66:
128:
25:
578:
692:
1510:
implementation provides a simple demonstration of the generator, in particular regarding the three bit selection methods. It is important to note that the requirements imposed upon the parameters
961:
334:
2720:
2636:
1024:
602:
1125:
1151:
1105:
1078:
1051:
988:
824:
798:
850:
870:
758:
makes the algorithm faster, doing so also reduces the security. A 2005 paper gives concrete, as opposed to asymptotic, security proof of BBS, for a given
2667:
149:
142:
494:
611:
1153:= 9, 81, 236, 36, 31, 202. The following table shows the output (in bits) for the different bit selection methods used to determine the output.
2605:
2715:
2594:
766:. The result can also be used to guide choices of the two numbers by balancing expected security against computational cost.
192:
164:
171:
229:
211:
109:
87:
52:
875:
80:
278:
2541:
Sidorenko, Andrey; Schoenmakers, Berry (2005). "Concrete
Security of the Blum-Blum-Shub Pseudorandom Generator".
703:
38:
178:
1181:
732:
250:
2648:
160:
731:
grows large, distinguishing the output bits from random should be at least as difficult as solving the
2662:
74:
475:
An interesting characteristic of the Blum Blum Shub generator is the possibility to calculate any
453:
138:
2657:
91:
993:
587:
1164:
1110:
185:
2574:
1129:
1083:
1056:
1029:
966:
803:
777:
8:
872:
is the seed). We can expect to get a large cycle length for those small numbers, because
829:
605:
437:
44:
855:
485:
742:
The performance of the BBS random-number generator depends on the size of the modulus
2628:
2590:
441:
2700:
2620:
2582:
2550:
266:
2586:
573:{\displaystyle x_{i}=\left(x_{0}^{2^{i}{\bmod {\lambda }}(M)}\right){\bmod {M}}}
707:
2709:
2632:
1545:"Returns the number of 1-valued bits in the integer-encoded BITS."
687:{\displaystyle \lambda (M)=\lambda (p\cdot q)=\operatorname {lcm} (p-1,q-1)}
349:
262:
1719:"Returns the least significant bit of the integer-encoded BITS."
2690:
2685:
2646:
1507:
445:
397:
258:
254:
2554:
2647:
Geisler, Martin; Krøigård, Mikkel; Danielsen, Andreas (December 2004),
2619:(2). Society for Industrial & Applied Mathematics (SIAM): 364–383.
1159:
449:
370:
2695:
713:
2624:
1848:
generator parameters P, Q, and S (seed), and returning three values:
1632:"Returns the even parity bit of the integer-encoded BITS."
127:
1845:
Blum-Blum-Shub pseudorandom number generator, configured to use the
1842:"Returns a function of no arguments which represents a simple
2524:
2522:
2691:
BlumBlumShub, a Java-language implementation by Mark
Rossmiller
2519:
2307:"~&Keys: E = even parity, L = least significant"
1863:
Please note that the parameters P, Q, and S are not checked in
561:
536:
359:. At each step of the algorithm, some output is derived from
317:
1866:
accordance to the conditions described in the article."
706:
of factoring. When the primes are chosen appropriately, and
2721:
Cryptographically secure pseudorandom number generators
2606:"A Simple Unpredictable Pseudo-Random Number Generator"
2686:
GMPBBS, a C-language implementation by Mark
Rossmiller
2540:
1132:
1113:
1086:
1059:
1032:
996:
969:
878:
858:
832:
806:
780:
614:
590:
497:
281:
2575:"Comparison of Two Pseudo-Random Number Generators"
2573:Blum, Lenore; Blum, Manuel; Shub, Michael (1983).
2163:;; Update the state such that x becomes the new x.
1145:
1119:
1099:
1072:
1045:
1018:
982:
955:
864:
844:
818:
792:
686:
596:
572:
448:which is also a quadratic residue), and should be
328:
2707:
383:or one or more of the least significant bits of
1184:implementation that does check for primality.
956:{\displaystyle {\rm {gcd}}((p-3)/2,(q-3)/2)=2}
702:There is a proof reducing its security to the
2603:
2572:
2528:
1857:(3) the least significant bit of the number.
329:{\displaystyle x_{n+1}=x_{n}^{2}{\bmod {M}}}
2581:. Boston, MA: Springer US. pp. 61–78.
53:Learn how and when to remove these messages
2661:
406:should be an integer that is co-prime to
230:Learn how and when to remove this message
212:Learn how and when to remove this message
110:Learn how and when to remove this message
2076:;; Compute the random bit(s) based on x.
1411:"The RNG has fallen into a loop at
440:to 3 (mod 4) (this guarantees that each
73:This article includes a list of general
472:) (this makes the cycle length large).
2708:
1854:(2) the even parity bit of the number,
148:Please improve this article by adding
2604:Blum, L.; Blum, M.; Shub, M. (1986).
746:and the number of bits per iteration
369:; the output is commonly either the
121:
59:
18:
963:. The generator starts to evaluate
13:
887:
884:
881:
79:it lacks sufficient corresponding
14:
2732:
2679:
727:are output, then in the limit as
34:This article has multiple issues.
2642:from the original on 2021-08-14.
2352:"~&--------------"
126:
64:
23:
2673:from the original on 2016-03-31
2490:"~&~6d | ~d | ~d"
2199:;; Print the exemplary outputs.
42:or discuss these issues on the
2716:Pseudorandom number generators
2534:
944:
933:
921:
907:
895:
892:
681:
657:
645:
633:
624:
618:
551:
545:
272:Blum Blum Shub takes the form
1:
2507:
733:quadratic residuosity problem
251:pseudorandom number generator
150:secondary or tertiary sources
16:Pseudorandom number generator
2512:
348:is the product of two large
7:
2587:10.1007/978-1-4757-0602-4_6
2337:"~&x | E | L"
720:) lower-order bits of each
697:
10:
2737:
2565:
2529:Blum, Blum & Shub 1986
769:
2696:An implementation in Java
2613:SIAM Journal on Computing
2112:get-least-significant-bit
1707:get-least-significant-bit
1026:and creates the sequence
1524:
1522:(seed) are not checked.
1186:
1019:{\displaystyle x_{-1}=s}
704:computational difficulty
597:{\displaystyle \lambda }
2543:Cryptography and Coding
1120:{\displaystyle \ldots }
94:more precise citations.
2579:Advances in Cryptology
1147:
1121:
1101:
1074:
1047:
1020:
984:
957:
866:
846:
820:
794:
688:
598:
574:
330:
137:relies excessively on
2499:least-significant-bit
2475:least-significant-bit
2388:least-significant-bit
2193:least-significant-bit
2157:least-significant-bit
2106:least-significant-bit
1165:Least significant bit
1148:
1146:{\displaystyle x_{5}}
1122:
1102:
1100:{\displaystyle x_{2}}
1075:
1073:{\displaystyle x_{1}}
1048:
1046:{\displaystyle x_{0}}
1021:
985:
983:{\displaystyle x_{0}}
958:
867:
847:
821:
795:
689:
599:
575:
331:
269:'s one-way function.
265:that is derived from
1686:get-number-of-1-bits
1533:get-number-of-1-bits
1130:
1111:
1084:
1057:
1030:
994:
967:
876:
856:
830:
819:{\displaystyle q=23}
804:
793:{\displaystyle p=11}
778:
612:
588:
495:
484:value directly (via
279:
253:proposed in 1986 by
2555:10.1007/11586821_24
2531:, pp. 364–383.
2376:multiple-value-bind
2217:make-blum-blum-shub
2094:get-even-parity-bit
1797:make-blum-blum-shub
1620:get-even-parity-bit
1180:The following is a
845:{\displaystyle s=3}
606:Carmichael function
555:
418:are not factors of
315:
1143:
1117:
1097:
1070:
1043:
1016:
980:
953:
862:
842:
816:
790:
684:
594:
570:
515:
425:) and not 1 or 0.
326:
301:
2650:About Random Bits
2596:978-1-4757-0604-8
1851:(1) the number x,
1178:
1177:
865:{\displaystyle s}
750:. While lowering
442:quadratic residue
436:, should both be
240:
239:
232:
222:
221:
214:
196:
120:
119:
112:
57:
2728:
2701:Randomness tests
2674:
2672:
2665:
2655:
2643:
2641:
2610:
2600:
2559:
2558:
2538:
2532:
2526:
2503:
2500:
2497:
2494:
2491:
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2485:
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2479:
2476:
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2470:
2467:
2464:
2461:
2458:
2455:
2452:
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2446:
2443:
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2437:
2434:
2431:
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2422:
2419:
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2413:
2410:
2407:
2404:
2401:
2398:
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2359:
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2239:
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2221:
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2212:
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2206:
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2200:
2197:
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2173:
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2167:
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2158:
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2140:
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2134:
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2128:
2125:
2122:
2119:
2116:
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2110:
2107:
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2101:
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2092:
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2086:
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2077:
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2068:
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2062:
2059:
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2053:
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2047:
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2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
2001:;; x = x^2 mod M
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
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1801:
1798:
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1792:
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1786:
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1777:
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1768:
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1720:
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1699:
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1570:
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1418:
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1403:
1400:
1397:
1394:
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1379:
1376:
1373:
1370:
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1364:
1361:
1358:
1355:
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1349:
1346:
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1340:
1337:
1334:
1331:
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1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1156:
1155:
1152:
1150:
1149:
1144:
1142:
1141:
1126:
1124:
1123:
1118:
1106:
1104:
1103:
1098:
1096:
1095:
1079:
1077:
1076:
1071:
1069:
1068:
1052:
1050:
1049:
1044:
1042:
1041:
1025:
1023:
1022:
1017:
1009:
1008:
989:
987:
986:
981:
979:
978:
962:
960:
959:
954:
940:
914:
891:
890:
871:
869:
868:
863:
851:
849:
848:
843:
825:
823:
822:
817:
799:
797:
796:
791:
693:
691:
690:
685:
608:. (Here we have
603:
601:
600:
595:
579:
577:
576:
571:
569:
568:
559:
554:
544:
543:
534:
533:
523:
507:
506:
428:The two primes,
335:
333:
332:
327:
325:
324:
314:
309:
297:
296:
267:Michael O. Rabin
235:
228:
217:
210:
206:
203:
197:
195:
161:"Blum Blum Shub"
154:
130:
122:
115:
108:
104:
101:
95:
90:this article by
81:inline citations
68:
67:
60:
49:
27:
26:
19:
2736:
2735:
2731:
2730:
2729:
2727:
2726:
2725:
2706:
2705:
2682:
2677:
2670:
2653:
2639:
2625:10.1137/0215025
2608:
2597:
2568:
2563:
2562:
2539:
2535:
2527:
2520:
2515:
2510:
2505:
2504:
2501:
2498:
2496:even-parity-bit
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2454:even-parity-bit
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2385:even-parity-bit
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2322:"~2%"
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
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2267:
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2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2190:even-parity-bit
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2136:even-parity-bit
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2088:even-parity-bit
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
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1970:
1967:
1964:
1961:
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1955:
1952:
1949:
1946:
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1934:
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1925:
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1916:
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1910:
1907:
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1901:
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1700:
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1562:
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1535:
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1529:
1526:
1504:
1503:
1500:
1497:
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1476:
1473:
1470:
1467:
1464:
1461:
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1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
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1374:
1371:
1368:
1365:
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1302:
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1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
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539:
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519:
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502:
498:
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486:Euler's theorem
483:
424:
405:
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147:
143:primary sources
131:
116:
105:
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86:Please help to
85:
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28:
24:
17:
12:
11:
5:
2734:
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2688:
2681:
2680:External links
2678:
2676:
2675:
2663:10.1.1.90.3779
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2569:
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2564:
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2517:
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1525:
1506:The following
1474:blum_blum_shub
1198:blum_blum_shub
1187:
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754:or increasing
723:
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243:Blum Blum Shub
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202:September 2013
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100:September 2013
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2523:
2518:
1935:;; M = p * q
1523:
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1509:
1185:
1183:
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1138:
1134:
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998:
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508:
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467:
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459:
455:
452:with a small
451:
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163: –
162:
158:
157:Find sources:
151:
145:
144:
140:
135:This article
133:
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2612:
2578:
2546:
2542:
2536:
1950:;; x0 = seed
1519:
1515:
1511:
1505:
1179:
1174:1 1 0 0 1 0
1171:0 1 1 0 1 0
773:
763:
759:
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263:Michael Shub
246:
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106:
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78:
50:
43:
37:
36:Please help
33:
2549:: 355–375.
1508:Common Lisp
1432:steps"
450:safe primes
446:square root
259:Manuel Blum
255:Lenore Blum
92:introducing
2710:Categories
2508:References
1354:iterations
1222:iterations
1160:Parity bit
371:bit parity
172:newspapers
139:references
75:references
39:improve it
2658:CiteSeerX
2633:0097-5397
2513:Citations
1115:…
1003:−
990:by using
928:−
902:−
676:−
664:−
655:
640:⋅
631:λ
616:λ
592:λ
541:λ
438:congruent
45:talk page
2668:archived
2637:Archived
2256:function
1803:&key
1605:logcount
698:Security
444:has one
2566:Sources
2463:declare
2442:declare
2421:integer
2409:declare
2397:funcall
2271:integer
2244:declare
2145:declare
2124:declare
2058:integer
2046:declare
1968:integer
1956:declare
1884:integer
1872:declare
1737:integer
1725:declare
1650:integer
1638:declare
1590:integer
1563:integer
1551:declare
1465:numbers
1444:numbers
1441:numbers
1423:numbers
1396:numbers
1333:numbers
1300:isprime
1273:isprime
852:(where
770:Example
735:modulo
604:is the
249:) is a
186:scholar
88:improve
2660:
2631:
2593:
2484:format
2364:repeat
2346:format
2331:format
2316:format
2301:format
2265:values
2196:))))))
2184:values
1995:lambda
1989:#'
1462:return
1450:append
1438:return
1291:assert
1264:assert
1246:assert
1228:assert
1189:import
1182:Python
584:where
410:(i.e.
350:primes
340:where
247:B.B.S.
188:
181:
174:
167:
159:
77:, but
2671:(PDF)
2654:(PDF)
2640:(PDF)
2609:(PDF)
1794:defun
1704:defun
1617:defun
1530:defun
1468:print
1402:print
1348:range
1294:sympy
1267:sympy
1192:sympy
193:JSTOR
179:books
2629:ISSN
2591:ISBN
2547:3796
2502:))))
2469:type
2448:type
2415:type
2361:loop
2250:type
2169:setf
2151:type
2130:type
2052:type
1962:type
1878:type
1785:bits
1773:byte
1749:bits
1731:type
1713:bits
1689:bits
1662:bits
1644:type
1626:bits
1608:bits
1575:bits
1557:type
1539:bits
1518:and
1456:seed
1390:seed
1369:seed
1360:seed
1216:seed
826:and
774:Let
762:and
716:log
432:and
414:and
398:seed
396:The
355:and
261:and
165:news
2621:doi
2583:doi
2551:doi
2472:bit
2451:bit
2400:bbs
2292:bbs
2286:bit
2283:bit
2238:)))
2211:bbs
2205:let
2154:bit
2133:bit
2118:)))
2082:let
2040:)))
2019:mod
2007:let
1911:let
1860:---
1788:)))
1767:ldb
1761:bit
1758:the
1698:)))
1680:mod
1674:bit
1671:the
1611:)))
1584:the
1498:100
1417:len
1339:for
1195:def
714:log
694:).
652:lcm
562:mod
537:mod
488:):
466:q-3
464:, (
458:p-3
454:gcd
373:of
318:mod
141:to
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2666:,
2656:,
2635:.
2627:.
2617:15
2615:.
2611:.
2589:.
2577:.
2545:.
2521:^
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2457:))
2436:))
2370:do
2295:))
2289:))
2259:()
2232::s
2229:23
2226::q
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2220::p
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2160:))
2139:))
2100:))
2085:((
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2010:((
1998:()
1986:))
1947:))
1932:))
1914:((
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1393:in
1387:if
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1345:in
1330:p2
1324:p1
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1306:p2
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1279:p1
1258:==
1249:p2
1240:==
1231:p1
1225:):
1210:p2
1204:p1
1107:,
1080:,
1053:,
814:23
800:,
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391:+1
381:+1
367:+1
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257:,
152:.
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2623::
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176:·
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107:(
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51:(
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