e)350
   5151 b(r)-52 b(e-encryption)350 b(of)f(the)27201 67011 y(same)367
   5152 b(plaintext.)28198 69170 y Fo(Consider)f(a)h(mix)h(serv)-28
   5153 b(er)367 b(who)g(receiv)-28 b(es)368 b(as)f(inputs)g
   5154 Fj(n)g Fo(ElGamal)27201 70332 y(ciphertexts)436 b(\()p
   5155 Fj(g)33487 69909 y Fi(r)33872 70049 y Fb(i)34279 70332
   5156 y Fj(;)171 b(m)35636 70443 y Fi(i)36277 70332 y Fg(\242)290
   5157 b Fj(y)37391 69909 y Fi(r)37776 70049 y Fb(i)38182 70332
   5158 y Fo(\),)459 b(and)434 b(outputs)h Fj(n)f Fo(ElGamal)i(cipher-)27201
   5159 71701 y(texts)349 b(\()p Fj(g)30667 71278 y Fi(r)31074
   5160 70999 y Fh(0)31052 71499 y Fb(i)31459 71701 y Fj(;)171
   5161 b(m)32816 71278 y Fr(0)32816 71886 y Fi(i)33399 71701
   5162 y Fg(\242)232 b Fj(y)34455 71278 y Fi(r)34862 70999 y
   5163 Fh(0)34840 71499 y Fb(i)35246 71701 y Fo(\).)476 b(Prop)28
   5164 b(osition)348 b(4.1)g(sho)-28 b(ws)347 b(that)h(an)-28
   5165 b(y)347 b(v)-28 b(eri\257er)p eop
   5166 %%Page: 5 5
   5167 5 4 bop -2022 -915 a Fo(can)448 b(compute)h(an)f(ElGamal)h(encryption)g
   5168 (\()p Fj(g)36 b(;)171 b(m)p Fo(\))449 b(of)19745 -1663
   5169 y Fc(Q)20882 -915 y Fj(m)21784 -804 y Fi(i)22136 -915
   5170 y Fo(,)475 b(and)-2022 247 y(an)345 b(ElGamal)h(encryption)g(\()p
   5171 Fj(g)9715 -176 y Fr(0)10014 247 y Fj(;)171 b(m)11371
   5172 -176 y Fr(0)11669 247 y Fo(\))346 b(of)13583 -500 y Fc(Q)14720
   5173 247 y Fj(m)15622 -176 y Fr(0)15622 432 y Fi(i)15974 247
   5174 y Fo(.)467 b(With)346 b(Prop)28 b(osition)-2022 1409
   5175 y(4.2,)453 b(the)431 b(mix)g(serv)-28 b(er)431 b(can)f(then)h(pro)-28
   5176 b(v)g(e)429 b(that)16589 662 y Fc(Q)17727 1409 y Fj(m)18629
   5177 1520 y Fi(i)19413 1409 y Fo(=)20642 662 y Fc(Q)21780
   5178 1409 y Fj(m)22682 986 y Fr(0)22682 1594 y Fi(i)23464
   5179 1409 y Fo(b)-28 b(y)-2022 2571 y(giving)306 b(a)f(zero-kno)-28
   5180 b(wledge)305 b(pro)28 b(of)305 b(that)g(log)15 b(\()p
   5181 Fj(g)15946 2148 y Fr(0)16244 2571 y Fj(=g)36 b Fo(\))285
   5182 b(=)g Fj(l)17 b(og)20351 2682 y Fi(y)20849 2571 y Fo(\()p
   5183 Fj(m)22149 2148 y Fr(0)22448 2571 y Fj(=m)p Fo(\).)-2022
   5184 5384 y Fp(5.)1328 b(OUR)333 b(NEW)f(MIX)g(NET)-1026 6823
   5185 y Fo(In)390 b(this)h(section,)404 b(w)-28 b(e)390 b(in)-28
   5186 b(tegrate)391 b(our)f(pro)28 b(of)391 b(of)f(almost)h(en)-28
   5187 b(tirely)-2022 7985 y(correct)404 b(mixing)h(in)g(the)g(design)f(of)g
   5188 (the)h(ElGamal)h(re-encryption)-2022 9147 y(mix)318 b(net)-28
   5189 b(w)g(ork)316 b(and)h(giv)-28 b(e)317 b(a)g(detailed)i(description)f
   5190 (of)f(the)g(resulting)-2022 10310 y(mix)342 b(net)-28
   5191 b(w)g(ork)340 b(proto)28 b(col.)-2022 12634 y Fn(Setup.)454
   5192 b Fo(The)339 b(decryption)g(serv)-28 b(ers)339 b(join)-28
   5193 b(tly)339 b(generate)h(the)f(param-)-2022 13797 y(eters)285
   5194 b(\()p Fj(q)36 b(;)170 b(g)36 b(;)171 b(x;)h(y)36 b Fo(\))284
   5195 b(of)h(an)f(ElGamal)i(cryptosystem)f(in)g(a)g(group)e
   5196 Fe(G)i Fo(of)-2022 14959 y(prime)364 b(order)f Fj(q)398
   5197 b Fo(generated)365 b(b)-28 b(y)363 b Fj(g)36 b Fo(.)522
   5198 b(The)364 b(priv)-57 b(ate)364 b(k)-28 b(ey)365 b Fj(x)f
   5199 Fo(suc)-28 b(h)362 b(that)-2022 16121 y Fj(y)533 b Fo(=)498
   5200 b Fj(g)833 15698 y Fi(x)1821 16121 y Fo(is)470 b(shared)f(among)g(all)h
   5201 (decryption)g(serv)-28 b(ers)468 b(in)i(a)f(\()p Fj(t;)171
   5202 b(k)29 b Fo(\))-2022 17283 y(secret)416 b(sharing)f(sc)-28
   5203 b(heme.)679 b(This)415 b(ma)-28 b(y)416 b(b)28 b(e)416
   5204 b(done)g(using)f(for)g(exam-)-2022 18446 y(ple)317 b(the)g(\()p
   5205 Fj(t;)171 b(k)29 b Fo(\)-threshold)318 b(k)-28 b(ey)317
   5206 b(generation)h(proto)28 b(col)317 b(of)f(P)-28 b(edersen.)-2022
   5207 19608 y(This)331 b(setup)h(step)g(is)f(executed)j(only)e(once.)452
   5208 b(After)332 b(that,)i(the)e(same)-2022 20770 y(parameters)262
   5209 b(can)g(b)28 b(e)262 b(used)h(to)f(mix)h(an)-28 b(y)262
   5210 b(n)-28 b(um)g(b)28 b(er)261 b(of)h(input)g(batc)-28
   5211 b(hes.)-2022 21933 y(The)338 b(parameters)i(of)e(the)i(ElGamal)g
   5212 (cryptosystem)g(need)g(only)g(b)28 b(e)-2022 23095 y(generated)240
   5213 b(anew)g(if)h(new)f(serv)-28 b(ers)240 b(join)g(the)g(mix)i(or)d
   5214 (existing)j(serv)-28 b(ers)-2022 24257 y(lea)g(v)g(e)342
   5215 b(the)g(mix.)-2022 26582 y Fn(Submission)394 b(of)g(inputs.)-542
   5216 28619 y Fg(\262)498 b Fo(The)e(serv)-28 b(ers)495 b(publish)h(the)g
   5217 (public)g(ElGamal)h(parameters)468 29781 y(that)342 b(w)-28
   5218 b(ere)341 b(generated)h(in)f(the)h(setup)g(phase.)-542
   5219 31896 y Fg(\262)498 b Fo(Users)478 b(submit)g(their)g(inputs)g(to)g
   5220 (the)g(mix)h(net)f(encrypted)468 33058 y(with)422 b(ElGamal.)697
   5221 b(Let)422 b Fj(m)10624 33169 y Fi(i)11397 33058 y Fo(b)28
   5222 b(e)422 b(the)g(input)f(of)g(user)h Fj(U)21718 33169
   5223 y Fi(i)22069 33058 y Fo(.)696 b(F)-85 b(or)468 34221
   5224 y(simplicit)-28 b(y)-85 b(,)381 b(w)-28 b(e)370 b(assume)i(that)f
   5225 Fj(m)13631 34332 y Fi(i)14316 34221 y Fg(2)334 b Fe(G)p
   5226 Fo(.)544 b(User)371 b Fj(U)20031 34332 y Fi(i)20753 34221
   5227 y Fo(encrypts)468 35383 y Fj(m)1370 35494 y Fi(i)2123
   5228 35383 y Fo(and)401 b(p)28 b(osts)402 b(the)g(resulting)g(ciphertext)h
   5229 (\()p Fj(g)18842 34960 y Fi(r)19304 35383 y Fj(;)172
   5230 b(m)20662 35494 y Fi(i)21281 35383 y Fg(\242)268 b Fj(y)22373
   5231 34960 y Fi(r)22835 35383 y Fo(\))401 b(to)468 36545 y(the)439
   5232 b(mix)h(net's)f(bulletin)h(b)28 b(oard.)747 b(Users)438
   5233 b(m)-28 b(ust)439 b(also)f(pro)-28 b(v)g(e)468 37708
   5234 y(kno)g(wledge)342 b(of)f Fj(m)7486 37819 y Fi(i)8179
   5235 37708 y Fo(\(see)h(section)g(2.1\))-542 39822 y Fg(\262)498
   5236 b Fo(The)349 b(mix)i(serv)-28 b(ers)349 b(agree)h(on)f(a)g(securit)-28
   5237 b(y)350 b(parameter)f Fj(\256)299 b(>)f Fo(0,)468 40985
   5238 y(where)348 b Fj(\256)g Fo(is)g(a)f(small)i(in)-28 b(teger)348
   5239 b(\(sa)-28 b(y)-85 b(,)349 b Fj(\256)296 b Fg(\267)f
   5240 Fo(5\).)474 b(Higher)348 b(v)-57 b(alues)468 42147 y(of)479
   5241 b Fj(\256)g Fo(pro)-28 b(vide)479 b(a)g(stronger)f(guaran)-28
   5242 b(tee)478 b(of)h(correct)f(mixing)468 43309 y(but)432
   5243 b(o\256er)h(less)g(priv)-57 b(acy)433 b(to)f(the)h(users.)728
   5244 b(W)-85 b(e)432 b(examine)i(this)468 44472 y(trade-o\256)342
   5245 b(in)g(detail)g(in)g(the)g(next)g(section.)-2022 46509
   5246 y Fn(Re-randomization)396 b(and)d(Mixing.)-2022 47671
   5247 y Fo(The)338 b(\257rst)g(mix)i(serv)-28 b(er)339 b(reads)f(users')h
   5248 (input)g(ciphertexts)h(from)e(the)-2022 48833 y(bulletin)504
   5249 b(b)28 b(oard,)544 b(re-randomizes)503 b(the)h(ciphertexts,)545
   5250 b(and)503 b(writes)-2022 49996 y(them)391 b(bac)-28 b(k)390
   5251 b(to)g(the)h(bulletin)h(b)28 b(oard)390 b(in)g(random)g(order.)602
   5252 b(One)391 b(b)-28 b(y)-2022 51158 y(one)364 b(all)h(other)f(mix)i(serv)
   5253 -28 b(ers)363 b(p)28 b(erform)365 b(the)f(same)h(op)28
   5254 b(eration.)525 b(The)-2022 52320 y(output)365 b(written)g(to)h(the)g
   5255 (bulletin)g(b)28 b(oard)365 b(b)-28 b(y)365 b(one)h(mix)g(serv)-28
   5256 b(er)366 b(b)28 b(e-)-2022 53483 y(comes)275 b(the)h(input)f(to)g(the)h
   5257 (next)f(mix)i(serv)-28 b(er,)288 b(un)-28 b(til)275 b(eac)-28
   5258 b(h)275 b(serv)-28 b(er)275 b(has)-2022 54645 y(p)28
   5259 b(erformed)341 b(the)h(follo)-28 b(wing)341 b(mix)i(step)e(exactly)j
   5260 (once:)-826 56682 y(1.)498 b(Mix)341 b(serv)-28 b(er)340
   5261 b Fj(M)6528 56793 y Fi(j)7302 56682 y Fo(reads)g(as)f(inputs)h
   5262 Fj(n)g Fo(ElGamal)h(ciphertexts)468 57844 y Fj(C)1199
   5263 57955 y Fi(i)1835 57844 y Fo(=)285 b(\()p Fj(g)3838 57421
   5264 y Fi(r)4223 57561 y Fb(i)4629 57844 y Fj(;)171 b(m)5986
   5265 57955 y Fi(i)6565 57844 y Fg(\242)228 b Fj(y)7617 57421
   5266 y Fi(r)8002 57561 y Fb(i)8408 57844 y Fo(\))342 b(from)f(the)g
   5267 (bulletin)i(b)28 b(oard.)-826 59959 y(2.)498 b Fj(M)1457
   5268 60070 y Fi(j)2302 59959 y Fo(re-randomizes)412 b(these)f(ciphertexts)i
   5269 (to)e(pro)28 b(duce)411 b Fj(C)23050 59536 y Fr(0)22979
   5270 60144 y Fi(i)23748 59959 y Fo(=)468 61319 y(\()p Fj(g)1390
   5271 60896 y Fi(r)1797 60617 y Fh(0)1775 61117 y Fb(i)2182
   5272 61319 y Fj(;)171 b(m)3539 60896 y Fr(0)3539 61504 y Fi(i)4118
   5273 61319 y Fg(\242)228 b Fj(y)5170 60896 y Fi(r)5577 60617
   5274 y Fh(0)5555 61117 y Fb(i)5961 61319 y Fo(\))-826 63434
   5275 y(3.)498 b Fj(M)1457 63545 y Fi(j)2131 63434 y Fo(outputs)241
   5276 b(these)g(new)f(ciphertexts)i(to)e(the)h(bulletin)h(b)28
   5277 b(oard)468 64596 y(in)283 b(random)g(order:)425 b Fj(C)9152
   5278 64173 y Fr(0)9081 64839 y Fi(')9620 64979 y Fb(j)10034
   5279 64839 y Fd(\()p Fi(i)p Fd(\))11025 64596 y Fo(,)295 b(where)283
   5280 b Fj(')15176 64707 y Fi(j)15892 64596 y Fo(is)g(a)g(random)f(p)28
   5281 b(erm)-28 b(u-)468 65890 y(tation)422 b(on)f Fj(n)g Fo(elemen)-28
   5282 b(ts)423 b(c)-28 b(hosen)421 b(b)-28 b(y)421 b(mix)h(serv)-28
   5283 b(er)421 b Fj(M)21368 66001 y Fi(j)21802 65890 y Fo(.)695
   5284 b(Mix)468 67052 y(serv)-28 b(er)283 b Fj(M)4368 67163
   5285 y Fi(j)5085 67052 y Fo(is)g(required)h(to)f(remem)-28
   5286 b(b)28 b(er)284 b(the)g(p)28 b(erm)-28 b(utation)283
   5287 b Fj(')24111 67163 y Fi(j)468 68214 y Fo(and)363 b(the)g
   5288 (re-randomization)h(factors)e Fj(r)16104 67791 y Fr(0)16076
   5289 68399 y Fi(i)16790 68214 y Fo(un)-28 b(til)364 b(the)f(v)-28
   5290 b(eri\257ca-)468 69376 y(tion)276 b(step)f(whic)-28 b(h)275
   5291 b(w)-28 b(e)274 b(describ)28 b(e)276 b(next)h(is)e(complete.)435
   5292 b(The)275 b(p)28 b(er-)468 70539 y(m)-28 b(utation)385
   5293 b Fj(')5589 70650 y Fi(j)6406 70539 y Fo(and)f(the)h(re-randomization)g
   5294 (factors)f(should)468 71701 y(of)341 b(course)g(b)28
   5295 b(e)343 b(k)-28 b(ept)341 b(secret.)27201 -915 y Fn(V)-98
   5296 b(eri\257cation.)27201 247 y Fo(Mix)491 b(serv)-28 b(ers)490
   5297 b(are)f(not)h(allo)-28 b(w)g(ed)490 b(to)g(ab)28 b(ort)489
   5298 b(at)h(an)-28 b(y)490 b(time)h(during)27201 1409 y(the)351
   5299 b(v)-28 b(eri\257cation.)484 b(A)350 b(mix)i(serv)-28
   5300 b(er)350 b(that)h(do)28 b(es)351 b(ab)28 b(ort)350 b(is)h(accused)g(of)
   5301 27201 2571 y(c)-28 b(heating.)28198 3734 y(Before)232
   5302 b(v)-28 b(eri\257cation)233 b(starts,)254 b(all)233 b(serv)-28
   5303 b(ers)232 b(join)-28 b(tly)233 b(generate)g(a)f(ran-)27201
   5304 4896 y(dom)413 b(string)g Fj(r)441 b Fo(whic)-28 b(h)412
   5305 b(will)h(b)28 b(e)414 b(used)f(to)f(generate)i(random)e(c)-28
   5306 b(hal-)27201 6058 y(lenges.)434 b(The)271 b(string)g
   5307 Fj(r)300 b Fo(is)271 b(generated)i(as)e(follo)-28 b(ws.)432
   5308 b(Eac)-28 b(h)270 b(mix)j(serv)-28 b(er)27201 7221 y
   5309 Fj(M)28190 7332 y Fi(j)29063 7221 y Fo(selects)441 b(a)e(random)g
   5310 (string)g Fj(r)40634 7332 y Fi(j)41507 7221 y Fo(and)g(commits)h(to)f
   5311 Fj(r)49606 7332 y Fi(j)50479 7221 y Fo(using)g(a)27201
   5312 8383 y(non-malleable)464 b(commitmen)-28 b(t)462 b(sc)-28
   5313 b(heme)462 b([8].)816 b(After)461 b(all)h(commit-)27201
   5314 9545 y(men)-28 b(ts)385 b(are)e(receiv)-28 b(ed,)397
   5315 b(they)385 b(are)f(op)28 b(ened.)584 b(The)384 b(random)f(string)h
   5316 Fj(r)27201 10708 y Fo(is)342 b(computed)g(as)f Fj(r)313
   5317 b Fo(=)285 b Fg(\251)36862 10819 y Fi(j)37296 10708 y
   5318 Fj(r)37758 10819 y Fi(j)38191 10708 y Fo(.)28198 11870
   5319 y(Next,)262 b(eac)-28 b(h)241 b(serv)-28 b(er)241 b(in)h(turn)f(m)-28
   5320 b(ust)241 b(pro)-28 b(v)g(e)240 b(that)i(it)g(re-randomized)27201
   5321 13032 y(and)384 b(mixed)i(the)f(ciphertexts)h(correctly)-85
   5322 b(.)584 b(The)385 b(follo)-28 b(wing)384 b(6)g(steps)27201
   5323 14195 y(are)425 b(rep)28 b(eated)425 b(individually)i(for)c(eac)-28
   5324 b(h)424 b(serv)-28 b(er.)705 b(The)424 b(v)-28 b(eri\257cation)27201
   5325 15357 y(step)382 b(for)e(serv)-28 b(er)381 b Fj(M)35013
   5326 15468 y Fi(j)35827 15357 y Fo(pro)28 b(ceeds)382 b(as)e(follo)-28
   5327 b(ws.)574 b(As)381 b(ab)28 b(o)-28 b(v)g(e,)391 b(let)382
   5328 b Fj(C)52270 15468 y Fi(i)52972 15357 y Fo(=)27201 16519
   5329 y(\()p Fj(g)28123 16096 y Fi(r)28508 16236 y Fb(i)28915
   5330 16519 y Fj(;)171 b(m)30272 16630 y Fi(i)30922 16519 y
   5331 Fg(\242)299 b Fj(y)32045 16096 y Fi(r)32430 16236 y Fb(i)32836
   5332 16519 y Fo(\))448 b(for)f(1)463 b Fg(\267)f Fj(i)h Fg(\267)f
   5333 Fj(n)448 b Fo(denote)g(the)h(input)f(ciphertexts)27201
   5334 17681 y(receiv)-28 b(ed)505 b(b)-28 b(y)503 b(serv)-28
   5335 b(er)503 b Fj(M)36996 17792 y Fi(j)37429 17681 y Fo(.)942
   5336 b(\(W)-85 b(e)503 b(omit)h(the)g(subscript)f Fj(j)559
   5337 b Fo(in)504 b(the)27201 18844 y(notation)328 b(of)f Fj(C)33166
   5338 18955 y Fi(i)33844 18844 y Fo(for)g(clarit)-28 b(y\).)451
   5339 b(Let)328 b Fj(C)41999 18421 y Fr(0)41928 19087 y Fi(')42467
   5340 19227 y Fb(j)42881 19087 y Fd(\()p Fi(i)p Fd(\))44199
   5341 18844 y Fo(b)28 b(e)328 b(the)f(set)h(of)f(outputs,)27201
   5342 20455 y(where)341 b Fj(C)30964 20032 y Fr(0)30893 20640
   5343 y Fi(i)31547 20455 y Fo(=)285 b(\()p Fj(g)33550 20032
   5344 y Fi(r)33957 19753 y Fh(0)33935 20253 y Fb(i)34341 20455
   5345 y Fj(;)171 b(m)35698 20032 y Fr(0)35698 20640 y Fi(i)36277
   5346 20455 y Fg(\242)228 b Fj(y)37329 20032 y Fi(r)37736 19753
   5347 y Fh(0)37714 20253 y Fb(i)38120 20455 y Fo(\).)28397
   5348 22424 y(1.)499 b(W)-85 b(e)354 b(\257rst)f(v)-28 b(erify)355
   5349 b(that)f(all)g Fj(C)40920 22001 y Fr(0)40849 22609 y
   5350 Fi(i)41572 22424 y Fo(are)g(prop)28 b(erly)354 b(formatted,)k(i.e.)
   5351 29692 23587 y(ev)-28 b(ery)409 b Fj(C)33266 23163 y Fr(0)33195
   5352 23772 y Fi(i)33973 23587 y Fo(consists)f(of)h(a)f(pair)g(\()p
   5353 Fj(s;)172 b(t)p Fo(\))396 b Fg(2)g Fe(G)46488 23163 y
   5354 Fd(2)46949 23587 y Fo(.)657 b(Observ)-28 b(e)409 b(that)29692
   5355 24749 y(this)371 b(can)f(b)28 b(e)371 b(done)f(e\261cien)-28
   5356 b(tly)-85 b(.)544 b(If)370 b Fe(G)g Fo(is)g(a)h(subgroup)e(of)h
   5357 Fe(Z)53002 24326 y Fr(\244)53002 24924 y Fi(p)53484 24749
   5358 y Fo(,)29692 25911 y(the)406 b(computational)h(cost)f(to)g(v)-28
   5359 b(erify)406 b(that)g(an)g(elemen)-28 b(t)407 b(b)28 b(e-)29692
   5360 27074 y(longs)382 b(to)g Fe(G)f Fo(is)h(one)g(exp)28
   5361 b(onen)-28 b(tiation,)394 b(but)382 b(the)g(v)-28 b(eri\257cation)29692
   5362 28236 y(for)371 b(all)h Fj(C)33543 27813 y Fr(0)33472
   5363 28421 y Fi(i)34213 28236 y Fo(can)g(b)28 b(e)372 b(batc)-28
   5364 b(hed)371 b(\(see)i([4]\),)381 b(resulting)372 b(in)g(a)f(cost)29692
   5365 29398 y(of)363 b(a)g(single)i(exp)28 b(onen)-28 b(tiation)365
   5366 b(to)e(v)-28 b(erify)364 b(all)g Fj(C)48007 28975 y Fr(0)47936
   5367 29583 y Fi(i)48305 29398 y Fo(.)521 b(If)363 b Fe(G)g
   5368 Fo(is)h(the)29692 30561 y(group)430 b(of)h(p)28 b(oin)-28
   5369 b(ts)431 b(of)g(an)g(elliptic)j(curv)-28 b(e)430 b(o)-28
   5370 b(v)g(er)431 b Fe(F)49037 30672 y Fi(p)49950 30561 y
   5371 Fo(of)g(prime)29692 31723 y(order)355 b Fj(q)36 b Fo(,)359
   5372 b(the)d(computational)h(cost)e(to)h(v)-28 b(erify)357
   5373 b(that)e(a)h(p)28 b(oin)-28 b(t)29692 32885 y(is)342
   5374 b(on)f(the)h(curv)-28 b(e)341 b(is)g(one)h(squaring)f(and)g(one)h
   5375 (cubing.)28397 34865 y(2.)499 b(Using)291 b(the)g(Chaum-P)-28
   5376 b(edersen)290 b(proto)28 b(col)291 b(\(prop)28 b(osition)291
   5377 b(4.2\),)29692 36027 y(mix)342 b(serv)-28 b(er)342 b
   5378 Fj(M)35670 36138 y Fi(j)36445 36027 y Fo(pro)-28 b(v)g(es)340
   5379 b(that)41829 35280 y Fc(Q)42796 35554 y Fi(n)42796 36332
   5380 y(i)p Fd(=1)44345 36027 y Fj(m)45247 36138 y Fi(i)45882
   5381 36027 y Fo(=)46963 35280 y Fc(Q)47930 35554 y Fi(n)47930
   5382 36332 y(i)p Fd(=1)49479 36027 y Fj(m)50381 35604 y Fr(0)50381
   5383 36212 y Fi(i)50733 36027 y Fo(.)28397 38007 y(3.)499
   5384 b(All)228 b(mix)h(serv)-28 b(ers)227 b(collab)28 b(orate)229
   5385 b(to)f(generate)g Fj(\256)g Fo(sets)g Fj(S)50094 38118
   5386 y Fd(1)50555 38007 y Fj(;)171 b(:)g(:)g(:)g(;)h(S)53457
   5387 38118 y Fi(\256)54046 38007 y Fo(,)29692 39170 y(where)405
   5388 b(eac)-28 b(h)406 b(set)g Fj(S)37375 39281 y Fi(i)38131
   5389 39170 y Fo(is)g(a)f(subset)h(of)f Fg(f)p Fo(1)p Fj(;)171
   5390 b(:)g(:)g(:)g(;)g(n)p Fg(g)p Fo(.)648 b(The)405 b(sets)29692
   5391 40332 y Fj(S)30318 40443 y Fi(i)31052 40332 y Fo(are)383
   5392 b(generated)h(indep)28 b(enden)-28 b(tly)385 b(of)e(one)g(another)g(in)
   5393 h(the)29692 41494 y(follo)-28 b(wing)369 b(manner.)537
   5394 b(Ev)-28 b(ery)369 b(index)g(1)330 b Fg(\267)h Fj(k)359
   5395 b Fg(\267)330 b Fj(n)369 b Fo(is)g(included)29692 42657
   5396 y(in)419 b Fj(S)31590 42768 y Fi(i)32359 42657 y Fo(indep)28
   5397 b(enden)-28 b(tly)420 b(at)e(random)g(with)g(probabilit)-28
   5398 b(y)419 b(1)p Fj(=)p Fo(2.)29692 43819 y(The)394 b(randomness)g(is)h
   5399 (deriv)-28 b(ed)395 b(from)g(the)g(random)f(string)g
   5400 Fj(r)29692 44981 y Fo(generated)306 b(join)-28 b(tly)307
   5401 b(b)-28 b(y)305 b(all)i(serv)-28 b(ers)305 b(at)h(the)g(b)28
   5402 b(eginning)307 b(of)f(the)29692 46143 y(v)-28 b(eri\257cation)422
   5403 b(step.)695 b(W)-85 b(e)421 b(examine)i(in)f(more)f(detail)h(ho)-28
   5404 b(w)420 b(to)29692 47306 y(generate)342 b(the)g(subsets)f
   5405 Fj(S)39725 47417 y Fi(i)40417 47306 y Fo(at)h(the)g(end)f(of)g(this)h
   5406 (section.)28397 49286 y(4.)499 b(The)341 b(sets)h Fj(S)34426
   5407 49397 y Fd(1)34887 49286 y Fj(;)171 b(:)g(:)g(:)g(;)g(S)37788
   5408 49397 y Fi(\256)38718 49286 y Fo(are)342 b(giv)-28 b(en)342
   5409 b(to)f(mix)i(serv)-28 b(er)341 b Fj(M)50331 49397 y Fi(j)50765
   5410 49286 y Fo(.)28397 51266 y(5.)499 b(Mix)407 b(serv)-28
   5411 b(er)406 b Fj(M)35884 51377 y Fi(j)36723 51266 y Fo(m)-28
   5412 b(ust)406 b(pro)28 b(duce)406 b Fj(\256)h Fo(subsets)e
   5413 Fj(S)48643 50842 y Fr(0)48586 51441 y Fd(1)49047 51266
   5414 y Fj(;)171 b(:)g(:)g(:)g(;)h(S)52006 50842 y Fr(0)51949
   5415 51441 y Fi(\256)52944 51266 y Fo(of)29692 52428 y Fg(f)p
   5416 Fo(1)p Fj(;)f(:)g(:)g(:)g(;)g(n)p Fg(g)512 b Fo(suc)-28
   5417 b(h)512 b(that)h(for)e(all)j(1)569 b Fg(\267)h Fj(i)g
   5418 Fg(\267)g Fj(\256)514 b Fg(j)p Fj(S)49648 52539 y Fi(i)49999
   5419 52428 y Fg(j)570 b Fo(=)g Fg(j)p Fj(S)53186 52005 y Fr(0)53129
   5420 52613 y Fi(i)53484 52428 y Fg(j)29692 53590 y Fo(and)31824
   5421 52843 y Fc(Q)32791 53895 y Fi(k)14 b Fr(2)p Fi(S)34300
   5422 54035 y Fb(i)34877 53590 y Fj(m)35779 53708 y Fi(k)36807
   5423 53590 y Fo(=)38124 52843 y Fc(Q)39091 53895 y Fi(k)g
   5424 Fr(2)p Fi(S)40635 53666 y Fh(0)40600 54164 y Fb(i)41176
   5425 53590 y Fj(m)42078 53167 y Fr(0)42078 53796 y Fi(k)42586
   5426 53590 y Fo(.)880 b(This)482 b(pro)28 b(duct)483 b(equalit)-28
   5427 b(y)29692 54966 y(is)342 b(pro)-28 b(v)g(ed)340 b(using)h(the)h
   5428 (Chaum-P)-28 b(edersen)341 b(proto)28 b(col.)28397 56946
   5429 y(6.)499 b(If)228 b(the)h(mix)h(serv)-28 b(er)228 b(fails)i(in)f(step)g
   5430 (5,)251 b(it)229 b(is)g(accused)g(of)g(c)-28 b(heating.)29692
   5431 58108 y(The)293 b Fm(de)-52 b(cryption)372 b Fo(serv)-28
   5432 b(ers)293 b(are)h(then)g(called)h(up)28 b(on)293 b(to)h(insp)28
   5433 b(ect)29692 59271 y(the)349 b(transcript)f(of)g(the)h(v)-28
   5434 b(eri\257cation)350 b(\(steps)f(3,)h(4)f(and)f(5\))h(on)29692
   5435 60433 y(the)496 b(bulletin)g(b)28 b(oard.)917 b(If)494
   5436 b(c)-28 b(heating)496 b(is)f(con\257rmed)g(b)-28 b(y)495
   5437 b(the)29692 61595 y(decryption)374 b(serv)-28 b(ers,)381
   5438 b(the)374 b(c)-28 b(heating)374 b(mix)h(serv)-28 b(er)373
   5439 b(is)h(banned)29692 62758 y(from)435 b(an)-28 b(y)435
   5440 b(future)f(mixing.)739 b(In)434 b(this)i(case,)459 b(the)436
   5441 b(remaining)29692 63920 y(honest)406 b(serv)-28 b(ers)406
   5442 b(restart)f(the)i(whole)f(mixing)h(from)f(the)g(b)28
   5443 b(e-)29692 65082 y(ginning)332 b(using)g(the)g(original)h(ciphertext)g
   5444 (inputs)f(p)28 b(osted)333 b(b)-28 b(y)29692 66245 y(users)341
   5445 b(to)g(the)h(bulletin)h(b)28 b(oard.)27201 68214 y Fn(Decryption.)27201
   5446 69377 y Fo(The)478 b(mix)i(net)-28 b(w)g(ork)477 b(pro)28
   5447 b(ceeds)479 b(to)f(the)h(decryption)g(step)f(only)h(if)27201
   5448 70539 y(the)332 b(v)-28 b(eri\257cation)331 b(step)g(did)g(not)g(exp)28
   5449 b(ose)332 b(an)-28 b(y)331 b(c)-28 b(heating)331 b(serv)-28
   5450 b(ers.)452 b(A)27201 71701 y(quorum)433 b(of)h(decryption)f(serv)-28
   5451 b(ers)433 b(join)-28 b(tly)434 b(p)28 b(erforms)433 b(a)g(threshold)p
   5452 eop
   5453 %%Page: 6 6
   5454 6 5 bop -2022 -915 a Fo(decryption)439 b(of)g(the)h(\257nal)f(output)g
   5455 (ciphertexts,)465 b(and)439 b(pro)-28 b(vides)439 b(a)-2022
   5456 247 y(zero-kno)-28 b(wledge)342 b(pro)28 b(of)341 b(of)g(correctness)g
   5457 (for)g(decryption.)-2022 2571 y Fn(P)-33 b(erfectly-correct)431
   5458 b(pro)33 b(of)p Fo(.)550 b(If)372 b(a)g(pro)28 b(of)372
   5459 b(that)g(the)h(mix)h(net)e(op-)-2022 3734 y(erated)387
   5460 b(p)28 b(erfectly)389 b(correctly)f(is)f(required,)399
   5461 b(w)-28 b(e)387 b(ma)-28 b(y)387 b(run)g(a)g(slo)-28
   5462 b(w)g(er)-2022 4896 y(p)28 b(erfectly-correct)362 b(v)-28
   5463 b(eri\257cation)362 b(step,)k(suc)-28 b(h)360 b(as)h(for)f(example)i
   5464 (that)-2022 6058 y(prop)28 b(osed)390 b(b)-28 b(y)391
   5465 b(Ne\256)h([20].)606 b(The)390 b(cost)i(of)f(this)g(additional)h(v)-28
   5466 b(eri\257ca-)-2022 7221 y(tion)341 b(is)h(not)f(included)h(in)g(the)g
   5467 (analysis)g(of)f(our)g(new)g(mix)h(net.)-1026 9545 y(This)414
   5468 b(completes)h(the)g(description)g(of)f(our)f(new)h(mix)i(net.)674
   5469 b(W)-85 b(e)-2022 10708 y(end)413 b(this)g(section)h(with)f(a)g
   5470 (description)g(of)g(ho)-28 b(w)412 b(to)h(generate)h(the)-2022
   5471 11870 y(sets)341 b Fj(S)606 11981 y Fd(1)1067 11870 y
   5472 Fj(;)171 b(:)g(:)g(:)g(;)g(S)3968 11981 y Fi(\256)4899
   5473 11870 y Fo(in)342 b(step)f(3)h(of)f(the)g(v)-28 b(eri\257cation)342
   5474 b(phase.)-2022 14195 y Fn(Optimization)395 b(for)f(the)f(generation)i
   5475 (of)f(c)-33 b(hallenges.)-2022 15357 y Fo(Recall)442
   5476 b(that)f(w)-28 b(e)441 b(denote)h(b)-28 b(y)442 b Fj(r)469
   5477 b Fo(the)442 b(randomness)f(join)-28 b(tly)442 b(gener-)-2022
   5478 16519 y(ated)415 b(b)-28 b(y)414 b(all)i(the)f(serv)-28
   5479 b(ers)415 b(b)28 b(efore)415 b(the)h(v)-28 b(eri\257cation)415
   5480 b(started.)676 b(Let)-2022 17681 y Fj(h)369 b Fo(:)h
   5481 Fg(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)2094 17258 y Fr(\244)2943
   5482 17681 y Fg(!)370 b(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)6840
   5483 17258 y Fd(160)8505 17681 y Fo(b)28 b(e)393 b(a)f(hash)g(function.)609
   5484 b(In)392 b(our)f(securit)-28 b(y)-2022 18844 y(analysis,)522
   5485 b(w)-28 b(e)485 b(mo)28 b(del)486 b Fj(h)f Fo(as)g(a)g(random)g
   5486 (oracle.)887 b(Let)486 b Fj(B)534 b Fo(b)28 b(e)486 b(the)-2022
   5487 20006 y(con)-28 b(ten)g(t)299 b(of)g(the)h(bulletin)g(b)28
   5488 b(oard)299 b(just)g(b)28 b(efore)300 b(the)g(sets)g Fj(S)20350
   5489 20117 y Fi(i)21000 20006 y Fo(are)f(gen-)-2022 21168
   5490 y(erated)347 b(\(i.e.)476 b(ev)-28 b(erything)348 b(that)g(has)f(b)28
   5491 b(een)349 b(p)28 b(osted)348 b(to)f(the)h(bulletin)-2022
   5492 22331 y(b)28 b(oard)459 b(up)h(to)g(that)h(p)28 b(oin)-28
   5493 b(t\).)812 b(W)-85 b(e)461 b(use)f(the)h(master)f(randomness)-2022
   5494 23493 y Fj(r)508 b Fo(and)479 b(the)h(string)f Fj(B)529
   5495 b Fo(together)480 b(with)g(the)g(hash)f(function)h Fj(h)f
   5496 Fo(to)-2022 24655 y(generate)358 b(the)h(sets)f Fj(S)6519
   5497 24766 y Fd(1)6980 24655 y Fj(;)171 b(:)g(:)g(:)g(;)g(S)9881
   5498 24766 y Fi(\256)10828 24655 y Fo(using)358 b(the)h(follo)-28
   5499 b(wing)358 b(rule:)489 b(index)-2022 25818 y Fj(k)k Fg(2)464
   5500 b(f)p Fo(1)p Fj(;)171 b(:)g(:)g(:)g(;)g(n)p Fg(g)449
   5501 b Fo(is)g(included)h(in)f Fj(S)12298 25929 y Fi(i)13098
   5502 25818 y Fo(for)g(mix)h(serv)-28 b(er)449 b Fj(M)20966
   5503 25929 y Fi(j)21849 25818 y Fo(if)g(and)-2022 26980 y(only)351
   5504 b(if)h(the)f(least)i(signi\257can)-28 b(t)351 b(bit)g(of)g
   5505 Fj(h)p Fo(\()p Fj(r)28 b Fg(jj)p Fj(B)50 b Fg(jj)p Fj(j)57
   5506 b Fg(jj)p Fj(i)p Fg(jj)p Fj(k)29 b Fo(\))355 b(is)d(1,)i(where)-2022
   5507 28142 y Fg(jj)342 b Fo(denotes)g(string)f(concatenation.)-2022
   5508 30778 y Fp(6.)1328 b(PR)-40 b(OPER)-53 b(TIES)-1026 32217
   5509 y Fo(In)312 b(this)h(section,)319 b(w)-28 b(e)312 b(examine)j(the)e
   5510 (prop)28 b(erties)313 b(of)f(our)g(new)h(mix)-2022 33379
   5511 y(net)489 b(in)h(terms)g(of)f(soundness,)526 b(e\261ciency)-85
   5512 b(,)529 b(robustness,)d(priv)-57 b(acy)-85 b(,)-2022
   5513 34541 y(correctness)341 b(and)g(\257nally)h(univ)-28
   5514 b(ersal)342 b(v)-28 b(eri\257abilit)g(y)-85 b(.)-2022
   5515 37741 y Fp(6.1)1328 b(Soundness)-915 40146 y Ff(Pr)-22
   5516 b(oposition)553 b Fo(6.1.)499 b Fn(\(Soundness\))427
   5517 b Fm(Our)f(mix)g(net)g(is)f(sound,)-2022 41308 y(in)500
   5518 b(the)i(sense)g(that)f(a)g(server)i(who)d(do)-52 b(es)501
   5519 b(not)g(deviate)h(fr)-52 b(om)501 b(the)-2022 42470 y(pr)-52
   5520 b(oto)g(c)g(ol)365 b(c)-52 b(annot)365 b(fail)g(the)h(veri\257c)-52
   5521 b(ation)366 b(step.)-915 44210 y Ff(Pr)-22 b(oof.)498
   5522 b Fo(A)366 b(mix)i(net)g(who)e(submits)i Fj(S)14706 43787
   5523 y Fr(0)15330 44210 y Fo(=)328 b Fj(')17123 44321 y Fi(j)17556
   5524 44210 y Fo(\()p Fj(S)57 b Fo(\))367 b(can)g(not)g(fail)-2022
   5525 45372 y(the)429 b(v)-28 b(eri\257cation)430 b(test.)718
   5526 b(Recall)430 b(from)f(prop)28 b(osition)429 b(2.1)g(that)g(our)-2022
   5527 46535 y(adv)-28 b(ersarial)436 b(mo)28 b(del)438 b(allo)-28
   5528 b(ws)436 b(the)i(adv)-28 b(ersary)436 b(to)g(con)-28
   5529 b(trol)437 b(at)f(most)-2022 47697 y(all)381 b(but)g(one)g(of)f(the)h
   5530 (mix)h(serv)-28 b(ers,)391 b(and)380 b(up)h(to)f(a)h(minorit)-28
   5531 b(y)381 b(of)g(de-)-2022 48859 y(cryption)322 b(serv)-28
   5532 b(ers.)448 b(The)322 b(in)-28 b(v)g(olv)g(emen)g(t)323
   5533 b(of)e(the)i(decryption)f(serv)-28 b(ers)-2022 50022
   5534 y(when)332 b(c)-28 b(heating)334 b(is)f(alleged)i(\(step)f(6)f(of)g
   5535 (the)g(v)-28 b(eri\257cation\))334 b(guaran-)-2022 51184
   5536 y(tees)343 b(soundness)f(since)h(a)f(ma)57 b(jorit)-28
   5537 b(y)342 b(of)g(them)h(is)g(honest.)458 b(\(W)-85 b(ere)343
   5538 b(it)-2022 52346 y(not)239 b(for)h(decryption)g(serv)-28
   5539 b(ers,)260 b(a)240 b(ma)57 b(jorit)-28 b(y)240 b(of)g(c)-28
   5540 b(heating)240 b(mix)h(serv)-28 b(ers)-2022 53508 y(could)341
   5541 b(evict)i(a)e(minorit)-28 b(y)342 b(of)f(honest)h(mix)g(serv)-28
   5542 b(ers.\))5409 b Fa(\244)-2022 56708 y Fp(6.2)1328 b(Ef\243ciency)-915
   5543 59113 y Ff(Pr)-22 b(oposition)553 b Fo(6.2.)499 b Fn(\(E\261ciency\))
   5544 559 b Fm(The)e(c)-52 b(ost)557 b(of)768 b Fo(mixing)559
   5545 b Fj(n)-2022 60275 y Fm(items)527 b(is)f Fo(2)p Fj(n)g
   5546 Fm(exp)-52 b(onentiations)526 b(p)-52 b(er)526 b(mix)h(server.)954
   5547 b(The)527 b(c)-52 b(ost)526 b(of)-2022 61437 y Fo(pro)-28
   5548 b(ving)326 b Fm(that)f(the)i(mixing)f(is)h(almost)g(entir)-52
   5549 b(ely)326 b(c)-52 b(orr)g(e)g(ct)328 b(is)e Fo(2)p Fj(\256)q
   5550 Fo(\(2)p Fj(k)171 b Fg(\241)-2022 62600 y Fo(1\))335
   5551 b Fm(exp)-52 b(onentiations)334 b(p)-52 b(er)336 b(mix)g(server)h(and)e
   5552 (the)h(c)-52 b(ost)335 b(of)g(de)-52 b(crypting)-2022
   5553 63762 y Fj(n)405 b Fm(outputs)h(is)g Fo(\(2)257 b(+)g(4)p
   5554 Fj(k)29 b Fo(\))p Fj(n)p Fm(,)416 b(wher)-52 b(e)406
   5555 b Fj(k)436 b Fm(is)406 b(the)g(total)g(numb)-52 b(er)407
   5556 b(of)f(mix)-2022 64924 y(servers.)-1026 67052 y Fo(As)462
   5557 b(discussed)g(in)g(section)h(3,)493 b(our)461 b(mix)i(has)f(the)g(lo)
   5558 -28 b(w)g(est)462 b(total)-2022 68214 y(computational)373
   5559 b(o)-28 b(v)g(erhead)372 b(to)g(mix)h Fj(n)f Fo(inputs.)548
   5560 b(In)372 b(particular)g(the)-2022 69377 y(n)-28 b(um)g(b)28
   5561 b(er)238 b(of)h(exp)28 b(onen)-28 b(tiations)241 b(required)e(to)h(pro)
   5562 -28 b(v)g(e)238 b(that)h(mixing)i(has)-2022 70539 y(b)28
   5563 b(een)267 b(done)g(correctly)g(is)f(a)h(constan)-28 b(t)266
   5564 b(indep)28 b(enden)-28 b(t)267 b(of)f(the)h(n)-28 b(um)g(b)28
   5565 b(er)-2022 71701 y(of)341 b(inputs.)27201 -915 y Fp(6.3)1329
   5566 b(Rob)-27 b(ustness)28308 1543 y Ff(Pr)-22 b(oposition)554
   5567 b Fo(6.3.)499 b Fn(\(Robustness\))263 b Fm(Like)g(any)d(r)-52
   5568 b(e-encryption)27201 2705 y(mix)308 b(network,)319 b(our)308
   5569 b(c)-52 b(onstruction)307 b(pr)-52 b(o)g(duc)g(es)308
   5570 b(an)f(output)g(as)g(long)h(as)27201 3868 y(a)412 b(quorum)h(of)f(de)
   5571 -52 b(cryption)411 b(servers)k(is)d(available)h(to)f(pr)-52
   5572 b(o)g(c)g(e)g(e)g(d)412 b(with)27201 5030 y(the)366 b(de)-52
   5573 b(cryption)365 b(phase.)27201 7212 y Fp(6.4)1329 b(Pri)-13
   5574 b(v)g(acy)28308 9671 y Ff(Pr)-22 b(oposition)554 b Fo(6.4.)499
   5575 b Fn(\(Priv)-65 b(acy\))263 b Fm(Every)f(input)e(is)h(hidden)g(among)
   5576 27201 10833 y Fj(n=)p Fo(2)28843 10410 y Fi(\256)29799
   5577 10833 y Fm(outputs)364 b(on)i(aver)-52 b(age.)28308 13015
   5578 y Ff(Pr)-22 b(oof.)498 b Fo(In)304 b(the)h(v)-28 b(eri\257cation)305
   5579 b(step,)312 b(eac)-28 b(h)305 b(mix)g(serv)-28 b(er)304
   5580 b(m)-28 b(ust)304 b(re-)27201 14177 y(v)-28 b(eal)234
   5581 b(the)g(image)g(b)-28 b(y)233 b(his)g(secret)h(p)28 b(erm)-28
   5582 b(utation)234 b(of)f Fj(\256)h Fo(sets)g Fj(S)49533 14288
   5583 y Fd(1)49993 14177 y Fj(;)171 b(:)h(:)f(:)g(;)g(S)52895
   5584 14288 y Fi(\256)53484 14177 y Fo(,)27201 15339 y(eac)-28
   5585 b(h)384 b(of)g(size)i(on)e(a)-28 b(v)g(erage)384 b Fj(n=)p
   5586 Fo(2.)583 b(Ev)-28 b(ery)385 b(input)f(b)28 b(elongs)385
   5587 b(either)g(to)27201 16502 y Fj(S)27827 16613 y Fi(i)28543
   5588 16502 y Fo(or)365 b(to)h(the)g(complemen)-28 b(t)366
   5589 b(of)g Fj(S)40443 16613 y Fi(i)40793 16502 y Fo(,)372
   5590 b(and)365 b(th)-28 b(us)365 b(the)h(corresp)28 b(onding)27201
   5591 17664 y(output)459 b(b)28 b(elongs)459 b(either)g(to)g(the)f(image)i
   5592 (of)e Fj(S)45722 17775 y Fi(i)46531 17664 y Fo(or)g(to)g(the)h(image)
   5593 27201 18826 y(of)368 b(the)g(complemen)-28 b(t)369 b(of)f
   5594 Fj(S)37747 18937 y Fi(i)38097 18826 y Fo(.)535 b(The)368
   5595 b(in)-28 b(tersection)368 b(of)g(the)g(images)h(of)27201
   5596 19989 y Fj(\256)237 b Fo(sets)f Fj(S)30617 20100 y Fi(i)31203
   5597 19989 y Fo(\(or)f(their)h(complemen)-28 b(ts\))237 b(is)f(on)f(a)-28
   5598 b(v)g(erage)235 b(of)h(size)g Fj(n=)p Fo(2)51849 19565
   5599 y Fi(\256)52438 19989 y Fo(.)250 b Fa(\244)28198 22313
   5600 y Fo(This)338 b(is)i(the)g(minim)-28 b(um)340 b(priv)-57
   5601 b(acy)340 b(guaran)-28 b(teed)339 b(b)-28 b(y)339 b(our)f(mix)j(net-)
   5602 27201 23475 y(w)-28 b(ork,)337 b(giv)-28 b(en)336 b(that)g(w)-28
   5603 b(e)335 b(consider)h(an)f(adv)-28 b(ersarial)336 b(mo)28
   5604 b(del)337 b(in)f(whic)-28 b(h)27201 24638 y(all)350 b(but)f(one)h(of)f
   5605 (the)h(mix)g(serv)-28 b(ers)349 b(ma)-28 b(y)349 b(b)28
   5606 b(e)350 b(con)-28 b(trolled)350 b(b)-28 b(y)349 b(the)g(ad-)27201
   5607 25800 y(v)-28 b(ersary)-85 b(.)439 b(If)292 b(w)-28 b(e)292
   5608 b(adopt)h(a)g(w)-28 b(eak)g(er)292 b(adv)-28 b(ersarial)293
   5609 b(mo)28 b(del)294 b(and)e(assume)27201 26962 y(that)324
   5610 b(a)f(ma)57 b(jorit)-28 b(y)323 b(of)f(mix)j(serv)-28
   5611 b(ers)322 b(are)h(honest,)328 b(w)-28 b(e)322 b(can)h(adapt)g(the)27201
   5612 28125 y(tec)-28 b(hniques)391 b(of)f([18])h(to)g(our)e(mix)i(net)-28
   5613 b(w)g(ork)389 b(to)i(guaran)-28 b(tee)390 b(p)28 b(erfect)27201
   5614 29287 y(priv)-57 b(acy)342 b(for)f(all)h(the)g(inputs)f(with)h(o)-28
   5615 b(v)g(erwhelming)341 b(probabilit)-28 b(y)-85 b(.)27201
   5616 31336 y Fp(6.5)1329 b(Corr)-24 b(ectness)28308 33794
   5617 y Ff(Pr)i(oposition)554 b Fo(6.5.)499 b Fn(\(Almost)264
   5618 b(En)-33 b(tirely)263 b(Correct)h(Mixing\))27201 34957
   5619 y Fm(If)399 b(the)g(set)g(of)f(outputs)g(pr)-52 b(o)g(duc)g(e)g(d)397
   5620 b(by)i(a)f(mix)h(server)i(is)e(not)f(a)g(p)-52 b(er-)27201
   5621 36119 y(mutation)443 b(of)g(the)g(inputs,)462 b(then)443
   5622 b(che)-52 b(ating)444 b(wil)52 b(l)444 b(b)-52 b(e)444
   5623 b(dete)-52 b(cte)g(d)444 b(with)27201 37281 y(pr)-52
   5624 b(ob)g(ability)394 b Fo(1)249 b Fg(\241)g Fo(\(5)p Fj(=)p
   5625 Fo(8\))36295 36858 y Fi(\256)36885 37281 y Fm(,)402 b(or)395
   5626 b(the)g(discr)-52 b(ete)396 b(lo)-52 b(garithm)395 b(pr)-52
   5627 b(oblem)396 b(in)27201 38443 y Fe(G)366 b Fm(c)-52 b(an)366
   5628 b(b)-52 b(e)367 b(solve)-52 b(d)366 b(in)g(p)-52 b(olynomial)365
   5629 b(time.)28198 40625 y Fo(The)442 b(pro)28 b(of)443 b(of)g(almost)h(en)
   5630 -28 b(tirely)444 b(correct)f(mixing)h(is)f(fairly)h(in-)27201
   5631 41788 y(v)-28 b(olv)g(ed)416 b(and)e(is)h(giv)-28 b(en)416
   5632 b(in)f(the)g(next)h(section.)677 b(Let)415 b(us)g(consider)f(a)27201
   5633 42950 y(concrete)426 b(example.)710 b(Consider)425 b(an)g(election)i
   5634 (with)f(160)p Fj(;)171 b Fo(000)425 b(bal-)27201 44112
   5635 y(lots.)905 b(A)491 b(securit)-28 b(y)491 b(parameter)g
   5636 Fj(\256)535 b Fo(=)f(6)491 b(guaran)-28 b(tees)490 b(that)i(ev)-28
   5637 b(ery)27201 45275 y(individual)374 b(ballot)g(is)f(hidden)f(among)h(2)p
   5638 Fj(;)171 b Fo(500)372 b(others.)549 b(By)373 b(prop)28
   5639 b(o-)27201 46437 y(sition)353 b(6.5,)j(the)d(probabilit)-28
   5640 b(y)352 b(that)g(the)h(output)f(set)h(computed)g(b)-28
   5641 b(y)27201 47599 y(the)291 b(mix)g(net)-28 b(w)g(ork)289
   5642 b(is)h(a)g(p)28 b(erm)-28 b(utation)291 b(of)e(the)i(inputs)f(is)g
   5643 (more)h(than)27201 48761 y(94\045.)27201 50810 y Fp(6.6)1329
   5644 b(Uni)-13 b(v)g(ersal)331 b(V)-133 b(eri\243ability)28198
   5645 52249 y Fo(Our)337 b(mix)i(o\256ers)g(no)e(guaran)-28
   5646 b(tee)338 b(of)g(univ)-28 b(ersal)338 b(v)-28 b(eri\257abilit)g(y)-85
   5647 b(.)456 b(W)-85 b(e)27201 53411 y(ha)-28 b(v)g(e)278
   5648 b(already)g(noted)g(that)g(a)g(slo)-28 b(w)g(er)276 b(pro)28
   5649 b(of)277 b(of)h(p)28 b(erfect)278 b(correctness)27201
   5650 54574 y(should)c(b)28 b(e)274 b(executed)h(in)f(parallel)g(with)f(our)g
   5651 (pro)28 b(of)273 b(and)g(w)-28 b(e)273 b(assume)27201
   5652 55736 y(that)380 b(univ)-28 b(ersal)380 b(v)-28 b(eri\257abilit)g(y)-85
   5653 b(,)390 b(if)380 b(required,)390 b(will)380 b(come)g(from)f(that)27201
   5654 56898 y(slo)-28 b(w)g(er)420 b(pro)28 b(of.)692 b(W)-85
   5655 b(e)421 b(note)f(that)h(while)g(it)g(is)g(imp)28 b(ortan)-28
   5656 b(t)420 b(that)h(the)27201 58061 y(results)242 b(of)f(the)h(election)h
   5657 (b)28 b(e)243 b(a)-28 b(v)-57 b(ailable)243 b(instan)-28
   5658 b(tly)242 b(\(with)f(our)g(pro)28 b(of)78 b(\),)27201
   5659 59223 y(it)267 b(is)g(acceptable)g(to)f(w)-28 b(ait)266
   5660 b(longer)g(\(a)h(da)-28 b(y\))266 b(for)f(a)h(pro)28
   5661 b(of)266 b(of)g(univ)-28 b(ersal)27201 60385 y(v)g(eri\257abilit)g(y)
   5662 343 b(\(with)f(another,)f(slo)-28 b(w)g(er)340 b(pro)28
   5663 b(of)78 b(\).)27201 63044 y Fp(7.)1329 b(PR)-40 b(OOF)359
   5664 b(OF)f(CORRECTNESS)g(\(PR)-40 b(OPOSI\255)29526 64593
   5665 y(TION)332 b(6.5\))28198 66032 y Fo(In)388 b(this)i(section)f(w)-28
   5666 b(e)389 b(pro)-28 b(v)g(e)388 b(Prop)28 b(osition)390
   5667 b(6.5)f(\(almost)h(en)-28 b(tirely)27201 67195 y(correct)481
   5668 b(mixing\).)875 b(Throughout)479 b(this)i(section,)517
   5669 b(w)-28 b(e)480 b(let)i Fe(G)e Fo(b)28 b(e)481 b(a)27201
   5670 68357 y(group)341 b(of)g(prime)h(order)e Fj(q)377 b Fo(and)341
   5671 b Fe(Z)40343 68468 y Fi(q)41086 68357 y Fo(=)285 b Fg(f)p
   5672 Fo(0)p Fj(;)171 b(:)g(:)g(:)g(;)g(q)263 b Fg(\241)228
   5673 b Fo(1)p Fg(g)p Fo(.)28308 70539 y Ff(Theorem)553 b Fo(7.1.)499
   5674 b Fm(L)-52 b(et)289 b Fo(0)284 b Fj(<)h(\262)f(<)40913
   5675 70131 y Fd(1)p 40913 70311 406 43 v 40913 70877 a(2)41740
   5676 70539 y Fm(b)-52 b(e)289 b(some)g(c)-52 b(onstant.)444
   5677 b(If)288 b(the)g(set)27201 71701 y(of)362 b(outputs)f(pr)-52
   5678 b(o)g(duc)g(e)g(d)361 b(by)h(a)g(mix)g(server)i(is)e(not)g(a)g(p)-52
   5679 b(ermutation)361 b(of)p eop
   5680 %%Page: 7 7
   5681 7 6 bop -2022 -915 a Fm(the)262 b(inputs,)283 b(then)262
   5682 b(a)h(single)h(chal)52 b(lenge)265 b(in)d(step)h Fo(5)g
   5683 Fm(of)f(the)h(veri\257c)-52 b(ation)-2022 247 y(pr)g(oto)g(c)g(ol)272
   5684 b(exp)-52 b(oses)274 b(che)-52 b(ating)273 b(with)g(pr)-52
   5685 b(ob)g(ability)272 b(at)h(le)-52 b(ast)19576 -161 y Fd(3)p
   5686 19576 19 406 43 v 19576 585 a(8)20141 247 y Fg(\241)26
   5687 b Fj(\262)p Fm(,)292 b(or)274 b(the)-2022 1409 y(discr)-52
   5688 b(ete)356 b(lo)-52 b(garithm)355 b(pr)-52 b(oblem)356
   5689 b(in)f Fe(G)g Fm(c)-52 b(an)355 b(b)-52 b(e)356 b(solve)-52
   5690 b(d)356 b(in)f(p)-52 b(olynomial)-2022 2571 y(time.)-1026
   5691 4656 y Fo(Before)316 b(pro)-28 b(ving)315 b(the)h(theorem,)322
   5692 b(w)-28 b(e)315 b(need)i(the)f(follo)-28 b(wing)315 b(simple)-2022
   5693 5818 y(fact:)-915 7903 y Ff(Lemma)552 b Fo(7.2.)499 b
   5694 Fm(L)-52 b(et)334 b Fj(u)7386 8014 y Fd(1)7847 7903 y
   5695 Fj(;)172 b(:)f(:)g(:)g(;)g(u)10712 8014 y Fi(n)p Fd(+1)12639
   5696 7903 y Fm(and)333 b Fj(v)15090 8014 y Fd(1)15552 7903
   5697 y Fj(;)171 b(:)g(:)g(:)g(;)g(v)18324 8014 y Fi(n)p Fd(+1)20251
   5698 7903 y Fm(b)-52 b(e)335 b(ve)-52 b(ctors)-2022 9065 y(in)466
   5699 b Fe(Z)-3 8642 y Fi(n)-3 9241 y(q)564 9065 y Fm(.)774
   5700 b(Supp)-52 b(ose)467 b(ther)-52 b(e)467 b(is)g(no)g Fj(n)p
   5701 Fm(-by-)p Fj(n)f Fm(matrix)h Fj(M)576 b Fg(2)468 b Fj(GL)22060
   5702 9176 y Fi(n)22626 9065 y Fo(\()p Fe(Z)23688 9176 y Fi(q)24146
   5703 9065 y Fo(\))-2022 10228 y Fm(such)427 b(that)g Fj(u)3107
   5704 10339 y Fi(i)3854 10228 y Fo(=)397 b Fj(M)380 b Fg(\242)273
   5705 b Fj(v)7470 10339 y Fi(i)8249 10228 y Fm(for)427 b(al)52
   5706 b(l)428 b Fj(i)397 b Fo(=)f(1)p Fj(;)171 b(:)g(:)g(:)g(;)g(n)272
   5707 b Fo(+)h(1)p Fm(.)655 b(Then)427 b(ther)-52 b(e)-2022
   5708 11390 y(is)512 b(a)f(p)-52 b(olynomial)511 b(time)i(algorithm)f(that)f
   5709 (\257nds)h Fj(n)333 b Fo(+)h(1)513 b Fm(elements)-2022
   5710 12552 y Fj(c)-1579 12663 y Fd(1)-1118 12552 y Fj(;)171
   5711 b(:)g(:)g(:)g(;)g(c)1600 12663 y Fi(n)p Fd(+1)3489 12552
   5712 y Fg(2)293 b Fe(Z)5129 12663 y Fi(q)5959 12552 y Fm(such)372
   5713 b(that)10387 11805 y Fc(P)11468 12079 y Fi(n)p Fd(+1)11468
   5714 12857 y Fi(i)p Fd(=1)13232 12552 y Fj(c)13675 12663 y
   5715 Fi(i)14027 12552 y Fj(v)14524 12663 y Fi(i)15170 12552
   5716 y Fo(=)294 b(0)371 b Fm(but)18874 11805 y Fc(P)19955
   5717 12079 y Fi(n)p Fd(+1)19955 12857 y Fi(i)p Fd(=1)21719
   5718 12552 y Fj(c)22162 12663 y Fi(i)22513 12552 y Fj(u)23102
   5719 12663 y Fi(i)23748 12552 y Fg(6)p Fo(=)-2022 13715 y(0)p
   5720 Fm(.)-915 15799 y Ff(Pr)-22 b(oof.)498 b Fo(By)228 b(relab)28
   5721 b(elling)229 b(the)g(v)-28 b(ectors)227 b Fj(v)14889
   5722 15910 y Fd(1)15351 15799 y Fj(;)171 b(:)g(:)g(:)g(;)g(v)18123
   5723 15910 y Fi(n)p Fd(+1)19945 15799 y Fo(as)227 b(required,)-2022
   5724 16962 y(w)-28 b(e)290 b(ma)-28 b(y)291 b(assume)g(that)g
   5725 Fj(V)8084 16538 y Fr(0)8667 16962 y Fo(=)284 b Fg(f)p
   5726 Fj(v)10756 17073 y Fd(1)11218 16962 y Fj(;)171 b(:)g(:)g(:)g(;)g(v)
   5727 13990 17080 y Fi(k)14499 16962 y Fg(g)291 b Fo(is)g(a)g(maximal)h
   5728 (linearly)-2022 18124 y(indep)28 b(enden)-28 b(t)458
   5729 b(subset)f(of)h Fj(V)705 b Fo(=)478 b Fg(f)p Fj(v)12021
   5730 18235 y Fd(1)12483 18124 y Fj(;)171 b(:)g(:)g(:)g(;)g(v)15255
   5731 18235 y Fi(n)p Fd(+1)16849 18124 y Fg(g)p Fo(.)804 b(Let)458
   5732 b Fj(M)565 b Fo(b)28 b(e)458 b(an)-2022 19286 y Fj(n)p
   5733 Fo(-b)-28 b(y-)p Fj(n)394 b Fo(matrix)h(suc)-28 b(h)393
   5734 b(that)h Fj(u)9976 19397 y Fi(i)10701 19286 y Fo(=)373
   5735 b Fj(M)d Fg(\242)263 b Fj(v)14273 19397 y Fi(i)15019
   5736 19286 y Fo(for)394 b(all)h Fj(v)18611 19397 y Fi(i)19335
   5737 19286 y Fg(2)372 b Fj(V)21215 18863 y Fr(0)21513 19286
   5738 y Fo(.)614 b(Suc)-28 b(h)-2022 20449 y(a)424 b(matrix)h(alw)-28
   5739 b(a)g(ys)423 b(exists)j(since)f(the)g(v)-28 b(ectors)424
   5740 b(in)h Fj(V)18699 20025 y Fr(0)19422 20449 y Fo(are)f(linearly)-2022
   5741 21611 y(indep)28 b(enden)-28 b(t.)698 b(No)-28 b(w,)441
   5742 b(for)422 b(an)-28 b(y)421 b Fj(i)e(>)h(k)451 b Fo(the)422
   5743 b(set)h Fj(V)18117 21188 y Fr(0)18697 21611 y Fg([)281
   5744 b(f)p Fj(v)20670 21722 y Fi(i)21021 21611 y Fg(g)422
   5745 b Fo(is)h(lin-)-2022 22978 y(early)469 b(dep)28 b(enden)-28
   5746 b(t)470 b(and)e(therefore)h(w)-28 b(e)469 b(can)f(\257nd)h
   5747 Fj(c)18522 22469 y Fd(\()p Fi(i)p Fd(\))18522 23231 y(0)19513
   5748 22978 y Fj(;)171 b(:)g(:)g(:)g(;)g(c)22231 22469 y Fd(\()p
   5749 Fi(i)p Fd(\))22231 23264 y Fi(k)23691 22978 y Fo(in)-2022
   5750 24382 y Fe(Z)-1358 24493 y Fi(q)-493 24382 y Fo(suc)-28
   5751 b(h)407 b(that)h Fj(c)4611 23873 y Fd(\()p Fi(i)p Fd(\))4611
   5752 24636 y(0)5602 24382 y Fj(v)6099 24493 y Fi(i)6722 24382
   5753 y Fo(+)7790 23635 y Fc(P)8871 23909 y Fi(k)8871 24687
   5754 y(j)39 b Fd(=1)10502 24382 y Fj(c)10945 23873 y Fd(\()p
   5755 Fi(i)p Fd(\))10945 24648 y Fi(j)11936 24382 y Fj(v)12433
   5756 24493 y Fi(j)13262 24382 y Fo(=)395 b(0.)654 b(Note)408
   5757 b(that)f Fj(c)21171 23873 y Fd(\()p Fi(i)p Fd(\))21171
   5758 24636 y(0)22557 24382 y Fg(6)p Fo(=)395 b(0.)-2022 25934
   5759 y(If)511 b(the)h(equalit)-28 b(y)513 b Fj(c)5616 25425
   5760 y Fd(\()p Fi(i)p Fd(\))5616 26187 y(0)6607 25934 y Fj(u)7196
   5761 26045 y Fi(i)7889 25934 y Fo(+)9027 25187 y Fc(P)10107
   5762 25460 y Fi(k)10107 26238 y(j)39 b Fd(=1)11739 25934 y
   5763 Fj(c)12182 25425 y Fd(\()p Fi(i)p Fd(\))12182 26199 y
   5764 Fi(j)13173 25934 y Fj(u)13762 26045 y Fi(j)14764 25934
   5765 y Fo(=)569 b(0)512 b(also)g(holds,)554 b(then)-2022 27096
   5766 y Fj(u)-1433 27207 y Fi(i)-661 27096 y Fo(=)421 b Fj(M)389
   5767 b Fg(\242)283 b Fj(v)2998 27207 y Fi(i)3349 27096 y Fo(.)701
   5768 b(Therefore,)443 b(b)-28 b(y)423 b(the)g(assumption)h(of)e(the)i
   5769 (lemma,)-2022 28408 y(there)441 b(exists)g Fj(`)450 b(>)g(k)470
   5770 b Fo(suc)-28 b(h)440 b(that)g Fj(c)11914 27899 y Fd(\()p
   5771 Fi(`)p Fd(\))11914 28661 y(0)12954 28408 y Fj(v)13451
   5772 28526 y Fi(`)14145 28408 y Fo(+)15235 27661 y Fc(P)16315
   5773 27935 y Fi(k)16315 28712 y(j)39 b Fd(=1)17947 28408 y
   5774 Fj(c)18390 27899 y Fd(\()p Fi(`)p Fd(\))18390 28673 y
   5775 Fi(j)19429 28408 y Fj(v)19926 28519 y Fi(j)20810 28408
   5776 y Fo(=)450 b(0)441 b(but)-2022 29960 y Fj(c)-1579 29450
   5777 y Fd(\()p Fi(`)p Fd(\))-1579 30213 y(0)-540 29960 y Fj(u)49
   5778 30078 y Fi(`)750 29960 y Fo(+)1848 29212 y Fc(P)2929
   5779 29486 y Fi(k)2929 30264 y(j)39 b Fd(=1)4560 29960 y Fj(c)5003
   5780 29450 y Fd(\()p Fi(`)p Fd(\))5003 30225 y Fi(j)6043 29960
   5781 y Fj(u)6632 30071 y Fi(j)7534 29960 y Fg(6)p Fo(=)469
   5782 b(0.)787 b(The)452 b(algorithm)h(w)-28 b(orks)450 b(b)-28
   5783 b(y)452 b(build-)-2022 31511 y(ing)431 b(the)h(v)-28
   5784 b(ectors)432 b(\()p Fj(c)6038 31002 y Fd(\()p Fi(i)p
   5785 Fd(\))6038 31764 y(0)7029 31511 y Fj(;)171 b(:)g(:)g(:)g(;)g(c)9747
   5786 31002 y Fd(\()p Fi(i)p Fd(\))9747 31798 y Fi(k)10739
   5787 31511 y Fo(\))431 b(for)g Fj(i)k Fo(=)g Fj(k)317 b Fo(+)288
   5788 b(1)p Fj(;)171 b(:)g(:)g(:)g(;)g(n)288 b Fo(+)g(1)431
   5789 b(and)-2022 32674 y(outputting)477 b(the)g(\257rst)f(one)i(satisfying)f
   5790 (the)h(requiremen)-28 b(t)477 b(of)g(the)-2022 33836
   5791 y(lemma.)22529 b Fa(\244)-1026 36160 y Fo(W)-85 b(e)356
   5792 b(no)-28 b(w)355 b(pro)-28 b(v)g(e)355 b(Theorem)h(7.1.)500
   5793 b(Assume)356 b(that)g(the)h(set)f(of)g(out-)-2022 37323
   5794 y(puts)294 b(pro)28 b(duced)295 b(b)-28 b(y)295 b(the)g(mix)h(serv)-28
   5795 b(er)295 b(is)g(not)g(a)g(p)28 b(erm)-28 b(utation)295
   5796 b(of)g(the)-2022 38485 y(set)395 b(of)f(inputs)h(but)g(c)-28
   5797 b(heating)395 b(is)g(detected)h(with)f(probabilit)-28
   5798 b(y)395 b(less)-2022 39647 y(than)567 39239 y Fd(3)p
   5799 567 39419 V 567 39985 a(8)1378 39647 y Fg(\241)273 b
   5800 Fj(\262)p Fo(.)658 b(W)-85 b(e)409 b(view)g(the)g(mix)h(serv)-28
   5801 b(er)409 b(as)f(a)h(p)28 b(olynomial-time)-2022 40810
   5802 y(randomized)407 b(algorithm,)424 b(and)407 b(construct)g(an)g
   5803 (algorithm)g Fg(A)g Fo(that)-2022 41972 y(uses)469 b(the)g(mix)h(serv)
   5804 -28 b(er)469 b(to)g(compute)h(discrete)g(logarithms)f(in)g
   5805 Fe(G)p Fo(.)-2022 43134 y(Algorithm)437 b Fg(A)f Fo(tak)-28
   5806 b(es)437 b(as)g(input)f(t)-28 b(w)g(o)436 b(v)-57 b(alues)437
   5807 b Fj(g)473 b Fo(and)436 b Fj(h)h Fo(in)g Fe(G)f Fo(and)-2022
   5808 44297 y(computes)341 b(log)3857 44546 y Fi(g)4511 44297
   5809 y Fj(h)g Fo(as)g(follo)-28 b(ws:)-826 46179 y(1.)498
   5810 b(Algorithm)323 b Fg(A)f Fo(creates)h(ElGamal)g(public)g(and)f(priv)-57
   5811 b(ate)323 b(k)-28 b(eys)468 47341 y(\(the)396 b(public)g(k)-28
   5812 b(ey)396 b(will)g(b)28 b(e)396 b(used)f(to)g(encrypt)h(the)f(inputs)g
   5813 (to)468 48504 y(the)311 b(mix)g(serv)-28 b(er\).)446
   5814 b(Algorithm)310 b Fg(A)h Fo(k)-28 b(eeps)310 b(the)h(priv)-57
   5815 b(ate)311 b(k)-28 b(ey)311 b(to)468 49666 y(itself)329
   5816 b(and)f(giv)-28 b(es)329 b(the)f(public)h(k)-28 b(ey)329
   5817 b(to)f(the)g(mix)h(serv)-28 b(er.)451 b(Note)468 50828
   5818 y(that)342 b Fg(A)f Fo(is)h(em)-28 b(ulating)342 b(the)g(decryption)g
   5819 (serv)-28 b(ers.)-826 52749 y(2.)498 b Fg(A)261 b Fo(creates)f
   5820 Fj(n)g Fo(inputs)h Fj(a)9361 52860 y Fi(i)9997 52749
   5821 y Fo(=)285 b Fj(g)11602 52326 y Fi(r)11987 52466 y Fb(i)12459
   5822 52749 y Fg(\242)66 b Fj(h)13399 52326 y Fi(s)13787 52466
   5823 y Fb(i)14477 52749 y Fg(2)284 b Fe(G)260 b Fo(for)g Fj(r)18427
   5824 52860 y Fi(i)18778 52749 y Fj(;)171 b(s)19710 52860 y
   5825 Fi(i)20346 52749 y Fg(2)284 b Fe(Z)21977 52860 y Fi(q)22696
   5826 52749 y Fo(c)-28 b(ho-)468 53912 y(sen)401 b(indep)28
   5827 b(enden)-28 b(tly)402 b(at)e(random.)631 b(Let)401 b
   5828 Fj(A)382 b Fo(=)h Fg(f)p Fj(a)19902 54023 y Fd(1)20364
   5829 53912 y Fj(;)171 b(:)g(:)g(:)g(;)g(a)23181 54023 y Fi(n)23748
   5830 53912 y Fg(g)p Fo(.)468 55074 y(Algorithm)379 b Fg(A)g
   5831 Fo(submits)g(ElGamal)h(encryptions)f(of)g(the)g(in-)468
   5832 56236 y(puts)341 b Fj(a)3291 56347 y Fd(1)3753 56236
   5833 y Fj(;)171 b(:)g(:)g(:)g(;)g(a)6570 56347 y Fi(n)7479
   5834 56236 y Fo(to)341 b(the)h(mix)g(serv)-28 b(er.)-826 58157
   5835 y(3.)498 b(The)367 b(mix)h(serv)-28 b(er)367 b(pro)28
   5836 b(duces)367 b(an)g(output)f(set)i(of)f Fj(n)f Fo(ElGamal)468
   5837 59319 y(ciphertexts.)732 b(Algorithm)433 b Fg(A)g Fo(decrypts)g(these)g
   5838 (ciphertexts)468 60482 y(to)342 b(obtain)f(the)h(set)g
   5839 Fj(B)334 b Fo(=)285 b Fg(f)p Fj(b)11409 60593 y Fd(1)11870
   5840 60482 y Fj(;)171 b(:)g(:)g(:)g(;)g(b)14584 60593 y Fi(n)15150
   5841 60482 y Fg(g)285 b(\265)g Fe(G)341 b Fo(of)g(outputs.)-826
   5842 62403 y(4.)498 b(Algorithm)287 b Fg(A)g Fo(generates)h(a)f(random)f
   5843 (subsets)g Fj(S)341 b Fg(\265)285 b Fj(A)h Fo(b)-28 b(y)287
   5844 b(in-)468 63565 y(cluding)404 b(ev)-28 b(ery)404 b(elemen)-28
   5845 b(t)405 b(of)e Fj(A)g Fo(indep)28 b(enden)-28 b(tly)405
   5846 b(at)e(random)468 64727 y(with)392 b(probabilit)-28 b(y)392
   5847 b(half.)608 b(It)392 b(c)-28 b(hallenges)392 b(the)h(mix)g(serv)-28
   5848 b(er)391 b(to)468 65890 y(rev)-28 b(eal)271 b(the)f(subset)g
   5849 Fj(F)142 b Fo(\()p Fj(S)57 b Fo(\))284 b Fg(\265)h Fj(B)319
   5850 b Fo(of)270 b(the)g(outputs)g(corresp)28 b(ond-)468 67052
   5851 y(ing)399 b(to)g Fj(S)57 b Fo(.)628 b(Recall)400 b(that)f(the)g(mix)h
   5852 (serv)-28 b(er)399 b(m)-28 b(ust)399 b(pro)28 b(duce)399
   5853 b(a)468 68214 y(subset)288 b Fj(F)142 b Fo(\()p Fj(S)57
   5854 b Fo(\))284 b Fg(\265)h Fj(B)338 b Fo(suc)-28 b(h)287
   5855 b(that)h Fg(j)p Fj(F)142 b Fo(\()p Fj(S)57 b Fo(\))p
   5856 Fg(j)285 b Fo(=)g Fg(j)p Fj(S)57 b Fg(j)288 b Fo(and)f(the)i(pro)28
   5857 b(d-)468 69376 y(uct)353 b(of)f(the)h(elemen)-28 b(ts)354
   5858 b(of)f Fj(F)142 b Fo(\()p Fj(S)57 b Fo(\))352 b(equals)h(the)g(pro)28
   5859 b(duct)353 b(of)f(the)468 70539 y(elemen)-28 b(ts)353
   5860 b(of)e Fj(S)57 b Fo(.)485 b(If)350 b(the)i(mix)h(serv)-28
   5861 b(er)351 b(do)28 b(es)352 b(not)f(reply)h(to)f(the)468
   5862 71701 y(c)-28 b(hallenge,)393 b(algorithm)382 b Fg(A)f
   5863 Fo(rewinds)g(it)h(and)f(queries)i(it)e(on)h(a)29692 -915
   5864 y(di\256eren)-28 b(t)343 b(random)e(subset)h(un)-28 b(til)343
   5865 b(the)f(mix)i(serv)-28 b(er)341 b(pro)28 b(duces)29692
   5866 247 y(a)370 b(reply)-85 b(.)544 b(Since)371 b(the)g(mix)h(serv)-28
   5867 b(er)370 b(answ)-28 b(ers)369 b(c)-28 b(hallenges)371
   5868 b(with)29692 1409 y(probabilit)-28 b(y)439 b(at)h(least)39000
   5869 1001 y Fd(5)p 39000 1181 V 39000 1747 a(8)39832 1409
   5870 y Fo(+)293 b Fj(\262)p Fo(,)464 b(the)439 b(algorithm)h
   5871 Fg(A)f Fo(needs)g(to)29692 2571 y(rewind)341 b(the)h(mix)g(serv)-28
   5872 b(er)341 b(less)h(than)f(t)-28 b(wice)342 b(on)f(a)-28
   5873 b(v)g(erage.)28397 4637 y(5.)499 b(Algorithm)291 b Fg(A)g
   5874 Fo(rep)28 b(eats)292 b(step)f(4)g(ab)28 b(o)-28 b(v)g(e)291
   5875 b Fj(n)127 b Fo(+)g(1)291 b(times,)302 b(rewind-)29692
   5876 5799 y(ing)431 b(the)h(mix)g(serv)-28 b(er)431 b(b)28
   5877 b(et)-28 b(w)g(een)431 b(queries.)726 b(It)430 b(obtains)h(inde-)29692
   5878 6961 y(p)28 b(enden)-28 b(t)229 b(random)g(subsets)g
   5879 Fj(S)41208 7072 y Fd(1)41668 6961 y Fj(;)171 b(:)g(:)h(:)f(;)g(S)44570
   5880 7072 y Fi(n)p Fd(+1)46392 6961 y Fo(of)228 b(the)i(inputs)f(and)29692
   5881 8124 y(the)259 b(corresp)28 b(onding)259 b(replies)h
   5882 Fj(F)142 b Fo(\()p Fj(S)42804 8235 y Fd(1)43265 8124
   5883 y Fo(\))p Fj(;)171 b(:)g(:)g(:)g(;)g(F)142 b Fo(\()p
   5884 Fj(S)47758 8235 y Fi(n)p Fd(+1)49351 8124 y Fo(\))259
   5885 b(from)g(the)29692 9286 y(mix)342 b(serv)-28 b(er.)28397
   5886 11351 y(6.)499 b(Let)447 b Fj(\302)p Fo(\()p Fj(S)57
   5887 b Fo(\))459 b Fg(2)g(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)37853
   5888 10928 y Fi(n)38866 11351 y Fo(b)28 b(e)447 b(the)g(c)-28
   5889 b(haracteristic)447 b(v)-28 b(ector)446 b(of)g Fj(S)29692
   5890 12513 y Fo(for)390 b(an)-28 b(y)390 b(subset)h Fj(S)447
   5891 b Fo(of)390 b Fj(A)h Fo(or)f Fj(B)50 b Fo(.)603 b(W)-85
   5892 b(e)391 b(view)g Fj(\302)p Fo(\()p Fj(S)57 b Fo(\))367
   5893 b Fg(2)f(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)53202 12090
   5894 y Fi(n)29692 13676 y Fo(as)467 b(a)f(v)-28 b(ector)467
   5895 b(in)g Fe(Z)37237 13252 y Fi(n)37237 13851 y(q)37804
   5896 13676 y Fo(.)831 b(If)467 b(there)g(exists)h(an)e Fj(n)p
   5897 Fo(-b)-28 b(y-)p Fj(n)467 b Fo(matrix)29692 14838 y Fj(M)540
   5898 b Fg(2)431 b Fj(GL)33834 14949 y Fi(n)34400 14838 y Fo(\()p
   5899 Fe(Z)35462 14949 y Fi(q)35921 14838 y Fo(\))f(suc)-28
   5900 b(h)429 b(that)h(for)f(all)h Fj(i)i Fo(=)h(1)p Fj(;)171
   5901 b(:)g(:)g(:)g(;)g(n)286 b Fo(+)h(1)430 b(the)29692 16000
   5902 y(equalit)-28 b(y)279 b Fj(\302)p Fo(\()p Fj(S)35187
   5903 16111 y Fi(i)35539 16000 y Fo(\))284 b(=)h Fj(M)208 b
   5904 Fg(\242)100 b Fj(\302)p Fo(\()p Fj(F)142 b Fo(\()p Fj(S)41739
   5905 16111 y Fi(i)42091 16000 y Fo(\)\))278 b(holds,)291 b(then)278
   5906 b(algorithm)g Fg(A)29692 17163 y Fo(rep)28 b(orts)341
   5907 b(failure.)28397 19228 y(7.)499 b(Otherwise,)305 b(there)296
   5908 b(is)g(no)f(matrix)g Fj(M)393 b Fg(2)284 b Fj(GL)46848
   5909 19339 y Fi(n)47414 19228 y Fo(\()p Fe(Z)48476 19339 y
   5910 Fi(q)48934 19228 y Fo(\))296 b(suc)-28 b(h)294 b(that)29692
   5911 20390 y Fj(\302)p Fo(\()p Fj(S)31354 20501 y Fi(i)31705
   5912 20390 y Fo(\))445 b(=)g Fj(M)400 b Fg(\242)292 b Fj(\302)p
   5913 Fo(\()p Fj(F)142 b Fo(\()p Fj(S)38610 20501 y Fi(i)38961
   5914 20390 y Fo(\)\))438 b(holds)f(for)g(all)h Fj(i)445 b
   5915 Fo(=)g(1)p Fj(;)171 b(:)g(:)h(:)f(;)g(n)291 b Fo(+)h(1.)29692
   5916 21552 y(By)411 b(Lemma)g(7.2,)428 b(w)-28 b(e)410 b(can)h(then)f
   5917 (\257nd)g(in)h(p)28 b(olynomial)412 b(time)29692 22715
   5918 y Fj(c)30135 22826 y Fd(1)30596 22715 y Fj(;)172 b(:)f(:)g(:)g(;)g(c)
   5919 33315 22826 y Fi(n)p Fd(+1)35193 22715 y Fg(2)284 b Fe(Z)36824
   5920 22826 y Fi(q)37624 22715 y Fo(suc)-28 b(h)340 b(that:)34234
   5921 24321 y Fi(n)p Fd(+1)34263 24624 y Fc(X)34341 26724 y
   5922 Fi(i)p Fd(=1)35943 25570 y Fj(c)36386 25681 y Fi(i)36737
   5923 25570 y Fj(\302)p Fo(\()p Fj(F)142 b Fo(\()p Fj(S)39593
   5924 25681 y Fi(i)39945 25570 y Fo(\)\))285 b(=)f(\(0)p Fj(;)172
   5925 b(:)f(:)g(:)g(;)g Fo(0\))285 b(mo)28 b(d)284 b Fj(q)29692
   5926 28343 y Fo(But)35030 29657 y Fi(n)p Fd(+1)35059 29960
   5927 y Fc(X)35137 32060 y Fi(i)p Fd(=1)36739 30906 y Fj(c)37182
   5928 31017 y Fi(i)37533 30906 y Fj(\302)p Fo(\()p Fj(S)39195
   5929 31017 y Fi(i)39547 30906 y Fo(\))g Fg(6)p Fo(=)h(\(0)p
   5930 Fj(;)171 b(:)g(:)h(:)f(;)g Fo(0\))284 b(mo)28 b(d)285
   5931 b Fj(q)29692 33847 y Fo(Let)527 b(\()p Fj(e)32586 33958
   5932 y Fd(1)33047 33847 y Fj(;)171 b(:)g(:)g(:)h(;)f(e)35799
   5933 33958 y Fi(n)36365 33847 y Fo(\))593 b(=)38745 33100
   5934 y Fc(P)39826 33374 y Fi(n)p Fd(+1)39826 34152 y Fi(i)p
   5935 Fd(=1)41590 33847 y Fj(c)42033 33958 y Fi(i)42384 33847
   5936 y Fj(\302)p Fo(\()p Fj(S)44046 33958 y Fi(i)44398 33847
   5937 y Fo(\))g Fg(2)f Fe(Z)47328 33424 y Fi(n)47328 34022
   5938 y(q)47895 33847 y Fo(.)1010 b(By)527 b(de\257ni-)29692
   5939 35009 y(tion)h(of)g(the)g(mix)h(serv)-28 b(er,)575 b(w)-28
   5940 b(e)528 b(kno)-28 b(w)527 b(that)47939 34262 y Fc(Q)48906
   5941 35314 y Fi(a)49354 35454 y Fb(j)49768 35314 y Fr(2)p
   5942 Fi(S)50824 35454 y Fb(i)51400 35009 y Fj(a)51942 35120
   5943 y Fi(j)52972 35009 y Fo(=)29692 35672 y Fc(Q)30659 36724
   5944 y Fi(b)31023 36864 y Fb(j)31436 36724 y Fr(2)p Fi(F)108
   5945 b Fd(\()p Fi(S)33435 36864 y Fb(i)33785 36724 y Fd(\))34331
   5946 36420 y Fj(b)34770 36531 y Fi(j)35621 36420 y Fo(for)416
   5947 b(all)i Fj(i)411 b Fo(=)g(1)p Fj(;)171 b(:)h(:)f(:)g(;)g(n)278
   5948 b Fo(+)g(1.)683 b(By)418 b(m)-28 b(ultiplying)29692 37774
   5949 y(these)342 b Fj(n)227 b Fo(+)h(1)342 b(relations,)g(w)-28
   5950 b(e)341 b(get)34594 39338 y Fi(n)34196 39637 y Fc(Y)34147
   5951 41737 y Fi(j)39 b Fd(=1)35553 40583 y Fo(\()p Fj(a)36493
   5952 40694 y Fi(j)36927 40583 y Fo(\))37325 40105 y Fi(e)37716
   5953 40245 y Fb(j)38470 40583 y Fo(=)39550 39334 y Fi(n)p
   5954 Fd(+1)39665 39637 y Fc(Y)39658 41737 y Fi(i)p Fd(=1)42328
   5955 39637 y Fc(Y)41259 41796 y Fi(b)41623 41936 y Fb(j)42037
   5956 41796 y Fr(2)p Fi(F)108 b Fd(\()p Fi(S)44036 41936 y
   5957 Fb(i)44385 41796 y Fd(\))44705 40583 y Fo(\()p Fj(b)45542
   5958 40694 y Fi(j)45976 40583 y Fo(\))46374 40105 y Fi(c)46745
   5959 40245 y Fb(i)47436 40583 y Fo(=)285 b(1)p Fj(:)29692
   5960 43691 y Fo(Recall)342 b(that)g Fj(a)35539 43802 y Fi(i)36175
   5961 43691 y Fo(=)285 b Fj(g)37780 43268 y Fi(r)38165 43408
   5962 y Fb(i)38798 43691 y Fg(\242)228 b Fj(h)39900 43268 y
   5963 Fi(s)40288 43408 y Fb(i)40695 43691 y Fo(.)455 b(Therefore)34733
   5964 45109 y Fi(n)34249 45408 y Fc(X)34286 47509 y Fi(j)39
   5965 b Fd(=1)35899 46355 y Fj(e)36375 46466 y Fi(j)36808 46355
   5966 y Fo(\()p Fj(r)37668 46466 y Fi(j)38330 46355 y Fo(+)228
   5967 b Fj(s)39831 46466 y Fi(j)40264 46355 y Fo(\(log)41985
   5968 46604 y Fi(g)42639 46355 y Fj(h)p Fo(\)\))284 b(=)h(0)g(mo)28
   5969 b(d)285 b Fj(q)36 b(:)29692 49336 y Fo(If)30880 48588
   5970 y Fc(P)31961 48862 y Fi(n)31961 49640 y(j)j Fd(=1)33592
   5971 49336 y Fj(e)34068 49447 y Fi(j)34502 49336 y Fj(s)34979
   5972 49447 y Fi(j)35970 49336 y Fo(=)558 b(0)505 b(then)h(algorithm)g
   5973 Fg(A)f Fo(rep)28 b(orts)505 b(failure.)29692 50627 y(This)283
   5974 b(happ)28 b(ens)284 b(with)g(probabilit)-28 b(y)284 b(at)f(most)h(1)p
   5975 Fj(=q)36 b Fo(.)436 b(Otherwise)29692 51789 y Fg(A)341
   5976 b Fo(outputs)33998 54453 y(log)35321 54702 y Fi(g)35974
   5977 54453 y Fj(h)285 b Fo(=)g Fg(\241)38726 53050 y Fc(\265)39964
   5978 53208 y Fi(n)39480 53507 y Fc(X)39516 55607 y Fi(j)39
   5979 b Fd(=1)41129 54453 y Fj(e)41605 54564 y Fi(j)42039 54453
   5980 y Fj(r)42501 54564 y Fi(j)42935 53050 y Fc(\266)43688
   5981 54453 y Fj(=)44200 53050 y Fc(\265)45438 53208 y Fi(n)44954
   5982 53507 y Fc(X)44991 55607 y Fi(j)g Fd(=1)46604 54453 y
   5983 Fj(e)47080 54564 y Fi(j)47513 54453 y Fj(s)47990 54564
   5984 y Fi(j)48424 53050 y Fc(\266)49178 54453 y Fj(:)28308
   5985 57818 y Ff(Pr)-22 b(oposition)554 b Fo(7.3.)499 b Fm(Supp)-52
   5986 b(ose)416 b(the)g(mix)h(server)h(pr)-52 b(o)g(duc)g(es)416
   5987 b(out-)27201 58980 y(puts)319 b(which)f(ar)-52 b(e)320
   5988 b(not)e(a)h(p)-52 b(ermutation)318 b(of)h(the)g(inputs,)327
   5989 b(yet)319 b(manages)27201 60143 y(to)496 b(r)-52 b(eply)496
   5990 b(to)f(chal)52 b(lenges)499 b(with)494 b(pr)-52 b(ob)g(ability)495
   5991 b(gr)-52 b(e)g(ater)497 b(than)51061 59734 y Fd(5)p 51061
   5992 59914 V 51061 60481 a(8)51922 60143 y Fo(+)322 b Fj(\262)p
   5993 Fm(.)27201 61305 y(Then)380 b(algorithm)g Fg(A)f Fm(suc)-52
   5994 b(c)g(e)g(e)g(ds)381 b(in)e(c)-52 b(omputing)379 b(discr)-52
   5995 b(ete)381 b(lo)-52 b(garithm)27201 62467 y(with)365 b(pr)-52
   5996 b(ob)g(ability)365 b(at)g(le)-52 b(ast)366 b Fj(\262)38319
   5997 62044 y Fd(2)38780 62467 y Fj(=)p Fo(128)228 b Fg(\241)g
   5998 Fo(1)p Fj(=q)36 b Fm(.)28308 64727 y Ff(Pr)-22 b(oof.)498
   5999 b Fo(By)302 b(assumption,)311 b(the)302 b(mix)g(serv)-28
   6000 b(er)302 b(correctly)g(answ)-28 b(ers)27201 65890 y(a)269
   6001 b(query)g(with)g(probabilit)-28 b(y)269 b(at)f(least)41813
   6002 65481 y Fd(5)p 41813 65661 V 41813 66228 a(8)42434 65890
   6003 y Fo(+)82 b Fj(\262)p Fo(,)283 b(where)269 b(the)g(probabilit)-28
   6004 b(y)27201 67052 y(is)467 b(tak)-28 b(en)466 b(o)-28 b(v)g(er)465
   6005 b(all)i(2)35644 66629 y Fi(n)36677 67052 y Fo(p)28 b(ossible)467
   6006 b(queries)g(and)e(the)i(random)e(bits)27201 68214 y(used)361
   6007 b(b)-28 b(y)361 b(the)g(mix)h(serv)-28 b(er.)514 b(By)362
   6008 b(a)f(standard)f(coun)-28 b(ting)360 b(argumen)-28 b(t,)27201
   6009 69376 y(if)370 b(w)-28 b(e)369 b(randomly)g(\257x)h(the)f(mix)i(serv)
   6010 -28 b(er's)369 b(random)g(bits,)377 b(then)369 b(with)27201
   6011 70539 y(probabilit)-28 b(y)365 b(at)e(least)i Fj(\262=)p
   6012 Fo(2)f(the)g(mix)h(serv)-28 b(er)363 b(answ)-28 b(ers)363
   6013 b(correctly)h(at)27201 71701 y(least)29729 71293 y Fd(5)p
   6014 29729 71473 V 29729 72039 a(8)30496 71701 y Fo(+)31652
   6015 71293 y Fd(13)p 31652 71473 812 43 v 31652 72039 a(16)32597
   6016 71701 y Fj(\262)341 b Fo(of)g(all)h(2)36453 71278 y Fi(n)37361
   6017 71701 y Fo(queries.)p eop
   6018 %%Page: 8 8
   6019 8 7 bop -1026 -915 a Fo(Once)238 b(w)-28 b(e)237 b(\257x)h(the)g(mix)h
   6020 (serv)-28 b(er's)237 b(random)g(bits,)259 b(the)238 b(set)g(of)g
   6021 (queries)-2022 247 y(that)268 b(the)h(mix)g(serv)-28
   6022 b(er)268 b(answ)-28 b(ers)267 b(correctly)i(is)g(a)f(w)-28
   6023 b(ell)269 b(de\257ned)f(subset)-2022 1409 y Fe(S)324
   6024 b Fg(\265)h(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)2479 986
   6025 y Fi(n)3046 1409 y Fo(.)527 b(F)-85 b(urthermore,)371
   6026 b(w)-28 b(e)365 b(can)g(view)h(the)g(mix)g(serv)-28 b(er)366
   6027 b(as)f(a)-2022 2571 y(deterministic)396 b(function)e
   6028 Fj(F)515 b Fo(:)374 b Fe(S)f Fg(!)g(f)p Fo(0)p Fj(;)171
   6029 b Fo(1)p Fg(g)14899 2148 y Fi(n)15860 2571 y Fo(mapping)394
   6030 b(subsets)h(of)-2022 3734 y(inputs)242 b(to)h(subsets)f(of)g(outputs.)
   6031 423 b(W)-85 b(e)242 b(kno)-28 b(w)242 b(that)h Fg(j)p
   6032 Fe(S)p Fg(j)285 b Fj(>)g Fo(2)20162 3311 y Fi(n)20729
   6033 3734 y Fo(\()21260 3326 y Fd(5)p 21260 3506 406 43 v
   6034 21260 4072 a(8)21828 3734 y Fo(+)22787 3326 y Fd(13)p
   6035 22787 3506 812 43 v 22787 4072 a(16)23732 3734 y Fj(\262)p
   6036 Fo(\))-2022 4896 y(with)463 b(probabilit)-28 b(y)463
   6037 b(at)g(least)h Fj(\262=)p Fo(2.)821 b(Assume)463 b(for)g(the)g(rest)g
   6038 (of)g(the)-2022 6058 y(pro)28 b(of)340 b(that)i Fg(j)p
   6039 Fe(S)p Fg(j)286 b Fj(>)f Fo(2)5872 5635 y Fi(n)6438 6058
   6040 y Fo(\()6969 5650 y Fd(5)p 6969 5830 406 43 v 6969 6396
   6041 a(8)7735 6058 y Fo(+)8892 5650 y Fd(13)p 8892 5830 812
   6042 43 v 8892 6396 a(16)9837 6058 y Fj(\262)p Fo(\).)-1026
   6043 7221 y(F)-85 b(or)405 b Fj(i)393 b Fo(=)g(1)p Fj(;)171
   6044 b(:)g(:)g(:)g(;)g(n)p Fo(,)423 b(let)407 b Fj(e)8939
   6045 7332 y Fi(i)9682 7221 y Fg(2)392 b(f)p Fo(0)p Fj(;)171
   6046 b Fo(1)p Fg(g)13260 6797 y Fi(n)14233 7221 y Fo(b)28
   6047 b(e)407 b(the)f Fj(i)p Fo(-th)h(unit)f(v)-28 b(ector)-2022
   6048 8383 y(\(i.e.)721 b(zero)28 b(es)430 b(ev)-28 b(erywhere)430
   6049 b(and)f(a)h(one)f(in)h(the)g Fj(i)p Fo(-th)g(co)28 b(ordinate\).)-2022
   6050 9545 y(The)341 b(follo)-28 b(wing)341 b(lemma)i(will)f(b)28
   6051 b(e)342 b(used)f(sev)-28 b(eral)342 b(times:)-915 11750
   6052 y Ff(Lemma)552 b Fo(7.4.)499 b Fm(Supp)-52 b(ose)366
   6053 b(that)g Fg(j)p Fe(S)p Fg(j)288 b Fj(>)e Fo(2)14054 11327
   6054 y Fi(n)14620 11750 y Fj(=)p Fo(2)p Fm(.)473 b(L)-52 b(et)367
   6055 b Fj(i)286 b Fg(2)g(f)p Fo(1)p Fj(;)171 b(:)g(:)g(:)g(;)g(n)p
   6056 Fg(g)p Fm(.)-2022 12912 y(Ther)-52 b(e)366 b(exist)g
   6057 Fj(v)3795 13023 y Fd(0)4257 12912 y Fj(;)171 b(v)5209
   6058 13023 y Fd(1)5955 12912 y Fg(2)284 b Fe(S)366 b Fm(such)g(that)f
   6059 Fj(v)12755 13023 y Fd(1)13501 12912 y Fo(=)285 b Fj(v)15079
   6060 13023 y Fd(0)15768 12912 y Fo(+)228 b Fj(e)17268 13023
   6061 y Fi(i)17619 12912 y Fm(.)-915 15117 y Ff(Pr)-22 b(oof.)498
   6062 b Fo(Let)287 b Fj(U)394 b Fo(b)28 b(e)288 b(the)f(subset)g(of)f
   6063 Fg(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)15637 14694 y Fi(n)16491
   6064 15117 y Fo(con)-28 b(taining)286 b(all)i(v)-28 b(ec-)-2022
   6065 16279 y(tors)525 b(with)h(a)h(one)f(in)g(p)28 b(osition)528
   6066 b Fj(i)p Fo(.)1010 b(Let)527 b Fe(S)15520 16390 y Fd(0)16574
   6067 16279 y Fo(=)593 b Fe(S)351 b Fg(\\)20075 16027 y Fo(\271)19900
   6068 16279 y Fj(U)635 b Fo(and)526 b(let)-2022 17441 y Fe(S)-1469
   6069 17552 y Fd(1)-531 17441 y Fo(=)476 b Fe(S)305 b Fg(\\)f
   6070 Fj(U)108 b Fo(.)801 b(W)-85 b(e)457 b(de\257ne)g Fj(f)585
   6071 b Fo(:)476 b Fe(S)11823 17552 y Fd(0)12761 17441 y Fg(!)h
   6072 Fj(U)564 b Fo(to)457 b(b)28 b(e)457 b(the)g(map)f(that)-2022
   6073 18604 y(sends)390 b Fj(v)1266 18715 y Fd(0)2119 18604
   6074 y Fo(to)h Fj(v)3917 18715 y Fd(0)4639 18604 y Fo(+)261
   6075 b Fj(e)6172 18715 y Fi(i)6524 18604 y Fo(.)604 b(Since)391
   6076 b Fg(j)p Fj(f)108 b Fo(\()p Fe(S)11975 18715 y Fd(0)12438
   6077 18604 y Fo(\))p Fg(j)261 b Fo(+)g Fg(j)p Fe(S)15275 18715
   6078 y Fd(1)15738 18604 y Fg(j)367 b Fo(=)h Fg(j)p Fe(S)p
   6079 Fg(j)g Fj(>)f Fg(j)p Fj(U)108 b Fg(j)p Fo(,)405 b(there)-2022
   6080 19766 y(exist)287 b Fj(v)843 19877 y Fd(1)1589 19766
   6081 y Fg(2)d Fj(f)108 b Fo(\()p Fe(S)4112 19877 y Fd(0)4575
   6082 19766 y Fo(\))118 b Fg(\\)g Fe(S)6445 19877 y Fd(1)6906
   6083 19766 y Fo(.)437 b(Then)286 b(\()p Fj(v)11141 19877 y
   6084 Fd(1)11720 19766 y Fg(\241)118 b Fj(e)13110 19877 y Fi(i)13462
   6085 19766 y Fj(;)171 b(v)14414 19877 y Fd(1)14876 19766 y
   6086 Fo(\))286 b(is)h(the)g(pair)f(of)g(desired)-2022 20928
   6087 y(v)-28 b(ectors.)22349 b Fa(\244)-1026 23253 y Fo(Next,)380
   6088 b(w)-28 b(e)371 b(b)28 b(ound)372 b(the)g(probabilit)-28
   6089 b(y)372 b(that)g(algorithm)g Fg(A)g Fo(ab)28 b(orts)-2022
   6090 24415 y(in)464 b(step)g(6.)823 b(W)-85 b(e)465 b(separate)f(our)f
   6091 (analysis)i(of)f(the)g(b)28 b(ound)464 b(in)g(t)-28 b(w)g(o)-2022
   6092 25578 y(cases,)461 b(dep)28 b(ending)437 b(on)g(whether)f
   6093 Fj(F)142 b Fo(\()p Fe(S)p Fo(\))437 b(spans)f(all)i(of)e
   6094 Fe(Z)20429 25154 y Fi(n)20429 25753 y(q)21432 25578 y
   6095 Fo(or)g(not.)-2022 26740 y(W)-85 b(e)341 b(pro)-28 b(v)g(e)341
   6096 b(\257rst)g(that)g Fe(S)h Fo(m)-28 b(ust)341 b(span)g(all)h(of)f
   6097 Fe(Z)15913 26317 y Fi(n)15913 26915 y(q)16821 26740 y
   6098 Fo(when)g Fg(j)p Fe(S)p Fg(j)286 b Fj(>)f Fo(2)22495
   6099 26317 y Fi(n)23061 26740 y Fj(=)p Fo(2.)-915 28945 y
   6100 Ff(Lemma)552 b Fo(7.5.)499 b Fm(If)319 b Fg(j)p Fe(S)p
   6101 Fg(j)285 b Fj(>)g Fo(\(2)9488 28521 y Fi(n)10055 28945
   6102 y Fo(\))p Fj(=)p Fo(2)318 b Fm(then)h(the)f(ve)-52 b(ctors)320
   6103 b(in)e Fe(S)h Fm(sp)-52 b(an)318 b(al)52 b(l)-2022 30107
   6104 y(of)365 b Fe(Z)-156 29684 y Fi(n)-156 30282 y(q)411
   6105 30107 y Fm(.)-915 32311 y Ff(Pr)-22 b(oof.)498 b Fo(Let)266
   6106 b Fj(i)285 b Fg(2)f(f)p Fo(1)p Fj(;)171 b(:)g(:)g(:)g(;)g(n)p
   6107 Fg(g)p Fo(.)430 b(By)267 b(the)f(previous)g(lemma,)283
   6108 b(there)-2022 33474 y(exist)348 b Fj(v)904 33585 y Fd(0)1366
   6109 33474 y Fj(;)171 b(v)2318 33585 y Fd(1)3074 33474 y Fg(2)295
   6110 b Fe(S)348 b Fo(suc)-28 b(h)346 b(that)i Fj(e)9969 33585
   6111 y Fi(i)10615 33474 y Fo(=)295 b Fj(v)12203 33585 y Fd(1)12897
   6112 33474 y Fg(\241)232 b Fj(v)14422 33585 y Fd(0)14883 33474
   6113 y Fo(.)474 b(Therefore)347 b Fj(e)20765 33585 y Fd(1)21227
   6114 33474 y Fj(;)171 b(:)g(:)g(:)g(;)g(e)23978 33585 y Fi(n)-2022
   6115 34636 y Fo(are)341 b(spanned)g(b)-28 b(y)341 b Fe(S)h
   6116 Fo(and)f(so)g Fe(S)h Fo(spans)e Fe(Z)13597 34213 y Fi(n)13597
   6117 34811 y(q)14164 34636 y Fo(.)9300 b Fa(\244)-915 38003
   6118 y Ff(Lemma)552 b Fo(7.6.)499 b Fm(L)-52 b(et)518 b Fj(\262)560
   6119 b Fg(2)f Fo([0)p Fj(;)10583 37595 y Fd(1)p 10583 37775
   6120 406 43 v 10583 38341 a(2)11121 38003 y Fo(])519 b Fm(and)e
   6121 Fe(S)560 b Fg(\265)g(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)19034
   6122 37580 y Fi(n)19601 38003 y Fm(.)925 b(If)518 b Fg(j)p
   6123 Fe(S)p Fg(j)561 b Fj(>)-2022 39288 y Fo(2)-1510 38865
   6124 y Fi(n)-944 39288 y Fo(\()-413 38880 y Fd(1)p -413 39060
   6125 V -413 39626 a(2)400 39288 y Fo(+)275 b Fj(\262)p Fo(\))p
   6126 Fm(,)446 b(then)431 b Fj(n)f Fm(r)-52 b(andom)430 b(and)g(indep)-52
   6127 b(endent)430 b(ve)-52 b(ctors)432 b(fr)-52 b(om)430 b
   6128 Fe(S)-2022 40450 y Fm(wil)52 b(l)366 b(sp)-52 b(an)365
   6129 b Fe(Z)2928 40027 y Fi(n)2928 40626 y(q)3861 40450 y
   6130 Fm(with)f(pr)-52 b(ob)g(ability)365 b(at)h(le)-52 b(ast)366
   6131 b Fj(\262=)p Fo(4)p Fm(.)-915 42655 y Ff(Pr)-22 b(oof.)498
   6132 b Fo(Let)351 b Fj(v)5359 42766 y Fd(1)5820 42655 y Fj(;)171
   6133 b(:)g(:)h(:)f(;)g(v)8593 42766 y Fi(c)9320 42655 y Fg(2)299
   6134 b Fe(Z)10966 42232 y Fi(n)10966 42830 y(q)11884 42655
   6135 y Fo(b)28 b(e)351 b(linearly)h(indep)28 b(enden)-28 b(t)352
   6136 b(v)-28 b(ec-)-2022 43817 y(tors)454 b(where)i Fj(v)3720
   6137 43928 y Fd(1)4181 43817 y Fj(;)171 b(:)g(:)g(:)g(;)h(v)6954
   6138 43928 y Fi(c)7855 43817 y Fg(2)474 b(f)p Fo(0)p Fj(;)171
   6139 b Fo(1)p Fg(g)11515 43394 y Fi(n)12082 43817 y Fo(.)797
   6140 b(W)-85 b(e)456 b(kno)-28 b(w)455 b(that)g(the)h(linear)-2022
   6141 44979 y(space)400 b(spanned)h(b)-28 b(y)400 b Fj(v)6799
   6142 45090 y Fd(1)7261 44979 y Fj(;)171 b(:)g(:)g(:)g(;)g(v)10033
   6143 45090 y Fi(c)10861 44979 y Fo(con)-28 b(tains)400 b(at)h(most)g(2)19327
   6144 44556 y Fi(c)20154 44979 y Fo(v)-28 b(ectors)400 b(in)-2022
   6145 46142 y Fg(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)481 45719
   6146 y Fi(n)1047 46142 y Fo(.)773 b(Therefore,)473 b(if)448
   6147 b(w)-28 b(e)446 b(are)h(giv)-28 b(en)448 b Fj(c)f Fo(linearly)i(indep)
   6148 28 b(enden)-28 b(t)-2022 47304 y(v)g(ectors)421 b Fj(v)2033
   6149 47415 y Fd(1)2494 47304 y Fj(;)172 b(:)f(:)g(:)g(;)g(v)5267
   6150 47415 y Fi(c)6115 47304 y Fo(as)421 b(ab)28 b(o)-28 b(v)g(e,)442
   6151 b(the)422 b(probabilit)-28 b(y)422 b(that)f(a)g(random)-2022
   6152 48466 y(v)-28 b(ector)357 b(from)f Fe(S)i Fo(is)f(linearly)h(dep)28
   6153 b(enden)-28 b(t)358 b(on)f Fj(v)16047 48577 y Fd(1)16509
   6154 48466 y Fj(;)171 b(:)g(:)g(:)g(;)g(v)19281 48577 y Fi(c)20065
   6155 48466 y Fo(is)357 b(at)g(most)-2022 49629 y(2)-1510 49205
   6156 y Fi(c)-1084 49629 y Fj(=)p Fg(j)p Fe(S)p Fg(j)p Fo(.)572
   6157 b(If)380 b(follo)-28 b(ws)379 b(that)h Fj(n)f Fo(random)h(v)-28
   6158 b(ectors)379 b(from)h Fe(S)g Fo(span)f(all)i(of)-2022
   6159 50791 y Fe(Z)-1358 50368 y Fi(n)-1358 50966 y(q)-450
   6160 50791 y Fo(with)341 b(probabilit)-28 b(y)342 b(at)f(least)2064
   6161 53382 y Fj(p)285 b Fo(=)3944 51979 y Fc(\265)4697 53382
   6162 y Fo(1)228 b Fg(\241)6899 52737 y Fo(2)p 6594 53154 1123
   6163 43 v 6594 54057 a Fg(j)p Fe(S)p Fg(j)7849 51979 y Fc(\266\265)9356
   6164 53382 y Fo(1)g Fg(\241)11327 52737 y Fo(2)11839 52314
   6165 y Fd(2)p 11253 53154 V 11253 54057 a Fg(j)p Fe(S)p Fg(j)12508
   6166 51979 y Fc(\266)13432 53382 y Fg(\242)171 b(\242)g(\242)14797
   6167 51979 y Fc(\265)15551 53382 y Fo(1)228 b Fg(\241)17447
   6168 52737 y Fo(2)17959 52314 y Fi(n)p Fr(\241)p Fd(1)p 17447
   6169 53154 2124 43 v 17948 54057 a Fg(j)p Fe(S)p Fg(j)19704
   6170 51979 y Fc(\266)-2022 55811 y Fo(In)378 b(what)h(follo)-28
   6171 b(ws,)388 b(all)380 b(the)f(logarithms)h(are)f(tak)-28
   6172 b(en)379 b(base)g(2.)569 b(Since)-2022 56973 y(log)-700
   6173 57222 y Fd(2)-239 56973 y Fo(\(1)228 b Fg(\241)g Fj(x)p
   6174 Fo(\))285 b Fg(\270)g(\241)p Fo(2)p Fj(x)342 b Fo(for)f
   6175 Fj(x)284 b Fg(2)g Fo([0)p Fj(;)11280 56565 y Fd(1)p 11280
   6176 56745 406 43 v 11280 57311 a(2)11818 56973 y Fo(])342
   6177 b(and)f Fg(j)p Fe(S)p Fg(j)286 b Fj(>)f Fo(2)17435 56550
   6178 y Fi(n)p Fr(\241)p Fd(1)19047 56973 y Fo(,)341 b(w)-28
   6179 b(e)341 b(ha)-28 b(v)g(e)891 59890 y(log)186 b Fj(p)996
   6180 b Fo(=)h(log)7010 58487 y Fc(\265)7764 59890 y Fo(1)227
   6181 b Fg(\241)9660 59245 y Fo(2)10172 58822 y Fi(n)p Fr(\241)p
   6182 Fd(1)p 9660 59662 2124 43 v 10161 60566 a Fg(j)p Fe(S)p
   6183 Fg(j)11917 58487 y Fc(\266)12898 59890 y Fo(+)13922 58641
   6184 y Fi(n)p Fr(\241)p Fd(2)13961 58943 y Fc(X)14039 61044
   6185 y Fi(i)p Fd(=1)15649 59890 y Fo(log)16972 58487 y Fc(\265)17726
   6186 59890 y Fo(1)g Fg(\241)19751 59245 y Fo(2)20263 58822
   6187 y Fi(i)p 19622 59662 1123 43 v 19622 60566 a Fg(j)p Fe(S)p
   6188 Fg(j)20877 58487 y Fc(\266)3895 63385 y Fg(\270)997 b
   6189 Fo(log)7010 61983 y Fc(\265)7764 63385 y Fo(1)227 b Fg(\241)9660
   6190 62741 y Fo(2)10172 62318 y Fi(n)p Fr(\241)p Fd(1)p 9660
   6191 63157 2124 43 v 10161 64061 a Fg(j)p Fe(S)p Fg(j)11917
   6192 61983 y Fc(\266)12898 63385 y Fg(\241)h Fo(2)14605 62136
   6193 y Fi(n)p Fr(\241)p Fd(2)14644 62439 y Fc(X)14722 64539
   6194 y Fi(i)p Fd(=1)16594 62741 y Fo(2)17106 62318 y Fi(i)p
   6195 16465 63157 1123 43 v 16465 64061 a Fg(j)p Fe(S)p Fg(j)3895
   6196 66588 y(\270)997 b Fo(log)7010 65186 y Fc(\265)7764 66588
   6197 y Fo(1)227 b Fg(\241)9660 65944 y Fo(2)10172 65521 y
   6198 Fi(n)p Fr(\241)p Fd(1)p 9660 66360 2124 43 v 10161 67264
   6199 a Fg(j)p Fe(S)p Fg(j)11917 65186 y Fc(\266)12898 66588
   6200 y Fg(\241)14077 65944 y Fo(2)14589 65521 y Fi(n)p 14055
   6201 66360 1123 43 v 14055 67264 a Fg(j)p Fe(S)p Fg(j)-2022
   6202 69162 y Fo(Since)373 b(w)-28 b(e)372 b(assume)h Fg(j)p
   6203 Fe(S)p Fg(j)338 b Fj(>)f Fo(\(2)9294 68738 y Fi(n)9861
   6204 69162 y Fo(\)\()10790 68753 y Fd(1)p 10790 68933 406
   6205 43 v 10790 69500 a(2)11577 69162 y Fo(+)249 b Fj(\262)p
   6206 Fo(\),)381 b(w)-28 b(e)372 b(get)i Fj(p)336 b Fg(\270)20211
   6207 68753 y Fi(\262)p 19494 68933 1781 43 v 19494 69500 a
   6208 Fd(2+4)p Fi(\262)21407 69162 y Fo(.)550 b(With)-2022
   6209 70539 y Fj(\262)284 b(<)-110 70131 y Fd(1)p -110 70311
   6210 406 43 v -110 70877 a(2)429 70539 y Fo(,)342 b(w)-28
   6211 b(e)341 b(get)h(the)g(desired)f(result.)11441 b Fa(\244)28308
   6212 -915 y Ff(Lemma)553 b Fo(7.7.)499 b Fm(If)462 b Fg(j)p
   6213 Fe(S)p Fg(j)e Fj(>)g Fo(2)38807 -1339 y Fi(n)39373 -915
   6214 y Fo(\()39904 -1324 y Fd(1)p 39904 -1144 V 39904 -577
   6215 a(2)40740 -915 y Fo(+)298 b Fj(\262)p Fo(\))462 b Fm(but)g
   6216 Fj(F)142 b Fo(\()p Fe(S)p Fo(\))463 b Fm(do)-52 b(es)462
   6217 b(not)g(sp)-52 b(an)27201 247 y(al)52 b(l)464 b(of)e
   6218 Fe(Z)30725 -176 y Fi(n)30725 422 y(q)31292 247 y Fm(,)487
   6219 b(then)463 b(algorithm)g Fg(A)g Fm(do)-52 b(es)463 b(not)f(ab)-52
   6220 b(ort)463 b(in)g(step)g Fo(6)g Fm(with)27201 1409 y(pr)-52
   6221 b(ob)g(ability)365 b(at)h(le)-52 b(ast)366 b Fj(\262=)p
   6222 Fo(4)p Fm(.)28308 3449 y Ff(Pr)-22 b(oof.)498 b Fo(By)272
   6223 b(Lemma)g(7.6,)286 b(w)-28 b(e)271 b(kno)-28 b(w)270
   6224 b(that)i(with)f(probabilit)-28 b(y)271 b(at)27201 4611
   6225 y(least)353 b Fj(\262=)p Fo(4,)i(the)e(\257rst)f Fj(n)f
   6226 Fo(v)-28 b(ectors)353 b Fj(\302)p Fo(\()p Fj(S)41705
   6227 4722 y Fd(1)42166 4611 y Fo(\))p Fj(;)171 b(:)g(:)g(:)g(;)h(\302)p
   6228 Fo(\()p Fj(S)46502 4722 y Fi(n)47068 4611 y Fo(\))303
   6229 b Fg(2)f(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)51257 4188
   6230 y Fi(n)52176 4611 y Fo(will)27201 5773 y(span)283 b(all)g(of)g
   6231 Fe(Z)32673 5350 y Fi(n)32673 5949 y(q)33240 5773 y Fo(.)436
   6232 b(But)283 b(since)h Fj(F)142 b Fo(\()p Fe(S)p Fo(\))283
   6233 b(do)28 b(es)284 b(not)f(span)f(all)i(of)f Fe(Z)50300
   6234 5350 y Fi(n)50300 5949 y(q)50866 5773 y Fo(,)295 b(the)284
   6235 b Fj(n)27201 6936 y Fo(v)-28 b(ectors)331 b Fj(\302)p
   6236 Fo(\()p Fj(F)142 b Fo(\()p Fj(S)33525 7047 y Fd(1)33986
   6237 6936 y Fo(\)\))p Fj(;)172 b(:)f(:)g(:)g(;)g(\302)p Fo(\()p
   6238 Fj(F)142 b Fo(\()p Fj(S)39914 7047 y Fi(n)40481 6936
   6239 y Fo(\)\))285 b Fg(2)f(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)45032
   6240 6512 y Fi(n)45928 6936 y Fo(do)330 b(not)h(span)e(all)j(of)27201
   6241 8098 y Fe(Z)27865 7675 y Fi(n)27865 8273 y(q)28432 8098
   6242 y Fo(.)528 b(Therefore)365 b(there)h(is)g(no)g(matrix)g
   6243 Fj(M)433 b Fg(2)324 b Fj(GL)46337 8209 y Fi(n)46903 8098
   6244 y Fo(\()p Fe(Z)47965 8209 y Fi(q)48424 8098 y Fo(\))366
   6245 b(that)f(maps)27201 9260 y Fj(\302)p Fo(\()p Fj(F)142
   6246 b Fo(\()p Fj(S)30057 9371 y Fi(i)30409 9260 y Fo(\)\))376
   6247 b(to)h Fj(\302)p Fo(\()p Fj(S)34530 9371 y Fi(i)34881
   6248 9260 y Fo(\))f(for)g(all)h Fj(i)343 b Fo(=)f(1)p Fj(;)171
   6249 b(:)g(:)h(:)f(;)g(n)375 b Fo(and)h(so)g(the)h(algorithm)27201
   6250 10422 y(do)28 b(es)342 b(not)g(ab)28 b(ort)341 b(in)g(step)h(6.)14721
   6251 b Fa(\244)28198 12747 y Fo(Next,)507 b(w)-28 b(e)472
   6252 b(w)-28 b(an)g(t)472 b(to)h(b)28 b(ound)473 b(the)h(probabilit)-28
   6253 b(y)473 b(of)g(ab)28 b(orting)473 b(in)27201 13909 y(step)414
   6254 b(6)e(when)h Fj(F)142 b Fo(\()p Fe(S)p Fo(\))413 b(spans)g(all)g(of)g
   6255 Fe(Z)41935 13486 y Fi(n)41935 14085 y(q)42502 13909 y
   6256 Fo(.)670 b(F)-85 b(or)412 b(that,)431 b(w)-28 b(e)413
   6257 b(\257rst)f(need)27201 15072 y(the)418 b(follo)-28 b(wing)417
   6258 b(t)-28 b(w)g(o)416 b(lemmas)i(whic)-28 b(h)416 b(sho)-28
   6259 b(w)416 b(that)i Fj(F)559 b Fo(cannot)417 b(b)28 b(e)417
   6260 b(a)27201 16234 y(linear)495 b(map)g(on)f Fe(S)p Fo(.)916
   6261 b(Recall)495 b(that)g(a)f(p)28 b(erm)-28 b(utation)495
   6262 b(matrix)h(is)e(a)27201 17396 y(p)28 b(erm)-28 b(utation)256
   6263 b(of)g(the)g(ro)-28 b(ws)253 b(of)j(the)g(iden)-28 b(tit)g(y)256
   6264 b(matrix.)428 b(Let)256 b(us)f(de\257ne)27201 18559 y(the)495
   6265 b Fj(L)g Fo(norm)f(of)g(a)h(v)-28 b(ector)495 b Fj(b)539
   6266 b Fo(=)h(\()p Fj(b)41844 18670 y Fd(1)42306 18559 y Fj(;)171
   6267 b(:)g(:)g(:)g(;)g(b)45020 18670 y Fi(n)45586 18559 y
   6268 Fo(\))540 b Fg(2)g(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)50250
   6269 18135 y Fi(n)51311 18559 y Fo(to)495 b(b)28 b(e)27201
   6270 19721 y Fj(L)p Fo(\()p Fj(b)p Fo(\))285 b(=)30499 18974
   6271 y Fc(P)31579 19247 y Fi(n)31579 20025 y(i)p Fd(=1)33128
   6272 19721 y Fj(b)33567 19832 y Fi(i)33918 19721 y Fo(.)28308
   6273 21760 y Ff(Lemma)553 b Fo(7.8.)499 b Fm(L)-52 b(et)395
   6274 b Fg(j)p Fe(S)p Fg(j)338 b Fj(>)f Fo(\(5)p Fj(=)p Fo(8\)2)41518
   6275 21337 y Fi(n)42479 21760 y Fm(and)394 b Fj(F)479 b Fo(:)337
   6276 b Fe(S)g Fg(!)f(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)51001
   6277 21337 y Fi(n)51962 21760 y Fm(b)-52 b(e)396 b(a)27201
   6278 22923 y(line)-52 b(ar)440 b(function)e(such)h(that)g
   6279 Fj(F)142 b Fo(\()p Fe(S)p Fo(\))440 b Fm(sp)-52 b(ans)438
   6280 b(al)52 b(l)440 b(of)f Fe(Z)47730 22500 y Fi(n)47730
   6281 23098 y(q)48736 22923 y Fm(and)f Fj(F)581 b Fm(pr)-52
   6282 b(e-)27201 24085 y(serves)495 b(the)e Fj(L)f Fm(norm)h(\(i.e.)850
   6283 b Fj(L)p Fo(\()p Fj(F)142 b Fo(\()p Fj(v)36 b Fo(\)\))515
   6284 b(=)f Fj(L)p Fo(\()p Fj(v)36 b Fo(\))493 b Fm(for)g(al)52
   6285 b(l)493 b Fj(v)550 b Fg(2)514 b Fe(S)p Fm(\).)27201 25247
   6286 y(Then)394 b(ther)-52 b(e)393 b(exists)h(a)e(p)-52 b(ermutation)393
   6287 b(matrix)g Fj(P)476 b Fg(2)333 b Fj(GL)48930 25358 y
   6288 Fi(n)49496 25247 y Fo(\()p Fe(Z)50558 25358 y Fi(q)51017
   6289 25247 y Fo(\))393 b Fm(such)27201 26410 y(that)365 b
   6290 Fj(F)142 b Fo(\()p Fj(v)36 b Fo(\))285 b(=)g Fj(P)370
   6291 b Fg(\242)227 b Fj(v)403 b Fm(for)365 b(al)52 b(l)367
   6292 b Fj(v)320 b Fg(2)284 b Fe(S)p Fm(.)28308 28449 y Ff(Pr)-22
   6293 b(oof.)498 b Fo(If)236 b Fj(F)426 b Fo(:)285 b Fe(S)g
   6294 Fg(!)g(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)39459 28026
   6295 y Fi(n)40261 28449 y Fo(is)237 b(a)f(linear)h(function,)258
   6296 b(w)-28 b(e)236 b(can)g(\257nd)27201 29611 y(an)374 b
   6297 Fj(n)p Fo(-b)-28 b(y-)p Fj(n)374 b Fo(matrix)h Fj(P)516
   6298 b Fo(suc)-28 b(h)373 b(that)i(for)e(all)i Fj(S)395 b
   6299 Fg(2)339 b Fe(S)p Fo(,)383 b Fj(F)142 b Fo(\()p Fj(S)57
   6300 b Fo(\))339 b(=)h Fj(P)391 b Fg(\242)250 b Fj(S)57 b
   6301 Fo(.)27201 30774 y(Note)253 b(that)g(the)h(matrix)f Fj(P)395
   6302 b Fo(m)-28 b(ust)252 b(b)28 b(e)254 b(of)e(full)h(rank)g(since)g
   6303 Fj(F)142 b Fo(\()p Fe(S)p Fo(\))254 b(spans)27201 31936
   6304 y(all)342 b(of)e Fe(Z)30452 31513 y Fi(n)30452 32111
   6305 y(q)31019 31936 y Fo(.)455 b(F)-85 b(or)340 b Fj(i;)171
   6306 b(j)341 b Fg(2)284 b Fo(1)p Fj(;)171 b(:)g(:)h(:)f(;)g(n)p
   6307 Fo(,)340 b(w)-28 b(e)340 b(write)h Fj(p)44796 32047 y
   6308 Fi(i;j)46118 31936 y Fo(the)g(en)-28 b(try)341 b(of)f
   6309 Fj(P)482 b Fo(at)27201 33098 y(ro)-28 b(w)340 b Fj(i)i
   6310 Fo(and)f(column)h Fj(j)57 b Fo(.)28198 34261 y(W)-85
   6311 b(e)435 b(start)g(b)-28 b(y)435 b(sho)-28 b(wing)434
   6312 b(that)h Fj(p)40941 34372 y Fi(i;j)42363 34261 y Fg(2)440
   6313 b(f\241)p Fo(1)p Fj(;)172 b Fo(0)p Fj(;)f Fo(1)p Fg(g)435
   6314 b Fo(for)f(all)i Fj(i;)171 b(j)498 b Fg(2)27201 35423
   6315 y Fo(1)p Fj(;)171 b(:)g(:)h(:)f(;)g(n)p Fo(.)873 b(Recall)482
   6316 b(that)f(for)f Fj(i)517 b Fo(=)g(1)p Fj(;)171 b(:)g(:)g(:)g(;)g(n)481
   6317 b Fo(w)-28 b(e)480 b(denote)h(the)h Fj(i)p Fo(-th)27201
   6318 36585 y(unit)461 b(v)-28 b(ector)460 b(b)-28 b(y)460
   6319 b Fj(e)34692 36696 y Fi(i)35526 36585 y Fg(2)482 b(f)p
   6320 Fo(0)p Fj(;)171 b Fo(1)p Fg(g)39194 36162 y Fi(n)39761
   6321 36585 y Fo(.)812 b(By)460 b(Lemma)h(7.4,)491 b(w)-28
   6322 b(e)460 b(can)g(\257nd)27201 37748 y Fj(v)27698 37859
   6323 y Fd(0)28160 37748 y Fj(;)171 b(v)29112 37859 y Fd(1)29866
   6324 37748 y Fg(2)293 b Fe(S)346 b Fo(suc)-28 b(h)346 b(that)g
   6325 Fj(v)36776 37859 y Fd(1)37531 37748 y Fo(=)293 b Fj(v)39117
   6326 37859 y Fd(0)39809 37748 y Fo(+)232 b Fj(e)41313 37859
   6327 y Fi(i)41664 37748 y Fo(.)470 b(Since)347 b Fj(v)45594
   6328 37859 y Fd(0)46056 37748 y Fj(;)171 b(v)47008 37859 y
   6329 Fd(1)47762 37748 y Fg(2)293 b Fe(S)p Fo(,)348 b(w)-28
   6330 b(e)346 b(kno)-28 b(w)27201 38910 y(that)263 b Fj(F)142
   6331 b Fo(\()p Fj(v)31032 39021 y Fd(0)31493 38910 y Fo(\))285
   6332 b(=)g Fj(P)142 b(v)34550 39021 y Fd(0)35274 38910 y Fo(and)261
   6333 b Fj(F)142 b Fo(\()p Fj(v)38876 39021 y Fd(1)39338 38910
   6334 y Fo(\))285 b(=)g Fj(P)142 b(v)42395 39021 y Fd(1)43118
   6335 38910 y Fo(are)262 b(in)h Fg(f)p Fo(0)p Fj(;)171 b Fo(1)p
   6336 Fg(g)48367 38487 y Fi(n)48933 38910 y Fo(.)429 b(But)263
   6337 b(since)27201 40072 y Fj(v)27698 40183 y Fd(1)28574 40072
   6338 y Fo(=)415 b Fj(v)30282 40183 y Fd(0)31023 40072 y Fo(+)280
   6339 b Fj(e)32575 40183 y Fi(i)33345 40072 y Fo(w)-28 b(e)419
   6340 b(ha)-28 b(v)g(e)419 b Fj(P)142 b(e)38642 40183 y Fi(i)39408
   6341 40072 y Fo(=)415 b Fj(P)142 b(v)41912 40183 y Fd(1)42653
   6342 40072 y Fg(\241)280 b Fj(P)142 b(v)45022 40183 y Fd(0)45483
   6343 40072 y Fo(.)689 b(This)419 b(sho)-28 b(ws)418 b(that)27201
   6344 41235 y(the)g(v)-28 b(ector)417 b Fj(P)142 b(e)33463
   6345 41346 y Fi(i)34231 41235 y Fo(can)417 b(b)28 b(e)418
   6346 b(expressed)g(as)f(the)h(di\256erence)g(b)28 b(et)-28
   6347 b(w)g(een)27201 42397 y(t)g(w)g(o)430 b(v)-28 b(ectors)431
   6348 b(in)g Fg(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)36580 41974
   6349 y Fi(n)37577 42397 y Fo(and)431 b(therefore)g(all)h(the)f(co)28
   6350 b(ordinates)432 b(of)27201 43559 y Fj(P)142 b(e)28473
   6351 43670 y Fi(i)29154 43559 y Fo(are)329 b(in)h Fg(f\241)p
   6352 Fo(1)p Fj(;)171 b Fo(0)p Fj(;)g Fo(1)p Fg(g)p Fo(.)452
   6353 b(But)330 b(the)f(v)-28 b(ector)330 b Fj(P)142 b(e)45144
   6354 43670 y Fi(i)45825 43559 y Fo(is)329 b(exactly)j(the)e
   6355 Fj(i)p Fo(-th)27201 44721 y(column)374 b(of)e(the)i(matrix)f
   6356 Fj(P)142 b Fo(.)550 b(Since)374 b(this)f(argumen)-28
   6357 b(t)373 b(w)-28 b(orks)371 b(for)i(all)27201 45884 y
   6358 Fj(i)480 b Fo(=)f(1)p Fj(;)171 b(:)g(:)g(:)g(;)h(n)p
   6359 Fo(,)487 b(w)-28 b(e)457 b(ha)-28 b(v)g(e)458 b(sho)-28
   6360 b(wn)457 b(that)h Fj(p)43657 45995 y Fi(i;j)45118 45884
   6361 y Fg(2)479 b(f\241)p Fo(1)p Fj(;)171 b Fo(0)p Fj(;)g
   6362 Fo(1)p Fg(g)458 b Fo(for)g(all)27201 47046 y Fj(i;)172
   6363 b(j)341 b Fg(2)284 b Fo(1)p Fj(;)171 b(:)g(:)g(:)g(;)g(n)p
   6364 Fo(.)28198 48208 y(Recall)287 b(the)f(de\257nition)h(of)f(the)g(norm)g
   6365 Fj(L)g Fo(giv)-28 b(en)286 b(ab)28 b(o)-28 b(v)g(e.)438
   6366 b(W)-85 b(e)286 b(ha)-28 b(v)g(e:)28164 50007 y Fj(L)p
   6367 Fo(\()p Fj(P)142 b(e)30531 50118 y Fi(i)30882 50007 y
   6368 Fo(\))996 b(=)h Fj(L)p Fo(\()p Fj(P)142 b(v)36457 50118
   6369 y Fd(1)37146 50007 y Fg(\241)228 b Fj(P)142 b(v)39463
   6370 50118 y Fd(0)39925 50007 y Fo(\))284 b(=)h Fj(L)p Fo(\()p
   6371 Fj(P)142 b(v)44076 50118 y Fd(1)44537 50007 y Fo(\))228
   6372 b Fg(\241)g Fj(L)p Fo(\()p Fj(P)142 b(v)48575 50118 y
   6373 Fd(0)49037 50007 y Fo(\))32276 51501 y(=)997 b Fj(L)p
   6374 Fo(\()p Fj(F)142 b Fo(\()p Fj(v)36855 51612 y Fd(1)37317
   6375 51501 y Fo(\)\))228 b Fg(\241)g Fj(L)p Fo(\()p Fj(F)142
   6376 b Fo(\()p Fj(v)42151 51612 y Fd(0)42612 51501 y Fo(\)\))285
   6377 b(=)g Fj(L)p Fo(\()p Fj(v)46366 51612 y Fd(1)46827 51501
   6378 y Fo(\))228 b Fg(\241)g Fj(L)p Fo(\()p Fj(v)50069 51612
   6379 y Fd(0)50531 51501 y Fo(\))284 b(=)h(1)27201 53299 y(This)341
   6380 b(implies)i(the)f(follo)-28 b(wing)341 b(equalit)-28
   6381 b(y)-85 b(,)344 b(whic)-28 b(h)340 b(w)-28 b(e)341 b(will)h(use)g
   6382 (later:)37438 54636 y Fi(n)36954 54935 y Fc(X)36991 57036
   6383 y Fi(j)39 b Fd(=1)39087 54636 y Fi(n)38603 54935 y Fc(X)38681
   6384 57036 y Fi(i)p Fd(=1)40253 55882 y Fj(p)40767 55993 y
   6385 Fi(i;j)42033 55882 y Fo(=)285 b Fj(n:)28198 58769 y Fo(Let)261
   6386 b(us)g(no)-28 b(w)260 b(consider)h(ro)-28 b(w)260 b Fj(R)39809
   6387 58880 y Fi(i)40444 58769 y Fo(=)285 b(\()p Fj(p)42437
   6388 58880 y Fi(i;)p Fd(1)43447 58769 y Fj(;)171 b(:)g(:)g(:)g(;)g(p)46236
   6389 58880 y Fi(i;n)47351 58769 y Fo(\))261 b(of)g(the)h(matrix)27201
   6390 59932 y Fj(P)518 b Fo(for)375 b Fj(i)342 b Fg(2)f Fo(1)p
   6391 Fj(;)172 b(:)f(:)g(:)g(;)g(n)p Fo(.)558 b(F)-85 b(or)375
   6392 b(an)-28 b(y)376 b Fj(v)i Fg(2)341 b Fe(S)p Fo(,)385
   6393 b(w)-28 b(e)375 b(kno)-28 b(w)375 b(that)h(the)h(scalar)27201
   6394 61094 y(pro)28 b(duct)294 b Fj(R)31771 61205 y Fi(i)32255
   6395 61094 y Fg(\242)133 b Fj(v)320 b Fg(2)284 b(f)p Fo(0)p
   6396 Fj(;)171 b Fo(1)p Fg(g)294 b Fo(since)h Fj(F)142 b Fo(\()p
   6397 Fj(v)36 b Fo(\))285 b(=)f Fj(P)142 b(v)321 b Fg(2)284
   6398 b(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)48289 60671 y Fi(n)48855
   6399 61094 y Fo(.)440 b(But)294 b Fg(j)p Fe(S)p Fg(j)286 b
   6400 Fj(>)27201 62256 y Fo(\(5)p Fj(=)p Fo(8\)2)30045 61833
   6401 y Fi(n)30924 62256 y Fo(b)-28 b(y)312 b(assumption.)446
   6402 b(This)312 b(implies)i(that)e(the)g(v)-28 b(ector)313
   6403 b Fj(R)51621 62367 y Fi(i)52284 62256 y Fo(has)27201
   6404 63419 y(at)426 b(most)g(11)g(non-zero)f(elemen)-28 b(ts)427
   6405 b(\(either)g(1)f(or)f Fg(\241)p Fo(1\).)709 b(Indeed,)447
   6406 b(if)27201 64581 y Fj(R)27976 64692 y Fi(i)28718 64581
   6407 y Fo(had)391 b Fj(z)410 b(>)368 b Fo(12)390 b(non-zero)h(elemen)-28
   6408 b(ts,)406 b(it)391 b(w)-28 b(ould)390 b(map)h(at)g(most)g(a)27201
   6409 65743 y(fraction)459 b(2)31616 64938 y Fc(\241)32085
   6410 65293 y Fi(z)26 b(=)p Fd(2)32085 66083 y Fi(z)g(=)p Fd(4)33338
   6411 64938 y Fc(\242)33807 65743 y Fj(=)p Fo(2)34831 65320
   6412 y Fi(z)g(=)p Fd(2)36619 65743 y Fj(<)481 b Fo(5)p Fj(=)p
   6413 Fo(8)459 b(of)g(v)-28 b(ectors)459 b(in)g Fg(f)p Fo(0)p
   6414 Fj(;)171 b Fo(1)p Fg(g)48586 65320 y Fi(n)49612 65743
   6415 y Fo(to)459 b Fg(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)p
   6416 Fo(.)27201 67052 y(Up)333 b(to)g(a)g(re-ordering)f(of)h(the)g(columns)h
   6417 (of)e(the)i(matrix)f Fj(P)142 b Fo(,)335 b(w)-28 b(e)332
   6418 b(can)27201 68214 y(assume)377 b(that)g(the)g Fj(z)387
   6419 b Fg(\267)343 b Fo(11)377 b(non-zero)f(co)28 b(ordinates)377
   6420 b(of)g Fj(R)49874 68325 y Fi(i)50601 68214 y Fo(are)g(the)27201
   6421 69377 y(\257rst)454 b Fj(z)498 b Fo(co)28 b(ordinates.)796
   6422 b(An)455 b(exhaustiv)-28 b(e)456 b(searc)-28 b(h)454
   6423 b(among)g(all)i(2)52208 68953 y Fd(1)52972 69377 y Fo(+)27201
   6424 70539 y(2)27713 70116 y Fd(2)28396 70539 y Fo(+)221 b
   6425 Fj(:)171 b(:)g(:)222 b Fo(+)f(2)32358 70116 y Fd(11)33510
   6426 70539 y Fo(=)285 b(2)35103 70116 y Fd(12)36191 70539
   6427 y Fg(\241)221 b Fo(2)338 b(p)28 b(ossibilities)341 b(of)d(all)h(v)-28
   6428 b(ectors)338 b(that)g(map)27201 71701 y(a)377 b(fraction)f(strictly)h
   6429 (greater)f(than)h(5)p Fj(=)p Fo(8)f(of)g(all)h(v)-28
   6430 b(ectors)376 b(in)h Fg(f)p Fo(0)p Fj(;)171 b Fo(1)p Fg(g)53202
   6431 71278 y Fi(n)p eop
   6432 %%Page: 9 9
   6433 9 8 bop -2022 -915 a Fo(to)391 b Fg(f)p Fo(0)p Fj(;)171
   6434 b Fo(1)p Fg(g)391 b Fo(rev)-28 b(eals)392 b(that)f(ev)-28
   6435 b(ery)392 b(ro)-28 b(w)390 b Fj(R)13401 -804 y Fi(i)14143
   6436 -915 y Fo(of)h Fj(P)533 b Fo(m)-28 b(ust)391 b(b)28 b(e)391
   6437 b(of)g(one)h(of)-2022 247 y(the)341 b(follo)-28 b(wing)342
   6438 b(4)f(t)-28 b(yp)28 b(es:)-542 2269 y Fg(\262)498 b Fo(T)-28
   6439 b(yp)28 b(e)458 b(1:)687 b(there)458 b(is)g(only)g(1)f(non-zero)g(co)28
   6440 b(ordinate)458 b(and)f(its)468 3432 y(v)-57 b(alue)343
   6441 b(is)e(1.)-542 5517 y Fg(\262)498 b Fo(T)-28 b(yp)28
   6442 b(e)231 b(2:)400 b(there)230 b(are)g(only)h(2)f(non-zero)h(co)28
   6443 b(ordinates)230 b(and)g(their)468 6679 y(v)-57 b(alues)343
   6444 b(are)e(1)p Fj(;)171 b Fo(1.)-542 8765 y Fg(\262)498
   6445 b Fo(T)-28 b(yp)28 b(e)231 b(3:)400 b(there)230 b(are)g(only)h(2)f
   6446 (non-zero)h(co)28 b(ordinates)230 b(and)g(their)468 9927
   6447 y(v)-57 b(alues)343 b(are)e(1)p Fj(;)171 b Fg(\241)p
   6448 Fo(1.)-542 12012 y Fg(\262)498 b Fo(T)-28 b(yp)28 b(e)231
   6449 b(4:)400 b(there)230 b(are)g(only)h(3)f(non-zero)h(co)28
   6450 b(ordinates)230 b(and)g(their)468 13175 y(v)-57 b(alues)343
   6451 b(are)e(1)p Fj(;)171 b Fo(1)p Fj(;)g Fg(\241)p Fo(1.)-2022
   6452 15197 y(Note)470 b(that)h(no)f(ro)-28 b(w)469 b(can)h(ha)-28
   6453 b(v)g(e)470 b(all)h(zero)f(co)28 b(ordinates)471 b(since)g(the)-2022
   6454 16359 y(matrix)401 b Fj(P)543 b Fo(is)401 b(of)g(full)g(rank.)634
   6455 b(Let)402 b Fj(t)11935 16470 y Fd(1)12396 16359 y Fj(;)171
   6456 b(t)13221 16470 y Fd(2)13682 16359 y Fj(;)g(t)14507 16470
   6457 y Fd(3)14968 16359 y Fj(;)g(t)15793 16470 y Fd(4)16655
   6458 16359 y Fo(b)28 b(e)401 b(the)h(n)-28 b(um)g(b)28 b(er)400
   6459 b(of)-2022 17521 y(ro)-28 b(ws)430 b(of)h Fj(P)574 b
   6460 Fo(of)432 b(t)-28 b(yp)28 b(e)432 b(1)p Fj(;)171 b Fo(2)p
   6461 Fj(;)g Fo(3)432 b(and)f(4)h(resp)28 b(ectiv)-28 b(ely)-85
   6462 b(.)729 b(Observ)-28 b(e)432 b(that)-2022 18684 y Fj(t)-1652
   6463 18795 y Fd(1)-964 18684 y Fo(+)228 b Fj(t)430 18795 y
   6464 Fd(2)1119 18684 y Fo(+)f Fj(t)2512 18795 y Fd(3)3201
   6465 18684 y Fo(+)h Fj(t)4595 18795 y Fd(4)5340 18684 y Fo(=)285
   6466 b Fj(n)p Fo(.)455 b(W)-85 b(e)342 b(also)g(kno)-28 b(w)340
   6467 b(that)4436 21274 y Fj(t)4806 21385 y Fd(1)5494 21274
   6468 y Fo(+)228 b(2)p Fj(t)7400 21385 y Fd(2)8089 21274 y
   6469 Fo(+)f Fj(t)9482 21385 y Fd(4)10228 21274 y Fo(=)11793
   6470 20029 y Fi(n)11309 20328 y Fc(X)11346 22428 y Fi(j)39
   6471 b Fd(=1)13442 20029 y Fi(n)12958 20328 y Fc(X)13036 22428
   6472 y Fi(i)p Fd(=1)14608 21274 y Fj(p)15122 21385 y Fi(i;j)16388
   6473 21274 y Fo(=)285 b Fj(n)-2022 24198 y Fo(and)442 b(therefore)g
   6474 Fj(t)4841 24309 y Fd(2)5755 24198 y Fo(=)453 b Fj(t)7374
   6475 24309 y Fd(3)7835 24198 y Fo(.)758 b(No)-28 b(w)441 b(supp)28
   6476 b(ose)443 b Fj(t)15633 24309 y Fd(2)16546 24198 y Fg(\270)453
   6477 b Fo(1.)759 b(The)442 b(matrix)-2022 25360 y Fj(P)519
   6478 b Fo(then)379 b(con)-28 b(tains)378 b(at)g(least)g(one)h(ro)-28
   6479 b(w)376 b Fj(v)13706 25471 y Fd(2)14546 25360 y Fo(of)h(t)-28
   6480 b(yp)28 b(e)379 b(2)f(and)g(one)g(ro)-28 b(w)-2022 26523
   6481 y Fj(v)-1525 26634 y Fd(3)-629 26523 y Fo(of)435 b(t)-28
   6482 b(yp)28 b(e)436 b(3.)737 b(Up)434 b(to)h(symmetries)i(and)e(p)28
   6483 b(erm)-28 b(utations)435 b(on)g(the)-2022 27685 y(columns)408
   6484 b(of)f Fj(P)142 b Fo(,)425 b(there)408 b(are)g(4)f(distinct)i(w)-28
   6485 b(a)g(ys)406 b(in)i(whic)-28 b(h)407 b(the)i(non-)-2022
   6486 28847 y(zero)271 b(co)28 b(ordinates)272 b(of)f Fj(v)7036
   6487 28958 y Fd(2)7769 28847 y Fo(and)g Fj(v)10187 28958 y
   6488 Fd(3)10920 28847 y Fo(can)g(b)28 b(e)273 b(arranged)d(relativ)-28
   6489 b(e)273 b(to)e(one)-2022 30010 y(another:)-2022 30889
   6490 y Fc(\265)-771 31694 y Fo(1)1793 b(1)-771 32857 y(1)997
   6491 b Fg(\241)p Fo(1)2544 30889 y Fc(\266\265)4550 31694
   6492 y Fo(1)f(1)g(0)1793 b(0)4550 32857 y(0)996 b(0)g(1)g
   6493 Fg(\241)p Fo(1)10881 30889 y Fc(\266\265)12887 31694
   6494 y Fo(1)g(1)1792 b(0)12887 32857 y(1)996 b(0)g Fg(\241)p
   6495 Fo(1)17709 30889 y Fc(\266\265)20511 31694 y Fo(1)h(1)f(0)19715
   6496 32857 y Fg(\241)p Fo(1)h(0)f(1)24538 30889 y Fc(\266)-2022
   6497 34824 y Fo(All)337 b(of)f(these)i(arrangemen)-28 b(ts)336
   6498 b(map)g(at)h(most)g(a)f(fraction)h(5)p Fj(=)p Fo(8)f(of)h(all)-2022
   6499 35986 y(v)-28 b(ectors)468 b(in)h Fg(f)p Fo(0)p Fj(;)171
   6500 b Fo(1)p Fg(g)5408 35563 y Fi(n)6443 35986 y Fo(to)469
   6501 b(v)-28 b(ectors)469 b(in)f Fg(f)p Fo(0)p Fj(;)171 b
   6502 Fo(1)p Fg(g)15252 35563 y Fi(n)15819 35986 y Fo(.)837
   6503 b(This)468 b(means)h(that)-2022 37148 y Fj(t)-1652 37259
   6504 y Fd(2)-907 37148 y Fo(=)285 b Fj(t)544 37259 y Fd(3)1289
   6505 37148 y Fo(=)g(0.)-1026 38310 y(It)504 b(remains)h(to)g(sho)-28
   6506 b(w)503 b(that)h Fj(t)11095 38421 y Fd(4)12113 38310
   6507 y Fo(=)556 b(0.)945 b(W)-85 b(e)505 b(pro)28 b(ceed)505
   6508 b(b)-28 b(y)504 b(con-)-2022 39473 y(tradiction.)674
   6509 b(Assume)414 b(there)g(is)h(at)f(least)h(one)f(ro)-28
   6510 b(w)413 b Fj(v)19257 39584 y Fd(4)20132 39473 y Fo(of)h(t)-28
   6511 b(yp)28 b(e)415 b(4.)-2022 40635 y(Up)440 b(to)g(a)g(re-ordering)h(of)f
   6512 (the)h(columns)f(of)g Fj(P)142 b Fo(,)466 b(w)-28 b(e)440
   6513 b(ma)-28 b(y)440 b(assume)-2022 41797 y Fj(v)-1525 41908
   6514 y Fd(4)-648 41797 y Fo(=)417 b(\()p Fg(\241)p Fo(1)p
   6515 Fj(;)172 b Fo(1)p Fj(;)f Fo(1)p Fj(;)g Fo(0)p Fj(;)g
   6516 Fo(0)p Fj(;)g(:)g(:)g(:)g(;)g Fo(0\).)693 b(Since)421
   6517 b Fj(L)p Fo(\()p Fj(P)142 b(e)15422 41908 y Fd(1)15884
   6518 41797 y Fo(\))416 b(=)h(1,)440 b(there)421 b(are)f(at)-2022
   6519 42960 y(least)342 b(t)-28 b(w)g(o)341 b(other)g(ro)-28
   6520 b(ws)340 b(of)i Fj(P)484 b Fo(that)342 b(ha)-28 b(v)g(e)341
   6521 b(a)h(1)f(in)h(the)h(\257rst)e(column.)-2022 44122 y(These)445
   6522 b(cannot)f(b)28 b(oth)446 b(b)28 b(e)445 b(of)g(t)-28
   6523 b(yp)28 b(e)446 b(1)e(for)h(otherwise)g(they)g(w)-28
   6524 b(ould)-2022 45284 y(b)28 b(e)329 b(linearly)h(dep)28
   6525 b(enden)-28 b(t)329 b(and)f(that)h(cannot)f(happ)28 b(en)329
   6526 b(since)g Fj(P)471 b Fo(is)329 b(of)-2022 46447 y(full)452
   6527 b(rank.)785 b(Therefore)452 b(there)g(is)f(at)h(least)g(one)g(other)g
   6528 (ro)-28 b(w)450 b Fj(v)22843 46023 y Fr(0)22807 46622
   6529 y Fd(4)23720 46447 y Fo(of)-2022 47609 y(t)-28 b(yp)28
   6530 b(e)334 b(4)g(whic)-28 b(h)333 b(has)h(a)g(1)g(in)g(the)g(\257rst)g
   6531 (column.)453 b(Up)334 b(to)g(symmetries)-2022 48771 y(and)341
   6532 b(p)28 b(erm)-28 b(utations)342 b(on)g(the)g(columns)g(of)g
   6533 Fj(P)142 b Fo(,)342 b(there)g(are)g(4)g(distinct)-2022
   6534 49934 y(w)-28 b(a)g(ys)332 b(in)i(whic)-28 b(h)333 b(the)h(non-zero)g
   6535 (co)28 b(ordinates)334 b(of)g Fj(v)17552 50045 y Fd(4)18347
   6536 49934 y Fo(and)f Fj(v)20863 49510 y Fr(0)20827 50109
   6537 y Fd(4)21622 49934 y Fo(can)h(b)28 b(e)-2022 51096 y(arranged)340
   6538 b(relativ)-28 b(e)343 b(to)e(one)h(another:)2882 52170
   6539 y Fc(\265)4134 52974 y Fg(\241)p Fo(1)1793 b(1)996 b(1)4930
   6540 54137 y(1)h Fg(\241)p Fo(1)f(1)9753 52170 y Fc(\266\265)11759
   6541 52974 y Fg(\241)p Fo(1)g(1)h(1)1792 b(0)12555 54137 y(1)996
   6542 b(1)h(0)f Fg(\241)p Fo(1)18886 52170 y Fc(\266)1374 56063
   6543 y(\265)2626 56867 y Fg(\241)p Fo(1)1793 b(1)996 b(1)g(0)3422
   6544 58030 y(1)g Fg(\241)p Fo(1)h(0)f(1)9753 56063 y Fc(\266\265)11759
   6545 56867 y Fg(\241)p Fo(1)g(1)h(1)f(0)1793 b(0)12555 58030
   6546 y(1)996 b(0)h(0)f(1)g Fg(\241)p Fo(1)20395 56063 y Fc(\266)-2022
   6547 59934 y Fo(All)337 b(of)f(these)i(arrangemen)-28 b(ts)336
   6548 b(map)g(at)h(most)g(a)f(fraction)h(5)p Fj(=)p Fo(8)f(of)h(all)-2022
   6549 61096 y(v)-28 b(ectors)468 b(in)h Fg(f)p Fo(0)p Fj(;)171
   6550 b Fo(1)p Fg(g)5408 60673 y Fi(n)6443 61096 y Fo(to)469
   6551 b(v)-28 b(ectors)469 b(in)f Fg(f)p Fo(0)p Fj(;)171 b
   6552 Fo(1)p Fg(g)15252 60673 y Fi(n)15819 61096 y Fo(.)837
   6553 b(This)468 b(means)h(that)-2022 62258 y Fj(t)-1652 62369
   6554 y Fd(4)-878 62258 y Fo(=)313 b(0.)507 b(Therefore)358
   6555 b(all)h(the)g(ro)-28 b(ws)357 b(of)i Fj(P)500 b Fo(are)358
   6556 b(of)h(t)-28 b(yp)28 b(e)359 b(1)g(so)f(that)h Fj(P)-2022
   6557 63421 y Fo(is)341 b(a)g(p)28 b(erm)-28 b(utation)342
   6558 b(matrix.)14754 b Fa(\244)-915 66858 y Ff(Lemma)552 b
   6559 Fo(7.9.)499 b Fm(L)-52 b(et)478 b Fg(j)p Fe(S)p Fg(j)487
   6560 b Fj(>)f Fo(2)10343 66435 y Fi(n)10910 66858 y Fj(=)p
   6561 Fo(2)477 b Fm(and)f Fj(F)628 b Fo(:)486 b Fe(S)h Fg(!)f(f)p
   6562 Fo(0)p Fj(;)171 b Fo(1)p Fg(g)21613 66435 y Fi(n)22656
   6563 66858 y Fm(b)-52 b(e)478 b(a)-2022 68020 y(line)-52 b(ar)290
   6564 b(function)g(such)g(that)g Fj(F)142 b Fo(\()p Fe(S)p
   6565 Fo(\))291 b Fm(sp)-52 b(ans)290 b(al)52 b(l)291 b(of)f
   6566 Fe(Z)17316 67597 y Fi(n)17316 68196 y(q)17882 68020 y
   6567 Fm(.)445 b(If)291 b(ther)-52 b(e)290 b(exists)-2022 69183
   6568 y(a)443 b(p)-52 b(ermutation)442 b(matrix)h Fj(P)567
   6569 b Fg(2)424 b Fj(GL)11944 69294 y Fi(n)12510 69183 y Fo(\()p
   6570 Fe(Z)13572 69294 y Fi(q)14031 69183 y Fo(\))443 b Fm(such)h(that)e
   6571 Fj(F)142 b Fo(\()p Fj(v)36 b Fo(\))425 b(=)h Fj(P)142
   6572 b(v)-2022 70345 y Fm(for)458 b(al)52 b(l)460 b Fj(v)488
   6573 b Fg(2)453 b Fe(S)p Fm(,)482 b(then)459 b(the)f(outputs)g
   6574 Fj(B)508 b Fm(of)459 b(the)f(mix)h(server)i(ar)-52 b(e)459
   6575 b(a)-2022 71507 y(p)-52 b(ermutation)365 b(of)g(the)h(inputs)f
   6576 Fj(A)p Fm(.)28308 -915 y Ff(Pr)-22 b(oof.)498 b Fo(Consider)227
   6577 b(the)h(unit)g(v)-28 b(ector)228 b Fj(e)43537 -804 y
   6578 Fi(j)44198 -915 y Fo(for)f Fj(j)341 b Fg(2)284 b Fo(1)p
   6579 Fj(;)171 b(:)g(:)g(:)g(;)h(n)p Fo(.)417 b(Since)27201
   6580 247 y Fj(P)455 b Fo(is)312 b(a)h(p)28 b(erm)-28 b(utation)312
   6581 b(matrix)i(there)e(exists)i(a)e(unit)h(v)-28 b(ector)312
   6582 b Fj(e)51136 358 y Fi(i)51800 247 y Fo(suc)-28 b(h)27201
   6583 1409 y(that)410 b Fj(P)142 b(e)30760 1520 y Fi(i)31509
   6584 1409 y Fo(=)398 b Fj(e)33179 1520 y Fi(j)33613 1409 y
   6585 Fo(.)659 b(By)410 b(Lemma)g(7.4,)427 b(w)-28 b(e)409
   6586 b(can)h(\257nd)e Fj(v)36 b(;)172 b(w)424 b Fg(2)397 b
   6587 Fe(S)410 b Fo(suc)-28 b(h)27201 2571 y(that)408 b Fj(w)422
   6588 b Fo(=)395 b Fj(v)308 b Fo(+)272 b Fj(e)34182 2682 y
   6589 Fi(i)34534 2571 y Fo(.)654 b(Let)408 b(us)g(write)g(the)g(co)28
   6590 b(ordinates)408 b(of)g Fj(v)443 b Fo(and)408 b Fj(w)27201
   6591 3734 y Fo(as)441 b Fj(v)486 b Fo(=)450 b(\()p Fj(v)31682
   6592 3845 y Fd(1)32144 3734 y Fj(;)171 b(:)g(:)g(:)g(;)h(v)34917
   6593 3845 y Fi(n)35483 3734 y Fo(\))441 b(and)g Fj(w)476 b
   6594 Fo(=)451 b(\()p Fj(w)42002 3845 y Fd(1)42463 3734 y Fj(;)171
   6595 b(:)g(:)g(:)g(;)g(w)45472 3845 y Fi(n)46039 3734 y Fo(\).)753
   6596 b(W)-85 b(e)441 b(also)g(write)27201 4896 y Fj(F)142
   6597 b Fo(\()p Fj(v)36 b Fo(\))285 b(=)g(\()p Fj(v)31623 4473
   6598 y Fr(0)31587 5071 y Fd(1)32049 4896 y Fj(;)171 b(:)g(:)g(:)g(;)g(v)
   6599 34857 4473 y Fr(0)34821 5071 y Fi(n)35388 4896 y Fo(\))342
   6600 b(and)f Fj(F)142 b Fo(\()p Fj(w)27 b Fo(\))284 b(=)h(\()p
   6601 Fj(w)42996 4473 y Fr(0)42969 5071 y Fd(1)43430 4896 y
   6602 Fj(;)171 b(:)g(:)g(:)h(;)f(w)46467 4473 y Fr(0)46440
   6603 5071 y Fi(n)47006 4896 y Fo(\).)456 b(W)-85 b(e)341 b(ha)-28
   6604 b(v)g(e)32095 6792 y Fj(F)142 b Fo(\()p Fj(w)27 b Fo(\))284
   6605 b(=)h Fj(P)142 b(w)311 b Fo(=)285 b Fj(P)142 b(v)263
   6606 b Fo(+)228 b Fj(P)142 b(e)42587 6903 y Fi(i)43223 6792
   6607 y Fo(=)285 b Fj(F)142 b Fo(\()p Fj(v)36 b Fo(\))228 b(+)g
   6608 Fj(e)48157 6903 y Fi(j)48591 6792 y Fj(:)27201 8687 y
   6609 Fo(This)341 b(implies)i(that)35384 10121 y Fi(n)34986
   6610 10420 y Fc(Y)34900 12542 y Fi(k)14 b Fd(=1)36380 11367
   6611 y Fo(\()p Fj(a)37320 11485 y Fi(k)37828 11367 y Fo(\))38226
   6612 10888 y Fi(w)38819 11044 y Fb(k)39644 11367 y Fo(=)284
   6613 b Fj(a)41266 11478 y Fi(i)42273 10121 y(n)41874 10420
   6614 y Fc(Y)41789 12542 y Fi(k)14 b Fd(=1)43269 11367 y Fo(\()p
   6615 Fj(a)44209 11485 y Fi(k)44717 11367 y Fo(\))45115 10888
   6616 y Fi(v)45530 11044 y Fb(k)35498 14557 y Fi(n)35100 14856
   6617 y Fc(Y)35014 16978 y Fi(k)g Fd(=1)36494 15803 y Fo(\()p
   6618 Fj(b)37331 15921 y Fi(k)37839 15803 y Fo(\))38237 15324
   6619 y Fi(w)38848 15046 y Fh(0)38830 15559 y Fb(k)39654 15803
   6620 y Fo(=)285 b Fj(b)41174 15914 y Fi(j)42262 14557 y(n)41864
   6621 14856 y Fc(Y)41778 16978 y Fi(k)14 b Fd(=1)43258 15803
   6622 y Fo(\()p Fj(b)44095 15921 y Fi(k)44603 15803 y Fo(\))45001
   6623 15324 y Fi(v)45440 15046 y Fh(0)45416 15559 y Fb(k)27201
   6624 18873 y Fo(Since)229 b Fj(v)36 b(;)171 b(w)311 b Fg(2)284
   6625 b Fe(S)p Fo(,)33850 18126 y Fc(Q)34817 18399 y Fi(n)34817
   6626 19177 y(k)14 b Fd(=0)36352 18873 y Fo(\()p Fj(a)37292
   6627 18991 y Fi(k)37801 18873 y Fo(\))38199 18450 y Fi(v)38614
   6628 18606 y Fb(k)39438 18873 y Fo(=)40519 18126 y Fc(Q)41486
   6629 18399 y Fi(n)41486 19177 y(k)g Fd(=1)43021 18873 y Fo(\()p
   6630 Fj(b)43858 18984 y Fi(i)44209 18873 y Fo(\))44607 18450
   6631 y Fi(v)45046 18171 y Fh(0)45022 18684 y Fb(k)45789 18873
   6632 y Fo(and)47667 18126 y Fc(Q)48634 18399 y Fi(n)48634
   6633 19177 y(k)g Fd(=1)50169 18873 y Fo(\()p Fj(a)51109 18991
   6634 y Fi(k)51617 18873 y Fo(\))52015 18450 y Fi(w)52608 18606
   6635 y Fb(k)53433 18873 y Fo(=)27201 19550 y Fc(Q)28168 19824
   6636 y Fi(n)28168 20602 y(k)g Fd(=1)29703 20298 y Fo(\()p
   6637 Fj(b)30540 20416 y Fi(k)31049 20298 y Fo(\))31447 19874
   6638 y Fi(w)32058 19596 y Fh(0)32040 20109 y Fb(k)32579 20298
   6639 y Fo(.)442 b(Th)-28 b(us)297 b(w)-28 b(e)297 b(m)-28
   6640 b(ust)299 b(ha)-28 b(v)g(e)298 b Fj(b)42572 20409 y Fi(j)43290
   6641 20298 y Fo(=)284 b Fj(a)44912 20409 y Fi(i)45264 20298
   6642 y Fo(.)441 b(The)298 b(same)h(reason-)27201 21460 y(ing)314
   6643 b(holds)g(for)f(all)i(1)284 b Fg(\267)h Fj(j)341 b Fg(\267)285
   6644 b Fj(n)p Fo(,)320 b(whic)-28 b(h)313 b(sho)-28 b(ws)312
   6645 b(that)i(the)h(mix)g(serv)-28 b(er)27201 22622 y(didn't)342
   6646 b(c)-28 b(heat.)20111 b Fa(\244)28198 24947 y Fo(Lemmas)421
   6647 b(7.8)h(and)e(7.9)h(sho)-28 b(w)420 b(that)h(when)g Fg(j)p
   6648 Fe(S)p Fg(j)d Fj(>)g Fo(\(5)p Fj(=)p Fo(8\)2)51131 24524
   6649 y Fi(n)52119 24947 y Fo(and)27201 26109 y Fj(F)142 b
   6650 Fo(\()p Fe(S)p Fo(\))393 b(spans)e(all)h(of)f Fe(Z)35940
   6651 25686 y Fi(n)35940 26284 y(q)36507 26109 y Fo(,)404 b(then)392
   6652 b Fj(F)534 b Fo(cannot)391 b(b)28 b(e)392 b(a)g(linear)g(map)g(on)f
   6653 Fe(S)27201 27271 y Fo(unless)320 b(the)g(mix)g(serv)-28
   6654 b(er)319 b(did)h(not)f(c)-28 b(heat.)449 b(The)319 b(next)h(lemma)h
   6655 (sho)-28 b(ws)27201 28434 y(that)386 b(when)e Fj(F)142
   6656 b Fo(\()p Fe(S)p Fo(\))386 b(spans)f(all)g(of)g Fe(Z)40894
   6657 28011 y Fi(n)40894 28609 y(q)41461 28434 y Fo(,)396 b(step)386
   6658 b(6)f(succeeds)g(with)g(non-)27201 29596 y(negligible)344
   6659 b(probabilit)-28 b(y)-85 b(.)28308 31880 y Ff(Lemma)553
   6660 b Fo(7.10.)499 b Fm(If)382 b Fg(j)p Fe(S)p Fg(j)316 b
   6661 Fj(>)e Fo(2)38949 31457 y Fi(n)39516 31880 y Fo(\()40047
   6662 31472 y Fd(5)p 40047 31652 406 43 v 40047 32218 a(8)40825
   6663 31880 y Fo(+)41994 31472 y Fd(13)p 41994 31652 812 43
   6664 v 41994 32218 a(16)42938 31880 y Fj(\262)p Fo(\))383
   6665 b Fm(and)f Fj(F)142 b Fo(\()p Fe(S)p Fo(\))383 b Fm(sp)-52
   6666 b(ans)382 b(al)52 b(l)383 b(of)27201 33043 y Fe(Z)27865
   6667 32619 y Fi(n)27865 33218 y(q)28432 33043 y Fm(,)286 b(then)267
   6668 b(algorithm)g Fg(A)f Fm(do)-52 b(es)267 b(not)f(ab)-52
   6669 b(ort)266 b(in)h(step)f Fo(6)h Fm(with)e(pr)-52 b(ob)g(ability)27201
   6670 34205 y(at)366 b(le)-52 b(ast)366 b Fj(\262=)p Fo(64)p
   6671 Fm(.)28308 36489 y Ff(Pr)-22 b(oof.)498 b Fo(By)439 b(Lemma)h(7.6,)463
   6672 b(w)-28 b(e)438 b(kno)-28 b(w)438 b(that)h(with)f(probabilit)-28
   6673 b(y)27201 37652 y(at)406 b(least)g(1)p Fj(=)p Fo(32,)421
   6674 b(the)406 b(\257rst)e Fj(n)h Fo(v)-28 b(ectors)405 b(in)h(step)f(6)g
   6675 (of)g(algorithm)h Fg(A)p Fo(,)27201 38814 y Fj(\302)p
   6676 Fo(\()p Fj(S)28863 38925 y Fd(1)29325 38814 y Fo(\))p
   6677 Fj(;)171 b(:)g(:)g(:)g(;)g(\302)p Fo(\()p Fj(S)33660
   6678 38925 y Fi(n)34227 38814 y Fo(\))486 b Fg(2)g(f)p Fo(0)p
   6679 Fj(;)171 b Fo(1)p Fg(g)38783 38391 y Fi(n)39812 38814
   6680 y Fo(will)463 b(span)f(all)h(of)f Fe(Z)47877 38391 y
   6681 Fi(n)47877 38989 y(q)48444 38814 y Fo(.)819 b(Let)463
   6682 b Fj(M)594 b Fg(2)27201 39976 y Fj(GL)28700 40087 y Fi(n)29266
   6683 39976 y Fo(\()p Fj(Z)30361 40087 y Fi(q)30819 39976 y
   6684 Fo(\))285 b(b)28 b(e)285 b(the)g(unique)g(matrix)g(that)g(maps)f
   6685 Fj(\302)p Fo(\()p Fj(F)142 b Fo(\()p Fj(S)48730 40087
   6686 y Fi(i)49082 39976 y Fo(\)\))284 b(to)h Fj(\302)p Fo(\()p
   6687 Fj(S)53019 40087 y Fi(i)53370 39976 y Fo(\))27201 41138
   6688 y(for)341 b(all)h Fj(i)285 b Fo(=)g(1)p Fj(;)171 b(:)g(:)g(:)g(;)g(n)p
   6689 Fo(.)28198 42301 y(Let)459 b Fj(T)623 b Fg(\265)481 b
   6690 Fe(S)459 b Fo(b)28 b(e)460 b(the)f(subset)g(of)g(all)h(the)f(v)-28
   6691 b(ectors)459 b Fj(v)517 b Fg(2)480 b Fe(S)459 b Fo(suc)-28
   6692 b(h)27201 43463 y(that)370 b Fj(v)e Fo(=)332 b Fj(M)354
   6693 b Fg(\242)247 b Fj(F)142 b Fo(\()p Fj(v)36 b Fo(\).)540
   6694 b(By)370 b(Lemmas)h(7.8)f(and)f(7.9,)377 b(w)-28 b(e)369
   6695 b(kno)-28 b(w)369 b(that)27201 44625 y Fg(j)p Fj(T)142
   6696 b Fg(j)372 b(\267)e Fo(\(5)p Fj(=)p Fo(8\)2)32890 44202
   6697 y Fi(n)33457 44625 y Fo(.)610 b(Therefore)392 b(the)i(probabilit)-28
   6698 b(y)393 b(that)g(the)g Fj(n)262 b Fo(+)g(1'st)27201 45788
   6699 y(v)-28 b(ector)436 b Fj(\302)p Fo(\()p Fj(S)32032 45899
   6700 y Fi(n)p Fd(+1)33626 45788 y Fo(\))f(in)h(step)g(6)f(of)h(algorithm)g
   6701 Fg(A)f Fo(is)h(not)g(in)f Fj(T)578 b Fo(is)436 b(at)27201
   6702 46950 y(least)29781 46542 y Fd(13)p 29781 46722 V 29781
   6703 47288 a(16)30726 46950 y Fj(\262)p Fo(.)610 b(When)393
   6704 b(that)g(happ)28 b(ens,)407 b(step)393 b(6)g(will)g(not)g(ab)28
   6705 b(ort.)611 b(The)27201 48112 y(probabilit)-28 b(y)479
   6706 b(of)f(not)g(ab)28 b(orting)478 b(in)g(step)h(6)f(is)g(therefore)g(at)g
   6707 (least)27201 49275 y(\(1)p Fj(=)p Fo(32\)\()30576 48866
   6708 y Fd(13)p 30577 49046 V 30577 49613 a(16)31521 49275
   6709 y Fj(\262)p Fo(\))285 b Fg(\270)f Fj(\262=)p Fo(64)342
   6710 b(as)f(required.)11765 b Fa(\244)27201 51599 y Fn(Pro)33
   6711 b(of)394 b(of)g(Prop)33 b(osition)396 b(7.3)27201 52762
   6712 y Fo(T)-85 b(o)227 b(summarize,)251 b(w)-28 b(e)228 b(pro)-28
   6713 b(v)g(e)226 b(here)i(the)g(lo)-28 b(w)g(er)227 b(b)28
   6714 b(ound)227 b(giv)-28 b(en)228 b(in)g(Prop)28 b(o-)27201
   6715 53924 y(sition)420 b(7.3)f(on)f(the)i(probabilit)-28
   6716 b(y)419 b(that)g(algorithm)g Fg(A)g Fo(succeeds)g(in)27201
   6717 55086 y(outputting)361 b(log)33662 55335 y Fi(g)34316
   6718 55086 y Fj(h)p Fo(.)511 b(Algorithm)360 b Fg(A)g Fo(only)h(ev)-28
   6719 b(er)360 b(ab)28 b(orts)360 b(in)g(steps)g(6)27201 56248
   6720 y(and)349 b(7.)480 b(W)-85 b(e)350 b(kno)-28 b(w)349
   6721 b(that)g(the)h(probabilit)-28 b(y)350 b(that)g Fg(A)f
   6722 Fo(ab)28 b(orts)349 b(in)h(step)27201 57411 y(7)445 b(is)f(at)g(most)h
   6723 (1)p Fj(=q)36 b Fo(.)763 b(F)-85 b(urthermore,)470 b(w)-28
   6724 b(e)444 b(ha)-28 b(v)g(e)444 b(sho)-28 b(wn)443 b(that)h(with)27201
   6725 58573 y(probabilit)-28 b(y)384 b(at)g(least)h Fj(\262=)p
   6726 Fo(2,)394 b(w)-28 b(e)383 b(ha)-28 b(v)g(e)384 b Fg(j)p
   6727 Fe(S)p Fg(j)356 b Fj(>)f Fo(2)45422 58150 y Fi(n)45989
   6728 58573 y Fo(\()46520 58165 y Fd(5)p 46520 58345 406 43
   6729 v 46520 58911 a(8)47314 58573 y Fo(+)48499 58165 y Fd(13)p
   6730 48499 58345 812 43 v 48499 58911 a(16)49444 58573 y Fj(\262)p
   6731 Fo(\).)582 b(When)27201 59735 y(that)342 b(happ)28 b(ens:)28682
   6732 61764 y Fg(\262)498 b Fo(When)447 b Fj(F)142 b Fo(\()p
   6733 Fe(S)p Fo(\))448 b(do)28 b(es)447 b(not)f(span)h(all)g(of)g
   6734 Fe(Z)45681 61341 y Fi(n)45681 61939 y(q)46247 61764 y
   6735 Fo(,)474 b(the)447 b(probabilit)-28 b(y)29692 62926 y(that)462
   6736 b Fg(A)g Fo(do)28 b(es)462 b(not)g(ab)28 b(ort)461 b(in)h(step)h(6)e
   6737 (is)h(at)g(least)h(1)p Fj(=)p Fo(32)f(b)-28 b(y)29692
   6738 64088 y(Lemma)342 b(7.7.)28682 66186 y Fg(\262)498 b
   6739 Fo(When)241 b Fj(F)142 b Fo(\()p Fe(S)p Fo(\))241 b(spans)f(all)i(of)e
   6740 Fe(Z)40713 65763 y Fi(n)40713 66361 y(q)41280 66186 y
   6741 Fo(,)261 b(the)241 b(probabilit)-28 b(y)241 b(that)g
   6742 Fg(A)g Fo(do)28 b(es)29692 67348 y(not)341 b(ab)28 b(ort)342
   6743 b(in)f(step)h(6)f(is)h(at)f(least)h Fj(\262=)p Fo(64)f(b)-28
   6744 b(y)342 b(Lemma)g(7.10.)27201 69377 y(It)484 b(follo)-28
   6745 b(ws)483 b(that)g(the)h(probabilit)-28 b(y)484 b(of)g(success)f(of)g
   6746 Fg(A)h Fo(is)g(at)f(least)27201 70539 y Fj(\262)27616
   6747 70116 y Fd(2)28077 70539 y Fj(=)p Fo(128)228 b Fg(\241)g
   6748 Fo(1)p Fj(=q)36 b Fo(.)19793 b Fa(\244)p eop
   6749 %%Page: 10 10
   6750 10 9 bop -2022 -915 a Fp(8.)1328 b(CONCLUSION)-1026 524
   6751 y Fo(The)500 b(strongest)g(p)28 b(oin)-28 b(t)501 b(of)f(our)g(new)g
   6752 (mix)i(net)-28 b(w)g(ork)499 b(is)i(its)f(ex-)-2022 1686
   6753 y(ceptional)465 b(sp)28 b(eed.)824 b(The)464 b(real)g(cost)g(of)g(pro)
   6754 -28 b(ving)464 b(almost)g(en)-28 b(tirely)-2022 2848
   6755 y(correct)391 b(mixing)i(is)f(orders)f(of)g(magnitude)h(faster)g(than)f
   6756 (all)i(other)-2022 4011 y(mix)272 b(net)-28 b(w)g(orks.)432
   6757 b(An)271 b(almost)i(en)-28 b(tirely)273 b(correct)e(output)h(is)g(a)-28
   6758 b(v)-57 b(ailable)-2022 5173 y(instan)-28 b(tly)362 b(and)f(can)h(b)28
   6759 b(e)362 b(announced)g(long)f(b)28 b(efore)363 b(it)f(is)g(con\257rmed)
   6760 -2022 6335 y(b)-28 b(y)341 b(a)g(slo)-28 b(w)g(er)340
   6761 b(p)28 b(erfectly)343 b(correct)e(mix)i(net)-28 b(w)g(ork.)-1026
   6762 7497 y(In)357 b(practice,)364 b(our)357 b(new)h(mix)i(is)e(of)g
   6763 (particular)g(in)-28 b(terest)359 b(to)f(large)-2022
   6764 8660 y(electronic)429 b(elections)i(\(sa)-28 b(y)-85
   6765 b(,)450 b(a)428 b(million)j(ballots)e(or)f(more\),)451
   6766 b(where)-2022 9822 y(a)418 b(guaran)-28 b(tee)418 b(of)h(almost)g(en)
   6767 -28 b(tirely)420 b(correct)e(mixing)i(ma)-28 b(y)419
   6768 b(w)-28 b(ell)419 b(b)28 b(e)-2022 10984 y(su\261cien)-28
   6769 b(t)430 b(to)f(announce)g(the)h(outcome)g(of)g(an)f(election)i(p)28
   6770 b(ending)-2022 12147 y(con\257rmation)338 b(b)-28 b(y)338
   6771 b(a)g(slo)-28 b(w)g(er)337 b(p)28 b(erfectly)340 b(correct)e(mixnet.)
   6772 456 b(This)337 b(ad-)-2022 13309 y(ditional)505 b(pro)28
   6773 b(of)504 b(of)g(p)28 b(erfect)505 b(correctness)g(do)28
   6774 b(es)505 b(not)f(require)h(the)-2022 14471 y(ballots)307
   6775 b(to)g(b)28 b(e)308 b(mixed)g(again,)315 b(and)306 b(of)h(course)f(do)
   6776 28 b(esn't)308 b(require)g(an)-28 b(y)-2022 15634 y(in)g(v)g(olv)g
   6777 (emen)g(t)342 b(from)f(the)h(v)-28 b(oters.)-1026 16796
   6778 y(W)-85 b(e)295 b(prop)28 b(ose)294 b(the)h(\257rst)f(construction)h
   6779 (that)f(exploits)j(a)d(trade-o\256)-2022 17958 y(b)28
   6780 b(et)-28 b(w)g(een)325 b(e\261ciency)j(and)d(correctness.)450
   6781 b(An)325 b(in)-28 b(teresting)326 b(direction)-2022 19121
   6782 y(for)348 b(future)g(w)-28 b(ork)348 b(w)-28 b(ould)348
   6783 b(b)28 b(e)349 b(to)g(study)g(this)g(trade-o\256)g(further.)477
   6784 b(In)-2022 20283 y(particular,)412 b(it)398 b(w)-28 b(ould)397
   6785 b(b)28 b(e)398 b(in)-28 b(teresting)399 b(to)e(determine)j(whether)d
   6786 (it)-2022 21445 y(is)307 b(p)28 b(ossible)308 b(to)f(build)g(mix)h
   6787 (nets)g(that)f(span)f(the)i(en)-28 b(tire)307 b(con)-28
   6788 b(tin)g(uum)-2022 22607 y(of)341 b(the)h(trade-o\256)f(b)28
   6789 b(et)-28 b(w)g(een)342 b(e\261ciency)h(and)e(correctness.)-2022
   6790 25264 y Fp(9.)1328 b(A)-73 b(CKNO)-66 b(WLEDGMENTS)-1026
   6791 26703 y Fo(The)249 b(second)g(author)g(wishes)h(to)f(thank)h(Markus)f
   6792 (Jak)-28 b(obsson)249 b(and)-2022 27866 y(Ari)406 b(Juels)h(for)f
   6793 (helpful)i(con)-28 b(v)g(ersations)406 b(and)g(commen)-28
   6794 b(ts)407 b(on)g(early)-2022 29028 y(v)-28 b(ersions)341
   6795 b(of)g(this)h(pap)28 b(er.)-2022 31685 y Fp(10.)1328
   6796 b(REFERENCES)-1767 33146 y Fo([1])499 b(M.)329 b(Ab)28
   6797 b(e.)328 b(Univ)-28 b(ersally)328 b(v)-28 b(eri\257able)328
   6798 b(mix-net)i(with)d(v)-28 b(eri\257cation)-188 34308 y(w)g(ork)341
   6799 b(indep)28 b(enden)-28 b(t)342 b(of)f(the)h(n)-28 b(um)g(b)28
   6800 b(er)341 b(of)g(mix-serv)-28 b(ers.)343 b(In)-188 35470
   6801 y Fm(Pr)-52 b(o)g(c.)366 b(of)f(Eur)-52 b(o)g(crypt)365
   6802 b('98)p Fo(,)340 b(pp.)i(437-447.)g(Springer-V)-85 b(erlag,)-188
   6803 36633 y(1998.)342 b(LNCS)e(1403.)-1767 37906 y([2])499
   6804 b(M.)343 b(Ab)28 b(e.)342 b(Mix-net)-28 b(w)g(orks)342
   6805 b(on)f(p)28 b(erm)-28 b(utation)342 b(net)-28 b(w)g(orks.)340
   6806 b(In)-188 39068 y Fm(Pr)-52 b(o)g(c.)366 b(of)f(Asiacrypt)g('99)p
   6807 Fo(,)341 b(pp.)h(258-273,)f(1999.)h(LNCS)e(1716.)-1767
   6808 40341 y([3])499 b(M.)343 b(Ab)28 b(e.)342 b(Remarks)f(on)g(mix-net)-28
   6809 b(w)g(orks)342 b(based)f(on)-188 41503 y(p)28 b(erm)-28
   6810 b(utation)342 b(net)-28 b(w)g(orks.)-1767 42776 y([4])499
   6811 b(M.)343 b(Bellare,)g(J.)f(Gara)-28 b(y)341 b(and)g(T.)g(Rabin.)g(Batc)
   6812 -28 b(h)342 b(V)-85 b(eri\257cation)-188 43939 y(with)342
   6813 b(Applications)g(to)f(Cryptograph)-28 b(y)340 b(and)h(Chec)-28
   6814 b(king.)342 b(In)-188 45101 y Fm(Pr)-52 b(o)g(c.)366
   6815 b(of)f(Eur)-52 b(o)g(crypt)365 b('98)p Fo(,)340 b(pp.)i(170-182.)g
   6816 (Springer)e(V)-85 b(erlag,)-188 46263 y(1998.)342 b(LNCS)e(1380.)-1767
   6817 47536 y([5])499 b(D.)342 b(Chaum.)f(Un)-28 b(traceable)342
   6818 b(electronic)h(mail,)f(return)-188 48698 y(addresses,)335
   6819 b(and)e(digital)h(pseudon)-28 b(yms.)334 b(In)e Fm(Communic)-52
   6820 b(ations)-188 49861 y(of)366 b(the)g(A)-26 b(CM)p Fo(,)341
   6821 b(24\(2\):84-88,)i(1981.)-1767 51134 y([6])499 b(D.)342
   6822 b(Chaum)f(and)g(T.)g(P)-28 b(edersen.)342 b(W)-85 b(allet)342
   6823 b(databases)g(with)-188 52296 y(observ)-28 b(ers.)342
   6824 b(In)e Fm(Pr)-52 b(o)g(c.)366 b(of)f(Crypto'92)p Fo(,)340
   6825 b(pp.)h(89-105.)-188 53458 y(Springer-V)-85 b(erlag,)342
   6826 b(1993.)f(LNCS)g(740.)-1767 54731 y([7])499 b(Y.)342
   6827 b(Desmedt)g(and)f(K.)h(Kurosa)-28 b(w)g(a.)340 b(Ho)-28
   6828 b(w)340 b(to)i(break)f(a)-188 55894 y(practical)h(MIX)f(and)g(design)h
   6829 (a)f(new)g(one.)h(In)f Fm(Pr)-52 b(o)g(c.)365 b(of)-188
   6830 57056 y(Eur)-52 b(o)g(crypt'2000)p Fo(,)340 b(pp.)h(557-572.)h(LNCS)e
   6831 (1807.)-1767 58329 y([8])499 b(D.)342 b(Dolev,)h(C.)f(Dw)-28
   6832 b(ork,)340 b(M.)i(Naor.)f(Nonmalleable)-188 59491 y(Cryptograph)-28
   6833 b(y)-85 b(.)340 b(In)h(SIAM)g(J.)h(Comput.)f(30\(2\):)456
   6834 b(391-437)-188 60654 y(\(2000\))-1767 61927 y([9])499
   6835 b(J.)342 b(F)-85 b(uruk)-57 b(a)-28 b(w)g(a)340 b(and)h(K.)h(Sak)-28
   6836 b(o.)341 b(An)g(e\261cien)-28 b(t)343 b(sc)-28 b(heme)342
   6837 b(for)-188 63089 y(pro)-28 b(ving)341 b(a)g(sh)-28 b(u\262e.)342
   6838 b(In)f Fm(Pr)-52 b(o)g(c.)365 b(of)g(Crypto)g('01)p Fo(,)341
   6839 b(pp.)g(368-387.)-188 64251 y(Springer-V)-85 b(erlag,)342
   6840 b(2001.)f(LNCS)g(2139.)27201 -915 y([10])499 b(R.)342
   6841 b(Gennaro,)f(S.)h(Jarec)-28 b(ki,)342 b(H.)g(Kra)-28
   6842 b(w)g(czyk)340 b(and)h(T.)h(Rabin.)29036 247 y(Secure)g(Distributed)g
   6843 (Key)g(Generation)h(for)d(Discrete-Log)29036 1409 y(Based)i
   6844 (Cryptosystems.)g(In)f Fm(Pr)-52 b(o)g(c.)365 b(of)g(Eur)-52
   6845 b(o)g(crypt)365 b('99)p Fo(,)29036 2571 y(pp.)342 b(295-310.)f
   6846 (Springer-V)-85 b(erlag,)342 b(1999.)f(LNCS)g(1592.)27201
   6847 3844 y([11])499 b(P)-85 b(.)341 b(Golle,)j(S.)d(Zhong,)h(D.)f(Boneh,)h
   6848 (M.)h(Jak)-28 b(obsson)340 b(and)29036 5007 y(A.)i(Juels.)g(Optimistic)
   6849 i(Mixing)e(for)f(Exit-P)-28 b(olls.)343 b(T)-85 b(o)340
   6850 b(app)28 b(ear)29036 6169 y(in)342 b(Asiacrypt)f(2002.)27201
   6851 7442 y([12])499 b(M.)342 b(Hirt)g(and)f(K.)h(Sak)-28
   6852 b(o.)341 b(E\261cien)-28 b(t)342 b(receipt-free)h(v)-28
   6853 b(oting)29036 8604 y(based)341 b(on)h(homomorphic)f(encryption.)h(In)f
   6854 Fm(Pr)-52 b(o)g(c.)365 b(of)29036 9767 y(Eur)-52 b(o)g(crypt'00)p
   6855 Fo(,)340 b(pp.)h(539-556.)h(Springer-V)-85 b(erlag,)341
   6856 b(2000.)29036 10929 y(LNCS)f(1807.)27201 12202 y([13])499
   6857 b(M.)342 b(Jak)-28 b(obsson.)342 b(A)f(practical)h(mix.)g(In)f
   6858 Fm(Pr)-52 b(o)g(c.)365 b(of)h(Eur)-52 b(o)g(crypt)29036
   6859 13364 y('98)p Fo(,)341 b(pp.)g(448-461.)h(Springer-V)-85
   6860 b(erlag,)341 b(1998.)h(LNCS)e(1403.)27201 14637 y([14])499
   6861 b(M.)342 b(Jak)-28 b(obsson)341 b(and)g(D.)h(M'Ra)-114
   6862 b(\304)-398 b(\263hi.)343 b(Mix-based)g(electronic)29036
   6863 15800 y(pa)-28 b(ymen)g(ts.)342 b(In)f Fm(Pr)-52 b(o)g(c.)365
   6864 b(of)g(SA)-26 b(C'98)p Fo(,)341 b(pp.)g(157-173.)29036
   6865 16962 y(Springer-V)-85 b(erlag,)341 b(1998.)h(LNCS)e(1556.)27201
   6866 18235 y([15])499 b(M.)342 b(Jak)-28 b(obsson.)342 b(Flash)f(mixing.)i
   6867 (In)e Fm(Pr)-52 b(o)g(c.)365 b(of)h(PODC)f('99)p Fo(,)29036
   6868 19397 y(pp.)342 b(83-89.)f(A)-28 b(CM,)341 b(1999.)27201
   6869 20670 y([16])499 b(M.)342 b(Jak)-28 b(obsson)341 b(and)g(A.)h(Juels.)g
   6870 (Millimix:)459 b(mixing)343 b(in)e(small)29036 21833
   6871 y(batc)-28 b(hes.)342 b(DIMA)-28 b(CS)340 b(T)-85 b(ec)-28
   6872 b(hnical)341 b(Rep)28 b(ort)341 b(99-33.)27201 23106
   6873 y([17])499 b(M.)342 b(Jak)-28 b(obsson)341 b(and)g(A.)h(Juels.)g(An)f
   6874 (optimally)i(robust)29036 24268 y(h)-28 b(ybrid)341 b(mix)h(net)-28
   6875 b(w)g(ork.)341 b(In)g Fm(Pr)-52 b(o)g(c.)365 b(of)g(PODC'01)p
   6876 Fo(,)29036 25430 y(pp.)342 b(284-292.)f(A)-28 b(CM)341
   6877 b(Press.)g(2001.)27201 26703 y([18])499 b(M.)342 b(Jak)-28
   6878 b(obsson,)342 b(A.)f(Juels)h(and)f(R.)h(Riv)-28 b(est.)342
   6879 b(Making)g(mix)29036 27866 y(nets)g(robust)e(for)h(electronic)i(v)-28
   6880 b(oting)341 b(b)-28 b(y)342 b(randomized)f(partial)29036
   6881 29028 y(c)-28 b(hec)g(king.)342 b(In)f Fm(Pr)-52 b(o)g(c.)365
   6882 b(of)g(USENIX'02)p Fo(.)27201 30301 y([19])499 b(M.)342
   6883 b(Mitomo)h(and)e(K.)h(Kurosa)-28 b(w)g(a.)340 b(A)-28
   6884 b(ttac)g(k)341 b(for)f(\260ash)h(mix.)i(In)29036 31463
   6885 y Fm(Pr)-52 b(o)g(c.)365 b(of)h(Asiacrypt'00)p Fo(,)340
   6886 b(pp.)h(192-204.)h(LNCS)f(1976.)27201 32736 y([20])499
   6887 b(A.)342 b(Ne\256.)g(A)f(v)-28 b(eri\257able)342 b(secret)g(sh)-28
   6888 b(u\262e)341 b(and)g(its)h(application)29036 33898 y(to)g(E-V)-85
   6889 b(oting.)341 b(In)g Fm(Pr)-52 b(o)g(c.)365 b(of)g(A)-26
   6890 b(CM)366 b(CCS'01)p Fo(,)341 b(pp.)g(116-125.)29036 35061
   6891 y(A)-28 b(CM)341 b(Press,)g(2001.)27201 36334 y([21])499
   6892 b(W.)342 b(Ogata,)g(K.)g(Kurosa)-28 b(w)g(a,)340 b(K.)i(Sak)-28
   6893 b(o)341 b(and)g(K.)h(T)-85 b(ak)-57 b(atani.)29036 37496
   6894 y(F)-85 b(ault)342 b(toleran)-28 b(t)341 b(anon)-28 b(ymous)341
   6895 b(c)-28 b(hannel.)342 b(In)f Fm(Pr)-52 b(o)g(c.)365 b(of)g(ICICS)29036
   6896 38658 y('97)p Fo(,)341 b(pp.)g(440-444,)h(1997.)g(LNCS)e(1334.)27201
   6897 39931 y([22])499 b(C.)341 b(P)-28 b(ark,)341 b(K.)h(Itoh)f(and)g(K.)h
   6898 (Kurosa)-28 b(w)g(a.)340 b(E\261cien)-28 b(t)29036 41094
   6899 y(anon)g(ymous)341 b(c)-28 b(hannel)341 b(and)g(all/nothing)i(election)
   6900 g(Sc)-28 b(heme.)29036 42256 y(In)341 b Fm(Pr)-52 b(o)g(c.)365
   6901 b(of)g(Eur)-52 b(o)g(crypt)365 b('93)p Fo(,)341 b(pp.)g(248-259.)29036
   6902 43418 y(Springer-V)-85 b(erlag,)341 b(1993.)h(LNCS)e(765.)27201
   6903 44691 y([23])499 b(T.)341 b(P)-28 b(edersen.)342 b(A)f(Threshold)g
   6904 (cryptosystem)h(without)f(a)29036 45854 y(trusted)g(part)-28
   6905 b(y)-85 b(.)341 b(In)g Fm(Pr)-52 b(o)g(c.)365 b(of)g(Eur)-52
   6906 b(o)g(crypt'91)p Fo(,)340 b(pp.)i(522-526,)29036 47016
   6907 y(1991.)27201 48289 y([24])499 b(B.)342 b(P\257tzmann)f(and)g(A.)h
   6908 (P\257tzmann.)f(Ho)-28 b(w)341 b(to)g(break)h(the)29036
   6909 49451 y(direct)g(RSA-implemen)-28 b(tation)343 b(of)e(mixes.)i(In)e
   6910 Fm(Pr)-52 b(o)g(c.)365 b(of)29036 50614 y(Eur)-52 b(o)g(crypt)365
   6911 b('89)p Fo(,)341 b(pp.)g(373-381.)h(Springer-V)-85 b(erlag,)341
   6912 b(1989.)29036 51776 y(LNCS)f(434.)27201 53049 y([25])499
   6913 b(B.)342 b(P\257zmann.)g(Breaking)g(an)f(e\261cien)-28
   6914 b(t)342 b(anon)-28 b(ymous)29036 54211 y(c)g(hannel.)342
   6915 b(In)f Fm(Pr)-52 b(o)g(c.)365 b(of)g(Eur)-52 b(o)g(crypt'94)p
   6916 Fo(,)340 b(pp.)h(339-348.)27201 55484 y([26])499 b(K.)342
   6917 b(Sak)-28 b(o)341 b(and)g(J.)h(Kilian.)h(Receipt-free)g(mix-t)-28
   6918 b(yp)28 b(e)344 b(v)-28 b(oting)29036 56646 y(sc)g(heme.)342
   6919 b(In)f Fm(Pr)-52 b(o)g(c.)365 b(of)h(Eur)-52 b(o)g(crypt)364
   6920 b('95)p Fo(.)341 b(Springer-V)-85 b(erlag,)29036 57809
   6921 y(1995.)342 b(LNCS)e(921.)27201 59082 y([27])499 b(Y.)342
   6922 b(Tsiounis)f(and)g(M.)h(Y)-85 b(ung.)341 b(On)g(the)h(securit)-28
   6923 b(y)342 b(of)f(ElGamal)29036 60244 y(based)g(encryption.)i(In)e
   6924 Fm(Pr)-52 b(o)g(c.)365 b(of)g(PK)-26 b(C'98)p Fo(.)p
   6925 eop
   6926 %%Trailer
   6927 end
   6928 userdict /end-hook known{end-hook}if
   6929 %%EOF