592:
less. Without the aid of a computer, this number would be a strong prime in the cryptographic sense because 439351292910452432574786963588089477522344330 has the large prime factor 1747822896920092227343 (and in turn the number one less than that has the large prime factor 1683837087591611009), 439351292910452432574786963588089477522344332 has the large prime factor 864608136454559457049 (and in turn the number one less than that has the large prime factor 105646155480762397). Even using algorithms more advanced than
33:
135:
591:
It is possible for a prime to be a strong prime both in the cryptographic sense and the number theoretic sense. For the sake of illustration, 439351292910452432574786963588089477522344331 is a strong prime in the number theoretic sense because the arithmetic mean of its two neighboring primes is 62
1583:
283:. For example, 17 is the seventh prime: the sixth and eighth primes, 13 and 19, add up to 32, and half that is 16; 17 is greater than 16, so 17 is a strong prime.
815:
1186:
208:
of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). Or to put it algebraically, writing the
396:
1268:
1191:
1105:
654:
808:
2084:
116:
1442:
2089:
1523:
54:
97:
801:
50:
1645:
1303:
69:
638:
1670:
1136:
702:
76:
1578:
787:
601:
83:
728:
is by congruences to powers of a base, not by inequality to the arithmetic mean of neighboring pseudoprimes.
1211:
597:
43:
1728:
857:
65:
2065:
1655:
1308:
1216:
653:. However, strong primes do not protect against modulus factorisation using newer algorithms such as
1635:
1630:
1288:
740:
144:
1738:
1675:
1665:
1650:
1283:
1141:
725:
690:
658:
1062:
646:
1707:
1682:
1660:
1640:
1263:
1235:
928:
650:
149:
1617:
1607:
1602:
1539:
1386:
1253:
1156:
209:
389:, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499 (sequence
1318:
1278:
1161:
1126:
1090:
1045:
898:
886:
698:
622:
205:
90:
1723:
1697:
1594:
1462:
1313:
1273:
1258:
1130:
1021:
986:
941:
866:
848:
732:
678:
666:
618:
593:
629:
should be chosen as the product of two strong primes. This makes the factorization of
2078:
1733:
1498:
1362:
1335:
1171:
1036:
974:
965:
950:
913:
839:
782:
462:
197:
185:
2054:
2049:
2044:
2039:
2034:
2029:
2024:
2019:
2014:
2009:
2004:
1999:
1994:
1989:
1984:
1979:
1974:
1969:
1964:
1959:
1954:
1949:
1944:
1939:
1934:
1929:
1924:
1919:
1914:
1909:
1904:
1899:
1894:
1889:
1884:
1687:
1410:
1293:
1166:
1151:
1146:
1110:
824:
662:
458:
439:
386:
382:
378:
374:
370:
366:
362:
358:
354:
350:
346:
342:
338:
334:
330:
181:
180:
with certain special properties. The definitions of strong primes are different in
177:
1879:
1874:
1869:
1864:
1859:
1854:
1849:
1844:
1839:
1834:
1829:
1824:
1819:
1814:
1809:
1804:
1799:
1794:
1789:
1784:
1779:
1625:
1298:
1206:
1201:
1181:
1095:
998:
874:
645:
computationally infeasible. For this reason, strong primes are required by the
326:
322:
318:
314:
310:
306:
302:
298:
294:
290:
169:
32:
669:. Similar (and more technical) argument is also given by Rivest and Silverman.
1702:
1518:
1426:
1346:
1196:
1100:
718:
404:
1743:
1692:
1573:
453:
is sufficiently large to be useful in cryptography; typically this requires
731:
When a prime is equal to the mean of its neighboring primes, it's called a
17:
1245:
793:
788:
3.1.4 What are Strong Primes and are they
Necessary for the RSA System?
701:. Therefore, for cryptosystems based on discrete logarithm, such as
596:, these numbers would be difficult to factor by hand. For a modern
1240:
1226:
767:
457:
to be too large for plausible computational resources to enable a
446:
is said to be "strong" if the following conditions are satisfied.
661:
algorithm. Given the additional cost of generating strong primes
797:
128:
26:
724:
Note that the criteria for determining if a pseudoprime is a
709: − 1 have at least one large prime factor.
391:
600:, these numbers can be factored almost instantaneously. A
1774:
1769:
1764:
1759:
515: − 1 has large prime factors. That is,
475: − 1 has large prime factors. That is,
1752:
1716:
1616:
1593:
1567:
1334:
1327:
1225:
1119:
1083:
832:
142:It has been suggested that this article should be
57:. Unsourced material may be challenged and removed.
721:is likely to be a cryptographically strong prime.
555: + 1 has large prime factors. That is,
1456: = 0, 1, 2, 3, ...
790:- RSA Lab's explanation on strong vs weak primes
604:prime has to be much larger than this example.
766:, Cryptology ePrint Archive: Report 2001/007.
423:is always a strong prime, since 3 must divide
809:
8:
649:standard for use in generating RSA keys for
608:Application of strong primes in cryptography
204:is a prime number that is greater than the
1331:
816:
802:
794:
117:Learn how and when to remove this message
758:
756:
665:do not currently recommend their use in
752:
685: − 1 are less than log
427: − 2, which cannot be prime.
673:Discrete-logarithm-based cryptosystems
572: − 1 for some integer
7:
655:Lenstra elliptic curve factorization
55:adding citations to reliable sources
783:Guide to Cryptography and Standards
764:Are 'Strong' Primes Needed for RSA?
681:in 1978 that if all the factors of
677:It is shown by Stephen Pohlig and
25:
762:Ron Rivest and Robert Silverman,
1192:Supersingular (moonshine theory)
735:. When it's less, it's called a
617:Some people suggest that in the
535: + 1 for some integer
492: + 1 for some integer
286:The first few strong primes are
133:
31:
768:http://eprint.iacr.org/2001/007
42:needs additional citations for
1187:Supersingular (elliptic curve)
689:, then the problem of solving
643: − 1 algorithm
1:
968:2 ± 2 ± 1
613:Factoring-based cryptosystems
739:(not to be confused with a
625:cryptosystems, the modulus
192:Definition in number theory
2106:
434:Definition in cryptography
2063:
469:with other strong primes.
233:, ...) = (2, 3, 5, ...),
148:into multiple articles. (
2085:Classes of prime numbers
1574:Mega (1,000,000+ digits)
1443:Arithmetic progression (
717:A computationally large
602:cryptographically strong
598:computer algebra system
2090:Theory of cryptography
1729:Industrial-grade prime
1106:Newman–Shanks–Williams
705:, it is required that
2066:List of prime numbers
1524:Sophie Germain/Safe (
415: + 2) with
240:is a strong prime if
212:of prime numbers as (
1248:(10 − 1)/9
51:improve this article
1557: ± 7, ...
1084:By integer sequence
869:(2 + 1)/3
741:weakly prime number
713:Miscellaneous facts
419: > 5,
1739:Formula for primes
1372: + 2 or
1304:Smarandache–Wellin
726:strong pseudoprime
691:discrete logarithm
659:Number Field Sieve
651:digital signatures
2072:
2071:
1683:Carmichael number
1618:Composite numbers
1553: ± 3, 8
1549: ± 1, 4
1512: ± 1, …
1508: ± 1, 4
1504: ± 1, 2
1494:
1493:
1039:3·2 − 1
944:2·3 + 1
858:Double Mersenne (
442:, a prime number
166:
165:
127:
126:
119:
101:
16:(Redirected from
2097:
1603:Eisenstein prime
1558:
1534:
1513:
1485:
1457:
1437:
1421:
1405:
1400: + 6,
1396: + 2,
1381:
1376: + 4,
1357:
1332:
1249:
1212:Highly cototient
1074:
1073:
1067:
1057:
1040:
1031:
1016:
993:
992:·2 − 1
981:
980:·2 + 1
969:
960:
945:
936:
923:
908:
893:
881:
880:·2 + 1
870:
861:
852:
843:
818:
811:
804:
795:
770:
760:
579:and large prime
542:and large prime
499:and large prime
394:
282:
281:
279:
278:
275:
272:
161:
158:
137:
136:
129:
122:
115:
111:
108:
102:
100:
59:
35:
27:
21:
2105:
2104:
2100:
2099:
2098:
2096:
2095:
2094:
2075:
2074:
2073:
2068:
2059:
1753:First 60 primes
1748:
1712:
1612:
1595:Complex numbers
1589:
1563:
1541:
1525:
1500:
1499:Bi-twin chain (
1490:
1464:
1444:
1428:
1412:
1388:
1364:
1348:
1323:
1309:Strobogrammatic
1247:
1221:
1115:
1079:
1071:
1065:
1064:
1047:
1038:
1023:
1000:
988:
976:
967:
952:
943:
930:
922:# + 1
920:
915:
907:# ± 1
905:
900:
892:! ± 1
888:
876:
868:
860:2 − 1
859:
851:2 − 1
850:
842:2 + 1
841:
828:
822:
779:
774:
773:
761:
754:
749:
715:
675:
615:
610:
590:
585:
578:
571:
565:
548:
541:
534:
528:
521:
514:
505:
498:
491:
485:
436:
430:
390:
276:
273:
271:
261:
252:
251:
249:
246:
241:
238:
232:
225:
218:
206:arithmetic mean
194:
162:
156:
153:
138:
134:
123:
112:
106:
103:
60:
58:
48:
36:
23:
22:
15:
12:
11:
5:
2103:
2101:
2093:
2092:
2087:
2077:
2076:
2070:
2069:
2064:
2061:
2060:
2058:
2057:
2052:
2047:
2042:
2037:
2032:
2027:
2022:
2017:
2012:
2007:
2002:
1997:
1992:
1987:
1982:
1977:
1972:
1967:
1962:
1957:
1952:
1947:
1942:
1937:
1932:
1927:
1922:
1917:
1912:
1907:
1902:
1897:
1892:
1887:
1882:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1842:
1837:
1832:
1827:
1822:
1817:
1812:
1807:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1762:
1756:
1754:
1750:
1749:
1747:
1746:
1741:
1736:
1731:
1726:
1724:Probable prime
1720:
1718:
1717:Related topics
1714:
1713:
1711:
1710:
1705:
1700:
1698:Sphenic number
1695:
1690:
1685:
1680:
1679:
1678:
1673:
1668:
1663:
1658:
1653:
1648:
1643:
1638:
1633:
1622:
1620:
1614:
1613:
1611:
1610:
1608:Gaussian prime
1605:
1599:
1597:
1591:
1590:
1588:
1587:
1586:
1576:
1571:
1569:
1565:
1564:
1562:
1561:
1537:
1533: + 1
1521:
1516:
1495:
1492:
1491:
1489:
1488:
1460:
1440:
1436: + 6
1424:
1420: + 4
1408:
1404: + 8
1384:
1380: + 6
1360:
1356: + 2
1343:
1341:
1329:
1325:
1324:
1322:
1321:
1316:
1311:
1306:
1301:
1296:
1291:
1286:
1281:
1276:
1271:
1266:
1261:
1256:
1251:
1243:
1238:
1232:
1230:
1223:
1222:
1220:
1219:
1214:
1209:
1204:
1199:
1194:
1189:
1184:
1179:
1174:
1169:
1164:
1159:
1154:
1149:
1144:
1139:
1134:
1123:
1121:
1117:
1116:
1114:
1113:
1108:
1103:
1098:
1093:
1087:
1085:
1081:
1080:
1078:
1077:
1060:
1056: − 1
1043:
1034:
1019:
996:
984:
972:
963:
948:
939:
935: + 1
926:
918:
911:
903:
896:
884:
872:
864:
855:
846:
836:
834:
830:
829:
823:
821:
820:
813:
806:
798:
792:
791:
785:
778:
777:External links
775:
772:
771:
751:
750:
748:
745:
733:balanced prime
714:
711:
679:Martin Hellman
674:
671:
667:key generation
619:key generation
614:
611:
609:
606:
594:trial division
588:
587:
583:
576:
569:
563:
550:
546:
539:
532:
526:
519:
512:
507:
503:
496:
489:
483:
470:
435:
432:
401:
400:
266:
256:
244:
236:
230:
223:
216:
193:
190:
164:
163:
141:
139:
132:
125:
124:
66:"Strong prime"
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2102:
2091:
2088:
2086:
2083:
2082:
2080:
2067:
2062:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2001:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1956:
1953:
1951:
1948:
1946:
1943:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1757:
1755:
1751:
1745:
1742:
1740:
1737:
1735:
1734:Illegal prime
1732:
1730:
1727:
1725:
1722:
1721:
1719:
1715:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1632:
1629:
1628:
1627:
1624:
1623:
1621:
1619:
1615:
1609:
1606:
1604:
1601:
1600:
1598:
1596:
1592:
1585:
1582:
1581:
1580:
1579:Largest known
1577:
1575:
1572:
1570:
1566:
1560:
1556:
1552:
1548:
1544:
1538:
1536:
1532:
1528:
1522:
1520:
1517:
1515:
1511:
1507:
1503:
1497:
1496:
1487:
1484:
1481: +
1480:
1476:
1472:
1469: −
1468:
1461:
1459:
1455:
1451:
1448: +
1447:
1441:
1439:
1435:
1431:
1425:
1423:
1419:
1415:
1409:
1407:
1403:
1399:
1395:
1391:
1385:
1383:
1379:
1375:
1371:
1367:
1361:
1359:
1355:
1351:
1345:
1344:
1342:
1340:
1338:
1333:
1330:
1326:
1320:
1317:
1315:
1312:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1280:
1277:
1275:
1272:
1270:
1267:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1244:
1242:
1239:
1237:
1234:
1233:
1231:
1228:
1224:
1218:
1215:
1213:
1210:
1208:
1205:
1203:
1200:
1198:
1195:
1193:
1190:
1188:
1185:
1183:
1180:
1178:
1175:
1173:
1170:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1132:
1128:
1125:
1124:
1122:
1118:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1089:
1088:
1086:
1082:
1076:
1070:
1061:
1059:
1055:
1051:
1044:
1042:
1035:
1033:
1030:
1027: +
1026:
1020:
1018:
1015:
1012: −
1011:
1007:
1004: −
1003:
997:
995:
991:
985:
983:
979:
973:
971:
964:
962:
959:
956: +
955:
949:
947:
940:
938:
934:
929:Pythagorean (
927:
925:
921:
912:
910:
906:
897:
895:
891:
885:
883:
879:
873:
871:
865:
863:
856:
854:
847:
845:
838:
837:
835:
831:
826:
819:
814:
812:
807:
805:
800:
799:
796:
789:
786:
784:
781:
780:
776:
769:
765:
759:
757:
753:
746:
744:
742:
738:
734:
729:
727:
722:
720:
712:
710:
708:
704:
700:
696:
692:
688:
684:
680:
672:
670:
668:
664:
660:
656:
652:
648:
644:
642:
636:
633: =
632:
628:
624:
620:
612:
607:
605:
603:
599:
595:
582:
575:
568:
562:
559: =
558:
554:
551:
545:
538:
531:
525:
522: =
518:
511:
508:
502:
495:
488:
482:
479: =
478:
474:
471:
468:
464:
460:
456:
452:
449:
448:
447:
445:
441:
433:
431:
428:
426:
422:
418:
414:
410:
406:
398:
393:
388:
384:
380:
376:
372:
368:
364:
360:
356:
352:
348:
344:
340:
336:
332:
328:
324:
320:
316:
312:
308:
304:
300:
296:
292:
289:
288:
287:
284:
269:
265:
259:
255:
247:
239:
229:
222:
215:
211:
207:
203:
199:
198:number theory
191:
189:
187:
186:number theory
183:
179:
175:
171:
160:
151:
147:
146:
140:
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121:
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110:
99:
96:
92:
89:
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68: –
67:
63:
62:Find sources:
56:
52:
46:
45:
40:This article
38:
34:
29:
28:
19:
1688:Almost prime
1646:Euler–Jacobi
1554:
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1540:Cunningham (
1530:
1526:
1509:
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1465:consecutive
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1413:
1401:
1397:
1393:
1389:
1387:Quadruplet (
1377:
1373:
1369:
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1336:
1284:Full reptend
1176:
1142:Wolstenholme
1137:Wall–Sun–Sun
1068:
1053:
1049:
1028:
1024:
1013:
1009:
1005:
1001:
989:
977:
957:
953:
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916:
901:
889:
877:
825:Prime number
763:
736:
730:
723:
716:
706:
694:
686:
682:
676:
663:RSA Security
640:
634:
630:
626:
616:
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580:
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465:products of
459:cryptanalyst
454:
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440:cryptography
437:
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234:
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213:
202:strong prime
201:
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182:cryptography
178:prime number
174:strong prime
173:
167:
154:
143:
113:
107:October 2018
104:
94:
87:
80:
73:
61:
49:Please help
44:verification
41:
1671:Somer–Lucas
1626:Pseudoprime
1264:Truncatable
1236:Palindromic
1120:By property
899:Primorial (
887:Factorial (
621:process in
170:mathematics
2079:Categories
1708:Pernicious
1703:Interprime
1463:Balanced (
1254:Permutable
1229:-dependent
1046:Williams (
942:Pierpont (
867:Wagstaff
849:Mersenne (
833:By formula
747:References
737:weak prime
719:safe prime
647:ANSI X9.31
639:Pollard's
405:twin prime
77:newspapers
18:Weak prime
1744:Prime gap
1693:Semiprime
1656:Frobenius
1363:Triplet (
1162:Ramanujan
1157:Fortunate
1127:Wieferich
1091:Fibonacci
1022:Leyland (
987:Woodall (
966:Solinas (
951:Quartan (
463:factorise
270:+ 1
260:− 1
1636:Elliptic
1411:Cousin (
1328:Patterns
1319:Tetradic
1314:Dihedral
1279:Primeval
1274:Delicate
1259:Circular
1246:Repunit
1037:Thabit (
975:Cullen (
914:Euclid (
840:Fermat (
210:sequence
157:May 2024
1631:Catalan
1568:By size
1339:-tuples
1269:Minimal
1172:Regular
1063:Mills (
999:Cuban (
875:Proth (
827:classes
693:modulo
411:,
395:in the
392:A051634
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150:discuss
91:scholar
1676:Strong
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1651:Fermat
1427:Sexy (
1347:Twin (
1289:Unique
1217:Unique
1177:Strong
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1147:Wilson
1111:Perrin
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637:using
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1641:Euler
1294:Happy
1241:Emirp
1207:Higgs
1202:Super
1182:Stern
1152:Lucky
1096:Lucas
403:In a
248:>
176:is a
145:split
98:JSTOR
84:books
1584:list
1519:Chen
1299:Self
1227:Base
1197:Good
1131:pair
1101:Pell
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657:and
397:OEIS
200:, a
184:and
172:, a
70:news
2055:281
2050:277
2045:271
2040:269
2035:263
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2025:251
2020:241
2015:239
2010:233
2005:229
2000:227
1995:223
1990:211
1985:199
1980:197
1975:193
1970:191
1965:181
1960:179
1955:173
1950:167
1945:163
1940:157
1935:151
1930:149
1925:139
1920:137
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1910:127
1905:113
1900:109
1895:107
1890:103
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623:RSA
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1066:⌊
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