3995:
238:
92:
258:. Here the sender and the recipient of a message form a group for the ring signature, then the signature is valid to the recipient, but anyone else will be unsure whether the recipient or the sender was the actual signer. Thus, such a signature is convincing, but cannot be transferred beyond its intended recipient.
1109:
386:
The property of linkability allows one to determine whether any two signatures have been produced by the same member (under the same private key). The identity of the signer is nevertheless preserved. One of the possible applications can be an offline
250:
official", without revealing which official signed the message. Ring signatures are right for this application because the anonymity of a ring signature cannot be revoked, and because the group for a ring signature can be improvised.
229:. If a ring signature is properly computed, it should pass the check. On the other hand, it should be hard for anyone to create a valid ring signature on any message for any set without knowing any of the private keys for that set.
43:. Therefore, a message signed with a ring signature is endorsed by someone in a particular set of people. One of the security properties of a ring signature is that it should be computationally infeasible to determine
2017:
909:
245:
In the original paper, Rivest, Shamir, and Tauman described ring signatures as a way to leak a secret. For instance, a ring signature could be used to provide an anonymous signature from "a high-ranking
443:; i.e., the size of the resulting signature increases linearly with the size of input (number of public keys). That means that such schemes are impracticable for real use cases with sufficiently large
2244:
1239:
871:
607:
51:
but differ in two key ways: first, there is no way to revoke the anonymity of an individual signature; and second, any set of users can be used as a signing set without additional setup.
1479:
1864:
1392:
2154:
1522:
2112:
1741:
690:
1554:
760:
1328:
1901:
2056:
1800:
1771:
1681:
1654:
1601:
1293:
1266:
901:
787:
717:
498:
441:
1624:
1574:
1420:
1152:
1132:
647:
627:
461:
397:
In addition to the previous scheme the public key of the signer is revealed (if they issue more than one signatures under the same private key). An
4274:
4166:
4133:
4050:
503:
More efficient algorithms have appeared recently. There are schemes with the sublinear size of the signature, as well as with constant size.
4255:
Au, Man Ho; Liu, Joseph K.; Susilo, Willy; Yuen, Tsz Hon (2006). "Constant-Size ID-Based
Linkable and Revocable-iff-Linked Ring Signature".
1906:
1295:, can be freely chosen. Under the assumptions of RSA, this implies knowledge of at least one of the inverses of the trap door functions
1104:{\displaystyle C_{k,v}(y_{1},y_{2},\dots ,y_{n})=E_{k}(y_{n}\oplus E_{k}(y_{n-1}\oplus E_{k}(\dots \oplus E_{k}(y_{1}\oplus v)\dots )))}
4300:
103:
about the definition. This section currently describes some useful properties of a ring signature but not the mathematical definition.
4295:
3981:
2162:
1157:
795:
531:
1525:
4256:
1425:
1805:
1333:
874:
293:
277:
2120:
1488:
463:(for example, an e-voting with millions of participants). But for some application with relatively small
2260:
2061:
1690:
517:
255:
652:
3999:
4229:
4149:
Liu, Joseph K.; Wong, Duncan S. (2005). "Linkable Ring
Signatures: Security Models and New Schemes".
1535:
106:
722:
4176:
4089:
525:
521:
500:
ring signature scheme by
Fujisaki and Suzuki in p2p payments to achieve sender's untraceability.
40:
4003:
3994:
1298:
4220:
Fujisaki, Eiichiro (2011). "Sub-linear size traceable ring signatures without random oracles".
4270:
4162:
4129:
4046:
4026:
2256:
63:
47:
of the set's members' keys was used to produce the signature. Ring signatures are similar to
36:
4262:
4237:
4154:
4119:
4079:
4038:
1871:
2034:
1778:
1749:
1659:
1632:
1579:
1271:
1244:
879:
789:
is a trap-door function (i.e. an RSA public key in the case of RSA based ring signatures).
765:
695:
105:
Please expand the
Definition to include this information. Further details may exist on the
4188:
474:
417:
48:
4233:
4108:
4222:
IEICE Transactions on
Fundamentals of Electronics, Communications and Computer Sciences
3969:
1609:
1559:
1405:
1137:
1117:
632:
612:
446:
411:
261:
There were various works, introducing new features and based on different assumptions:
288:
users should collaborate to sign a message, this variant of a ring signature requires
4289:
4093:
1529:
28:
4066:
Debnath, Ashmita; Singaravelu, Pradheepkumar; Verma, Shekhar (19 December 2012).
247:
4241:
17:
4084:
4067:
4022:
4018:
1402:
Generating a ring signature involves six steps. The plaintext is signified by
873:
is called the ring equation, and is defined below. The equation is based on a
468:
59:
55:
4124:
4042:
4030:
237:
75:
67:
91:
4205:
Fujisaki, Eiichiro; Suzuki, Koutarou (2007). "Traceable Ring
Signature".
398:
4266:
4158:
4261:. Lecture Notes in Computer Science. Vol. 4329. pp. 364–378.
4118:. Lecture Notes in Computer Science. Vol. 2442. pp. 465–480.
4037:. Lecture Notes in Computer Science. Vol. 2248. pp. 552–565.
3965:
2012:{\displaystyle (P_{1},P_{2},\dots ,P_{n};v;x_{1},x_{2},\dots ,x_{n})}
464:
388:
39:
that can be performed by any member of a set of users that each have
4153:. Lecture Notes in Computer Science. Vol. 2. pp. 614–623.
126:
Suppose that a set of entities each have public/private key pairs, (
254:
Another application, also described in the original paper, is for
236:
4068:"Efficient spatial privacy preserving scheme for sensor network"
209:). Anyone can check the validity of a ring signature given σ,
85:
4109:"Threshold Ring Signatures and Applications to Ad-hoc Groups"
241:
Behaviour of the Rivest, Shamir, Tauman ring signature scheme
2132:
1541:
1500:
4151:
Computational
Science and Its Applications – ICCSA 2005
2307:"""RSA implementation."""
74:, comes from the ring-like structure of the signature
2165:
2123:
2064:
2037:
1909:
1874:
1808:
1781:
1752:
1693:
1662:
1635:
1612:
1582:
1562:
1538:
1491:
1428:
1408:
1336:
1301:
1274:
1247:
1160:
1140:
1120:
912:
882:
798:
768:
725:
698:
655:
635:
615:
534:
520:
based ring signature scheme, as well as one based on
477:
449:
420:
3632:
To sign and verify 2 messages in a ring of 4 users:
2239:{\displaystyle C_{k,v}(y_{1},y_{2},\dots ,y_{n})=v}
1234:{\displaystyle v=C_{k,v}(y_{1},y_{2},\dots ,y_{n})}
2238:
2148:
2106:
2050:
2011:
1895:
1858:
1794:
1765:
1735:
1687:igner's private key), and calculate corresponding
1675:
1648:
1618:
1595:
1568:
1548:
1516:
1473:
1414:
1386:
1322:
1287:
1260:
1233:
1146:
1126:
1103:
895:
865:
781:
754:
711:
684:
641:
621:
601:
492:
455:
435:
866:{\displaystyle C_{k,v}(y_{1},y_{2},\dots ,y_{n})}
602:{\displaystyle C_{k,v}(y_{1},y_{2},\dots ,y_{n})}
2027:Signature verification involves three steps.
8:
467:input size such estimate may be acceptable.
176:can compute a ring signature σ on a message
2259:implementation of the original paper using
2263:. Requires 3rd-party module PyCryptodome.
4123:
4107:E. Bresson; J. Stern; M. Szyd lo (2002).
4083:
2221:
2202:
2189:
2170:
2164:
2131:
2130:
2122:
2095:
2082:
2069:
2063:
2042:
2036:
2000:
1981:
1968:
1949:
1930:
1917:
1908:
1873:
1847:
1831:
1826:
1813:
1807:
1786:
1780:
1757:
1751:
1724:
1711:
1698:
1692:
1667:
1661:
1640:
1634:
1611:
1587:
1581:
1561:
1540:
1539:
1537:
1499:
1498:
1490:
1465:
1446:
1433:
1427:
1407:
1378:
1362:
1346:
1341:
1335:
1311:
1306:
1300:
1279:
1273:
1252:
1246:
1222:
1203:
1190:
1171:
1159:
1139:
1119:
1074:
1061:
1042:
1023:
1010:
997:
984:
968:
949:
936:
917:
911:
887:
881:
854:
835:
822:
803:
797:
773:
767:
743:
730:
724:
703:
697:
676:
660:
654:
634:
614:
590:
571:
558:
539:
533:
476:
448:
419:
1474:{\displaystyle P_{1},P_{2},\dots ,P_{n}}
4258:Progress in Cryptology - INDOCRYPT 2006
4072:Central European Journal of Engineering
4035:Advances in Cryptology — ASIACRYPT 2001
4010:
1859:{\displaystyle x_{s}=g_{s}^{-1}(y_{s})}
1387:{\displaystyle g_{i}^{-1}(y_{i})=x_{i}}
401:can be implemented using this protocol.
292:users to cooperate in the ring signing
4200:
4198:
4184:
4174:
2031:Apply the public key trap door on all
1241:can be solved as long as at least one
410:Most of the proposed algorithms have
7:
4116:Advances in Cryptology — CRYPTO 2002
2159:Verify that the ring equation holds
2149:{\displaystyle k={\mathcal {H}}(m)}
1656:for all ring members but yourself (
1517:{\displaystyle k={\mathcal {H}}(m)}
340:)-ring signature, σ, on a message,
2107:{\displaystyle y_{i}=g_{i}(x_{i})}
1736:{\displaystyle y_{i}=g_{i}(x_{i})}
25:
685:{\displaystyle y_{1},\dots y_{n}}
649:, and a list of arbitrary values
54:Ring signatures were invented by
3998: This article incorporates
3993:
3982:Witness-indistinguishable proofs
1802:using the signer's private key:
516:The original paper describes an
213:, and the public keys involved,
90:
1134:which is forced to be equal to
2227:
2182:
2143:
2137:
2101:
2088:
2006:
1910:
1890:
1875:
1868:The ring signature now is the
1853:
1840:
1730:
1717:
1549:{\displaystyle {\mathcal {H}}}
1511:
1505:
1368:
1355:
1228:
1183:
1098:
1095:
1092:
1086:
1067:
1048:
1016:
990:
974:
929:
860:
815:
749:
736:
596:
551:
487:
481:
430:
424:
233:Applications and modifications
1:
1683:will be calculated using the
875:symmetric encryption function
2117:Calculate the symmetric key
1746:Solve the ring equation for
1422:, the ring's public keys by
1330:(i.e. a private key), since
755:{\displaystyle g_{i}(x_{i})}
1526:cryptographic hash function
4317:
4242:10.1587/transfun.E95.A.151
1323:{\displaystyle g_{i}^{-1}}
1114:It outputs a single value
629:, an initialization value
4301:Digital signature schemes
4085:10.2478/s13531-012-0048-7
1606:Pick a random glue value
265:Threshold ring signatures
4125:10.1007/3-540-45708-9_30
4043:10.1007/3-540-45682-1_32
3634:
2265:
1576:will be used as key for
394:Traceable ring signature
383:Linkable ring signatures
4296:Public-key cryptography
4207:Public Key Cryptography
4031:"How to Leak a Secret"
2240:
2150:
2108:
2052:
2023:Signature verification
2013:
1897:
1896:{\displaystyle (2n+1)}
1860:
1796:
1767:
1737:
1677:
1650:
1620:
1597:
1570:
1550:
1528:. This step assumes a
1518:
1475:
1416:
1388:
1324:
1289:
1262:
1235:
1148:
1128:
1105:
897:
867:
783:
756:
713:
686:
643:
623:
603:
494:
457:
437:
242:
101:is missing information
3972:use this technology.
2251:Python implementation
2241:
2151:
2109:
2053:
2051:{\displaystyle x_{i}}
2014:
1898:
1861:
1797:
1795:{\displaystyle x_{s}}
1768:
1766:{\displaystyle y_{s}}
1738:
1678:
1676:{\displaystyle x_{s}}
1651:
1649:{\displaystyle x_{i}}
1621:
1598:
1596:{\displaystyle E_{k}}
1571:
1551:
1519:
1476:
1417:
1389:
1325:
1290:
1288:{\displaystyle x_{i}}
1263:
1261:{\displaystyle y_{i}}
1236:
1149:
1129:
1106:
898:
896:{\displaystyle E_{k}}
868:
784:
782:{\displaystyle g_{i}}
757:
714:
712:{\displaystyle y_{i}}
687:
644:
624:
604:
528:"combining function"
495:
458:
438:
240:
4002:available under the
2289:Crypto.PublicKey.RSA
2163:
2121:
2062:
2035:
1907:
1872:
1806:
1779:
1750:
1691:
1660:
1633:
1610:
1580:
1560:
1536:
1489:
1426:
1406:
1398:Signature generation
1334:
1299:
1272:
1245:
1158:
1138:
1118:
910:
880:
796:
766:
723:
696:
653:
633:
613:
532:
493:{\displaystyle O(n)}
475:
447:
436:{\displaystyle O(n)}
418:
70:in 2001. The name,
66:, and introduced at
4267:10.1007/11941378_26
4234:2012IEITF..95..151F
4159:10.1007/11424826_65
1839:
1354:
1319:
1268:, and by extension
278:threshold signature
256:deniable signatures
3968:and several other
3664:"world!"
2236:
2146:
2104:
2048:
2009:
1893:
1856:
1822:
1792:
1763:
1733:
1673:
1646:
1616:
1593:
1566:
1546:
1514:
1485:Calculate the key
1471:
1412:
1384:
1337:
1320:
1302:
1285:
1258:
1231:
1144:
1124:
1101:
893:
863:
779:
752:
709:
682:
639:
619:
609:which takes a key
599:
490:
453:
433:
243:
4276:978-3-540-49767-7
4168:978-3-540-25861-2
4135:978-3-540-44050-5
4052:978-3-540-42987-6
4019:Rivest, Ronald L.
3658:"hello"
3396:"utf-8"
3273:"utf-8"
1619:{\displaystyle v}
1569:{\displaystyle k}
1415:{\displaystyle m}
1147:{\displaystyle v}
1127:{\displaystyle z}
642:{\displaystyle v}
622:{\displaystyle k}
456:{\displaystyle n}
332:} can compute a (
268:Unlike standard "
124:
123:
64:Yael Tauman Kalai
37:digital signature
16:(Redirected from
4308:
4281:
4280:
4252:
4246:
4245:
4217:
4211:
4210:
4202:
4193:
4192:
4186:
4182:
4180:
4172:
4146:
4140:
4139:
4127:
4113:
4104:
4098:
4097:
4087:
4063:
4057:
4056:
4015:
3997:
3970:cryptocurrencies
3961:Cryptocurrencies
3956:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
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3875:
3872:
3869:
3866:
3863:
3860:
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3827:
3824:
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3806:
3803:
3800:
3797:
3794:
3791:
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3770:
3767:
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3758:
3755:
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3725:
3722:
3719:
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3707:
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3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
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3665:
3662:
3659:
3656:
3653:
3650:
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3112:
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2515:
2512:
2509:
2506:
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2500:
2497:
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2482:
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2470:
2467:
2464:
2461:
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2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2245:
2243:
2242:
2237:
2226:
2225:
2207:
2206:
2194:
2193:
2181:
2180:
2155:
2153:
2152:
2147:
2136:
2135:
2113:
2111:
2110:
2105:
2100:
2099:
2087:
2086:
2074:
2073:
2057:
2055:
2054:
2049:
2047:
2046:
2018:
2016:
2015:
2010:
2005:
2004:
1986:
1985:
1973:
1972:
1954:
1953:
1935:
1934:
1922:
1921:
1902:
1900:
1899:
1894:
1865:
1863:
1862:
1857:
1852:
1851:
1838:
1830:
1818:
1817:
1801:
1799:
1798:
1793:
1791:
1790:
1772:
1770:
1769:
1764:
1762:
1761:
1742:
1740:
1739:
1734:
1729:
1728:
1716:
1715:
1703:
1702:
1682:
1680:
1679:
1674:
1672:
1671:
1655:
1653:
1652:
1647:
1645:
1644:
1625:
1623:
1622:
1617:
1602:
1600:
1599:
1594:
1592:
1591:
1575:
1573:
1572:
1567:
1555:
1553:
1552:
1547:
1545:
1544:
1523:
1521:
1520:
1515:
1504:
1503:
1480:
1478:
1477:
1472:
1470:
1469:
1451:
1450:
1438:
1437:
1421:
1419:
1418:
1413:
1393:
1391:
1390:
1385:
1383:
1382:
1367:
1366:
1353:
1345:
1329:
1327:
1326:
1321:
1318:
1310:
1294:
1292:
1291:
1286:
1284:
1283:
1267:
1265:
1264:
1259:
1257:
1256:
1240:
1238:
1237:
1232:
1227:
1226:
1208:
1207:
1195:
1194:
1182:
1181:
1153:
1151:
1150:
1145:
1133:
1131:
1130:
1125:
1110:
1108:
1107:
1102:
1079:
1078:
1066:
1065:
1047:
1046:
1034:
1033:
1015:
1014:
1002:
1001:
989:
988:
973:
972:
954:
953:
941:
940:
928:
927:
902:
900:
899:
894:
892:
891:
872:
870:
869:
864:
859:
858:
840:
839:
827:
826:
814:
813:
788:
786:
785:
780:
778:
777:
761:
759:
758:
753:
748:
747:
735:
734:
718:
716:
715:
710:
708:
707:
691:
689:
688:
683:
681:
680:
665:
664:
648:
646:
645:
640:
628:
626:
625:
620:
608:
606:
605:
600:
595:
594:
576:
575:
563:
562:
550:
549:
524:. They define a
522:Rabin signatures
499:
497:
496:
491:
462:
460:
459:
454:
442:
440:
439:
434:
119:
116:
110:
99:This Definition
94:
86:
49:group signatures
21:
4316:
4315:
4311:
4310:
4309:
4307:
4306:
4305:
4286:
4285:
4284:
4277:
4254:
4253:
4249:
4219:
4218:
4214:
4204:
4203:
4196:
4183:
4173:
4169:
4148:
4147:
4143:
4136:
4111:
4106:
4105:
4101:
4065:
4064:
4060:
4053:
4017:
4016:
4012:
3990:
3978:
3963:
3958:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3630:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2253:
2217:
2198:
2185:
2166:
2161:
2160:
2119:
2118:
2091:
2078:
2065:
2060:
2059:
2038:
2033:
2032:
2025:
1996:
1977:
1964:
1945:
1926:
1913:
1905:
1904:
1870:
1869:
1843:
1809:
1804:
1803:
1782:
1777:
1776:
1753:
1748:
1747:
1720:
1707:
1694:
1689:
1688:
1663:
1658:
1657:
1636:
1631:
1630:
1608:
1607:
1583:
1578:
1577:
1558:
1557:
1534:
1533:
1487:
1486:
1461:
1442:
1429:
1424:
1423:
1404:
1403:
1400:
1374:
1358:
1332:
1331:
1297:
1296:
1275:
1270:
1269:
1248:
1243:
1242:
1218:
1199:
1186:
1167:
1156:
1155:
1154:. The equation
1136:
1135:
1116:
1115:
1070:
1057:
1038:
1019:
1006:
993:
980:
964:
945:
932:
913:
908:
907:
883:
878:
877:
850:
831:
818:
799:
794:
793:
769:
764:
763:
739:
726:
721:
720:
699:
694:
693:
672:
656:
651:
650:
631:
630:
611:
610:
586:
567:
554:
535:
530:
529:
514:
512:Original scheme
509:
473:
472:
445:
444:
416:
415:
408:
399:e-voting system
379:
370:
363:
354:
331:
322:
315:
306:
235:
228:
219:
208:
199:
192:
171:
162:
153:
146:
139:
132:
120:
114:
111:
104:
95:
84:
23:
22:
18:Ring signatures
15:
12:
11:
5:
4314:
4312:
4304:
4303:
4298:
4288:
4287:
4283:
4282:
4275:
4247:
4228:(1): 393–415.
4212:
4194:
4185:|journal=
4167:
4141:
4134:
4099:
4058:
4051:
4009:
3989:
3986:
3985:
3984:
3977:
3974:
3962:
3959:
3940:verify_message
3910:verify_message
3883:verify_message
3635:
2934:verify_message
2266:
2252:
2249:
2248:
2247:
2235:
2232:
2229:
2224:
2220:
2216:
2213:
2210:
2205:
2201:
2197:
2192:
2188:
2184:
2179:
2176:
2173:
2169:
2157:
2145:
2142:
2139:
2134:
2129:
2126:
2115:
2103:
2098:
2094:
2090:
2085:
2081:
2077:
2072:
2068:
2045:
2041:
2024:
2021:
2020:
2019:
2008:
2003:
1999:
1995:
1992:
1989:
1984:
1980:
1976:
1971:
1967:
1963:
1960:
1957:
1952:
1948:
1944:
1941:
1938:
1933:
1929:
1925:
1920:
1916:
1912:
1892:
1889:
1886:
1883:
1880:
1877:
1866:
1855:
1850:
1846:
1842:
1837:
1834:
1829:
1825:
1821:
1816:
1812:
1789:
1785:
1773:
1760:
1756:
1744:
1732:
1727:
1723:
1719:
1714:
1710:
1706:
1701:
1697:
1670:
1666:
1643:
1639:
1627:
1615:
1604:
1590:
1586:
1565:
1543:
1513:
1510:
1507:
1502:
1497:
1494:
1468:
1464:
1460:
1457:
1454:
1449:
1445:
1441:
1436:
1432:
1411:
1399:
1396:
1381:
1377:
1373:
1370:
1365:
1361:
1357:
1352:
1349:
1344:
1340:
1317:
1314:
1309:
1305:
1282:
1278:
1255:
1251:
1230:
1225:
1221:
1217:
1214:
1211:
1206:
1202:
1198:
1193:
1189:
1185:
1180:
1177:
1174:
1170:
1166:
1163:
1143:
1123:
1112:
1111:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1077:
1073:
1069:
1064:
1060:
1056:
1053:
1050:
1045:
1041:
1037:
1032:
1029:
1026:
1022:
1018:
1013:
1009:
1005:
1000:
996:
992:
987:
983:
979:
976:
971:
967:
963:
960:
957:
952:
948:
944:
939:
935:
931:
926:
923:
920:
916:
890:
886:
862:
857:
853:
849:
846:
843:
838:
834:
830:
825:
821:
817:
812:
809:
806:
802:
776:
772:
751:
746:
742:
738:
733:
729:
719:is defined as
706:
702:
679:
675:
671:
668:
663:
659:
638:
618:
598:
593:
589:
585:
582:
579:
574:
570:
566:
561:
557:
553:
548:
545:
542:
538:
513:
510:
508:
507:Implementation
505:
489:
486:
483:
480:
452:
432:
429:
426:
423:
407:
404:
403:
402:
395:
392:
384:
381:
375:
368:
359:
352:
327:
320:
311:
304:
266:
234:
231:
224:
217:
204:
197:
188:
167:
158:
151:
144:
137:
130:
122:
121:
98:
96:
89:
83:
80:
72:ring signature
33:ring signature
24:
14:
13:
10:
9:
6:
4:
3:
2:
4313:
4302:
4299:
4297:
4294:
4293:
4291:
4278:
4272:
4268:
4264:
4260:
4259:
4251:
4248:
4243:
4239:
4235:
4231:
4227:
4223:
4216:
4213:
4208:
4201:
4199:
4195:
4190:
4178:
4170:
4164:
4160:
4156:
4152:
4145:
4142:
4137:
4131:
4126:
4121:
4117:
4110:
4103:
4100:
4095:
4091:
4086:
4081:
4077:
4073:
4069:
4062:
4059:
4054:
4048:
4044:
4040:
4036:
4032:
4028:
4024:
4020:
4014:
4011:
4008:
4007:
4005:
4001:
3996:
3987:
3983:
3980:
3979:
3975:
3973:
3971:
3967:
3960:
3633:
2264:
2262:
2258:
2250:
2233:
2230:
2222:
2218:
2214:
2211:
2208:
2203:
2199:
2195:
2190:
2186:
2177:
2174:
2171:
2167:
2158:
2140:
2127:
2124:
2116:
2096:
2092:
2083:
2079:
2075:
2070:
2066:
2043:
2039:
2030:
2029:
2028:
2022:
2001:
1997:
1993:
1990:
1987:
1982:
1978:
1974:
1969:
1965:
1961:
1958:
1955:
1950:
1946:
1942:
1939:
1936:
1931:
1927:
1923:
1918:
1914:
1887:
1884:
1881:
1878:
1867:
1848:
1844:
1835:
1832:
1827:
1823:
1819:
1814:
1810:
1787:
1783:
1774:
1758:
1754:
1745:
1725:
1721:
1712:
1708:
1704:
1699:
1695:
1686:
1668:
1664:
1641:
1637:
1628:
1613:
1605:
1588:
1584:
1563:
1531:
1530:random oracle
1527:
1508:
1495:
1492:
1484:
1483:
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1466:
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1430:
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1397:
1395:
1379:
1375:
1371:
1363:
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1350:
1347:
1342:
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1312:
1307:
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1276:
1253:
1249:
1223:
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1215:
1212:
1209:
1204:
1200:
1196:
1191:
1187:
1178:
1175:
1172:
1168:
1164:
1161:
1141:
1121:
1089:
1083:
1080:
1075:
1071:
1062:
1058:
1054:
1051:
1043:
1039:
1035:
1030:
1027:
1024:
1020:
1011:
1007:
1003:
998:
994:
985:
981:
977:
969:
965:
961:
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955:
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855:
851:
847:
844:
841:
836:
832:
828:
823:
819:
810:
807:
804:
800:
792:The function
790:
774:
770:
744:
740:
731:
727:
704:
700:
677:
673:
669:
666:
661:
657:
636:
616:
591:
587:
583:
580:
577:
572:
568:
564:
559:
555:
546:
543:
540:
536:
527:
523:
519:
511:
506:
504:
501:
484:
478:
470:
466:
450:
427:
421:
413:
405:
400:
396:
393:
390:
389:e-cash system
385:
382:
378:
374:
367:
362:
358:
351:
347:
343:
339:
335:
330:
326:
319:
314:
310:
303:
299:
295:
291:
287:
283:
279:
275:
271:
267:
264:
263:
262:
259:
257:
252:
249:
239:
232:
230:
227:
223:
216:
212:
207:
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196:
191:
187:
183:
179:
175:
170:
166:
161:
157:
150:
143:
136:
129:
118:
108:
102:
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93:
88:
87:
81:
79:
77:
73:
69:
65:
61:
57:
52:
50:
46:
42:
38:
35:is a type of
34:
30:
19:
4257:
4250:
4225:
4221:
4215:
4206:
4150:
4144:
4115:
4102:
4075:
4071:
4061:
4034:
4027:Tauman, Yael
4013:
4004:CC BY-SA 4.0
3992:
3991:
3964:
3856:sign_message
3826:sign_message
3631:
2664:second_range
2628:second_range
2448:sign_message
2254:
2026:
1684:
1629:Pick random
1401:
1113:
791:
515:
502:
414:output size
409:
376:
372:
365:
360:
356:
349:
345:
344:, on input (
341:
337:
333:
328:
324:
317:
312:
308:
301:
297:
289:
285:
281:
273:
269:
260:
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214:
210:
205:
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189:
185:
181:
180:, on input (
177:
173:
168:
164:
159:
155:
148:
141:
134:
127:
125:
112:
100:
71:
53:
44:
32:
29:cryptography
26:
4078:(1): 1–10.
4023:Shamir, Adi
3952:signature_2
3922:signature_2
3895:signature_1
3844:signature_2
3814:signature_1
2676:whole_range
2658:first_range
2652:whole_range
2586:first_range
471:implements
248:White House
4290:Categories
4209:: 181–200.
3988:References
2255:Here is a
1775:Calculate
1524:, using a
469:CryptoNote
412:asymptotic
406:Efficiency
296:. Namely,
172:). Party
115:April 2022
82:Definition
60:Adi Shamir
56:Ron Rivest
4187:ignored (
4177:cite book
4094:137248994
3691:PublicKey
3432:hexdigest
3318:hexdigest
3180:functools
2295:functools
2212:…
1991:…
1940:…
1833:−
1456:…
1348:−
1313:−
1213:…
1090:…
1081:⊕
1055:⊕
1052:⋯
1036:⊕
1028:−
1004:⊕
959:…
845:…
670:…
581:…
154:), ..., (
107:talk page
76:algorithm
68:ASIACRYPT
4029:(2001).
4006:license.
3976:See also
3703:generate
3543:<<
2427:<<
2313:__init__
1556:, since
762:, where
300:parties
294:protocol
280:, where
272:-out-of-
4230:Bibcode
3721:urandom
3411:hashlib
3297:hashlib
3237:_permut
2979:_permut
2694:randint
2532:randint
2490:_permut
2277:hashlib
1903:-tuple
371:, ...,
355:, ...,
323:, ...,
307:, ...,
220:, ...,
200:, ...,
4273:
4165:
4132:
4092:
4049:
3966:Monero
3874:assert
3685:Crypto
3682:return
3627:result
3624:return
3615:result
3567:result
3489:divmod
3402:return
3390:encode
3384:"
3360:"
3267:encode
3222:return
3186:reduce
3147:return
3006:return
2922:return
2688:random
2526:random
2292:import
2286:import
2283:random
2280:import
2274:import
2268:import
2257:Python
465:median
62:, and
4112:(PDF)
4090:S2CID
3802:range
3745:range
3534:<=
3198:range
3081:range
2964:->
2640:range
2598:range
2349:->
2298:class
526:keyed
45:which
4271:ISBN
4189:help
4163:ISBN
4130:ISBN
4047:ISBN
4000:text
3946:msg1
3916:msg2
3889:msg1
3862:msg2
3832:msg1
3808:size
3781:Ring
3763:list
3751:size
3709:1024
3652:msg2
3646:msg1
3637:size
3609:else
3546:self
3453:self
3417:sha1
3372:self
3339:self
3303:sha1
3279:self
3243:self
3204:self
3150:self
3114:list
3045:self
3027:self
3009:self
2973:self
2967:bool
2940:self
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2352:None
2343:1024
2319:self
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1532:for
140:), (
41:keys
31:, a
4263:doi
4238:doi
4155:doi
4120:doi
4080:doi
4039:doi
3931:not
3928:and
3901:and
3793:for
3787:key
3769:key
3757:key
3739:_rn
3733:map
3727:key
3697:RSA
3670:_rn
3667:def
3585:pow
3444:def
3435:(),
3423:msg
3405:int
3351:msg
3330:def
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3309:msg
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2261:RSA
518:RSA
316:∈ {
284:of
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