1102:
332:
386:. Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve. In order to achieve this speed-up, the number field sieve has to perform computations and factorizations in
946:
747:
87:
1700:
developed a faster implementation of final processing as part of msieve, which is in the public domain. Both implementations feature the ability to be distributed among several nodes in a cluster with a sufficiently fast interconnect.
1621:
One such method was suggested by Murphy and Brent; they introduce a two-part score for polynomials, based on the presence of roots modulo small primes and on the average value that the polynomial takes over the sieving area.
1476:
1475:, with a "square root" which also can be computed. It should be remarked that the use of Gaussian elimination does not give the optimal run time of the algorithm. Instead, sparse matrix solving algorithms such as
2535:
570:
92:
796:
2539:
1020:
565:
2047:
327:{\displaystyle {\begin{aligned}&\exp \left(\left((64/9)^{1/3}+o(1)\right)\left(\log n\right)^{1/3}\left(\log \log n\right)^{2/3}\right)\\={}&L_{n}\left\end{aligned}}}
1086:
1051:
784:
471:
2080:
1550:, and these homomorphisms will map each "square root" (typically not represented as a rational number) into its integer representative. Now the product of the factors
1610:
The choice of polynomial can dramatically affect the time to complete the remainder of the algorithm. The method of choosing polynomials based on the expansion of
2595:
2404:
2040:
1708:
software written by
Kleinjung, or by msieve, and lattice sieving by GPL software written by Franke and Kleinjung; these are distributed in GGNFS.
2272:
1804:
382:(see below). The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of
2214:
2033:
2143:
2320:
349:: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from
2118:
2013:
1735:(which contains final-processing code, a polynomial selection optimized for smaller numbers and an implementation of the line sieve)
1142:
1693:
2229:
2267:
2204:
2148:
2111:
2409:
2300:
2219:
2209:
1124:
63:
2085:
2237:
2490:
1390:, and we have a better chance for them to be smooth at the same time. The current best-known approach for this search is
2485:
2414:
2315:
2452:
2366:
1863:
356:
The principle of the number field sieve (both special and general) can be understood as an improvement to the simpler
1120:
951:
The product of any two such values can be computed by taking the product as polynomials, then reducing any powers of
2531:
2521:
2480:
2256:
2224:
2095:
1750:
1670:
941:{\displaystyle a_{k-1}r^{k-1}+\cdots +a_{1}r^{1}+a_{0}r^{0},{\text{ where }}a_{0},\ldots ,a_{k-1}\in \mathbb {Q} .}
346:
2516:
2457:
1480:
1964:
2419:
2292:
2138:
2090:
390:. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve.
1677:. A project called NFSNET ran from 2002 through at least 2007. It used volunteer distributed computing on the
742:{\displaystyle {\begin{aligned}(a+bi)(c+di)&=ac+(ad+bc)i+(bd)i^{2}\\&=(ac-bd)+(ad+bc)i.\end{aligned}}}
985:
2434:
2325:
1587:
2545:
2495:
2475:
1186:
1175:
976:
753:
43:
2001:(eds.). "The development of the number field sieve". Lecture Notes in Math. (1993) 1554. Springer-Verlag.
1207:. An optimal strategy for choosing these polynomials is not known; one simple method is to pick a degree
1112:
2196:
2171:
2100:
1697:
1626:
51:
2555:
1618:
shown above is suboptimal in many practical situations, leading to the development of better methods.
2550:
2442:
2424:
2399:
2361:
2105:
1878:
1398:
1938:
1844:
2560:
2526:
2447:
2351:
2310:
2305:
2282:
2186:
1692:
Until 2007, the gold-standard implementation was a suite of software developed and distributed by
1562:
can be obtained as a square in two waysâone for each homomorphism. Thus, one can find two numbers
2391:
2338:
2335:
2176:
2075:
1674:
1526:
1201:
1194:
2132:
2125:
2511:
2467:
2181:
2158:
2009:
1800:
1696:
in the
Netherlands, which was available only under a relatively restrictive license. In 2007,
1413:
of squares in our number fields, but that condition can be achieved by this method too. Each
1060:
1025:
758:
454:
967:
as described above, yielding a value in the same form. To ensure that this field is actually
2356:
1886:
1454:
1054:
980:
2346:
2245:
1998:
1391:
1190:
1057:
with integer coefficients. In some cases, this ring of integers is equivalent to the ring
361:
1882:
1821:
2376:
2277:
2262:
2166:
2067:
2005:
1770:
1686:
787:
357:
338:
1409:
to be squares at the same time. A slightly stronger condition is neededâthat they are
2589:
2371:
2056:
1994:
1669:
Some implementations focus on a certain smaller class of numbers. These are known as
1367:
369:
31:
2381:
1774:
1722:
1717:
1682:
1503:
971:-dimensional and does not collapse to an even smaller field, it is sufficient that
438:
387:
1891:
17:
1449:, with a "square root" which can be determined (as a product of known factors in
350:
2025:
1913:
1410:
1156:
342:
2059:
2008:. Prime Numbers: A Computational Perspective (2001). 2nd edition, Springer.
59:
47:
2022:
Matthew E. Briggs: An
Introduction to the General Number Field Sieve, 1998
1732:
1678:
1394:; to get acceptable yields, it is necessary to use a large factor base.
1712:
1968:
1216:
1582:
and again with probability at least one half we get a factor of
2029:
1705:
1095:
1918:
Presentation at EIDMA-CWI Workshop on
Factoring Large Numbers
1864:"On polynomial selection for the general number field sieve"
1727:
1942:
1625:
The best reported results were achieved by the method of
506:. This equation can be used to reduce away any powers of
378:. The size of these values is exponential in the size of
1431:
and hence that the product of the corresponding factors
2576:
indicate that algorithm is for numbers of special forms
1738:
1651:
composed of small prime factors congruent to 1 modulo 2
1351:) is an integer. The goal is to find integer values of
1246:) as the polynomial with the smallest coefficients and
1185:
are chosen, which have integer coefficients, which are
1063:
1028:
988:
761:
457:
404:
or the number of bits in the binary representation of
364:. When using such algorithms to factor a large number
1822:"On quadratic polynomials for the number field sieve"
1117:. In particular, there are no examples or pseudocode.
799:
568:
90:
2016:. Section 6.2: Number field sieve, pp. 278â301.
1453:)âit will typically be represented as an irrational
2504:
2466:
2433:
2390:
2334:
2291:
2195:
2157:
2066:
1080:
1045:
1014:
979:over the rationals. Similarly, one may define the
940:
778:
741:
465:
326:
425:. The running time of the number field sieve is
372:(i.e. numbers with small prime factors) of order
1784:. Vol. 43, no. 12. pp. 1473â1485.
1689:and Richard Wackerbarth of Texas were involved.
559:. This allows us to define the complex product:
353:(which are trivial to factor by taking roots).
1704:Polynomial selection is normally performed by
2041:
8:
1211:for a polynomial, consider the expansion of
1123:. There might be a discussion about this on
1370:relative to the chosen basis of primes. If
2048:
2034:
2026:
1826:Australian Computer Science Communications
393:The size of the input to the algorithm is
1890:
1457:. Similarly, the product of the factors
1143:Learn how and when to remove this message
1065:
1064:
1062:
1030:
1029:
1027:
997:
996:
995:
991:
990:
987:
931:
930:
915:
896:
887:
878:
868:
855:
845:
820:
804:
798:
763:
762:
760:
668:
569:
567:
459:
458:
456:
305:
301:
289:
272:
258:
250:
228:
224:
188:
184:
134:
130:
118:
91:
89:
1015:{\textstyle \mathbb {O} _{\mathbb {Q} }}
1762:
752:In general, this leads directly to the
1088:. However, there are many exceptions.
1386:will be small too, about the size of
786:, which can be defined as the set of
498:as a linear combination of powers of
492:, which can be rearranged to express
7:
1862:Kleinjung, Thorsten (October 2006).
427:super-polynomial but sub-exponential
1912:Paul Leyland (December 12, 2003).
1401:, one can get products of certain
25:
2257:Special number field sieve (SNFS)
2251:General number field sieve (GNFS)
1820:Murphy, B.; Brent, R. P. (1998),
1657:and over leading coefficients of
2596:Integer factorization algorithms
1941:. April 23, 2007. Archived from
1673:techniques, such as used in the
1397:Having enough such pairs, using
1265:Consider the number field rings
1100:
368:, it is necessary to search for
345:. It is a generalization of the
1846:On RSA 200 and larger projects
1311:are integers, then so will be
1303:with integer coefficients, if
1193:, and which, when interpreted
1075:
1069:
1040:
1034:
1007:
1001:
773:
767:
726:
708:
702:
684:
661:
652:
643:
625:
606:
591:
588:
573:
298:
283:
156:
150:
127:
112:
1:
1892:10.1090/S0025-5718-06-01870-9
1287:are roots of the polynomials
2215:Lenstra elliptic curve (ECM)
1230:) for a number of different
550: + 1 = 0
473:(the rational numbers), and
1661:which are divisible by 60.
1606:Improving polynomial choice
408:. Any element of the order
2612:
2522:Exponentiation by squaring
2205:Continued fraction (CFRAC)
1871:Mathematics of Computation
1751:Special number field sieve
1671:special number field sieve
1222:(allowing digits between â
436:
429:in the size of the input.
347:special number field sieve
36:general number field sieve
2569:
1795:Ribenboim, Paulo (1972).
1405:and of the corresponding
1359:that simultaneously make
1081:{\textstyle \mathbb {Z} }
1046:{\textstyle \mathbb {Q} }
779:{\textstyle \mathbb {Q} }
466:{\textstyle \mathbb {Q} }
66:for factoring an integer
1914:"NFSNET: the first year"
1200:, have a common integer
451:-degree polynomial over
2435:Greatest common divisor
1588:greatest common divisor
27:Factorization algorithm
2546:Modular exponentiation
1799:. Wiley-Interscience.
1775:"A Tale of Two Sieves"
1082:
1047:
1016:
977:irreducible polynomial
942:
780:
754:algebraic number field
743:
541:is the imaginary unit
467:
328:
2273:Shanks's square forms
2197:Integer factorization
2172:Sieve of Eratosthenes
2004:Richard Crandall and
1843:Franke, Jens (2006),
1083:
1048:
1017:
943:
781:
744:
477:is a complex root of
468:
329:
81:bits) is of the form
2551:Montgomery reduction
2425:Function field sieve
2400:Baby-step giant-step
2246:Quadratic sieve (QS)
1647:, and searches over
1399:Gaussian elimination
1113:confusing or unclear
1061:
1026:
986:
797:
759:
566:
455:
88:
2561:Trachtenberg system
2527:Integer square root
2468:Modular square root
2187:Wheel factorization
2139:Quadratic Frobenius
2119:LucasâLehmerâRiesel
1945:on October 22, 2007
1939:"Welcome to NFSNET"
1883:2006MaCom..75.2037K
1121:clarify the section
1053:which are roots of
2453:Extended Euclidean
2392:Discrete logarithm
2321:SchönhageâStrassen
2177:Sieve of Pritchard
1999:H. W. Lenstra, Jr.
1877:(256): 2037â2047.
1782:Notices of the AMS
1698:Jason Papadopoulos
1675:Cunningham project
1627:Thorsten Kleinjung
1490:is a root of both
1078:
1043:
1012:
938:
776:
739:
737:
522:. For example, if
463:
418:is exponential in
324:
322:
52:factoring integers
18:Number Field Sieve
2583:
2582:
2182:Sieve of Sundaram
1806:978-0-471-71804-8
1797:Algebraic Numbers
1773:(December 1996).
1327:), which we call
1153:
1152:
1145:
1055:monic polynomials
1022:as the subset of
890:
889: where
16:(Redirected from
2603:
2532:Integer relation
2505:Other algorithms
2410:Pollard kangaroo
2301:Ancient Egyptian
2159:Prime-generating
2144:SolovayâStrassen
2057:Number-theoretic
2050:
2043:
2036:
2027:
1995:Arjen K. Lenstra
1981:
1980:
1978:
1976:
1967:. Archived from
1961:
1955:
1954:
1952:
1950:
1935:
1929:
1928:
1926:
1924:
1909:
1903:
1902:
1900:
1899:
1894:
1868:
1859:
1853:
1852:
1851:
1840:
1834:
1833:
1817:
1811:
1810:
1792:
1786:
1785:
1779:
1767:
1660:
1656:
1650:
1646:
1617:
1613:
1455:algebraic number
1378:are small, then
1148:
1141:
1137:
1134:
1128:
1104:
1103:
1096:
1087:
1085:
1084:
1079:
1068:
1052:
1050:
1049:
1044:
1033:
1021:
1019:
1018:
1013:
1011:
1010:
1000:
994:
981:ring of integers
974:
970:
966:
956:
947:
945:
944:
939:
934:
926:
925:
901:
900:
891:
888:
883:
882:
873:
872:
860:
859:
850:
849:
831:
830:
815:
814:
785:
783:
782:
777:
766:
748:
746:
745:
740:
738:
677:
673:
672:
558:
551:
544:
540:
536:
521:
511:
505:
501:
497:
491:
480:
476:
472:
470:
469:
464:
462:
450:
446:
424:
417:
413:
407:
403:
385:
381:
377:
367:
333:
331:
330:
325:
323:
319:
315:
314:
313:
309:
293:
276:
263:
262:
251:
242:
238:
237:
236:
232:
223:
219:
197:
196:
192:
183:
179:
163:
159:
143:
142:
138:
122:
94:
80:
78:
69:
57:
21:
2611:
2610:
2606:
2605:
2604:
2602:
2601:
2600:
2586:
2585:
2584:
2579:
2565:
2500:
2462:
2429:
2386:
2330:
2287:
2191:
2153:
2126:Proth's theorem
2068:Primality tests
2062:
2054:
2019:
1990:
1985:
1984:
1974:
1972:
1963:
1962:
1958:
1948:
1946:
1937:
1936:
1932:
1922:
1920:
1911:
1910:
1906:
1897:
1895:
1866:
1861:
1860:
1856:
1849:
1842:
1841:
1837:
1819:
1818:
1814:
1807:
1794:
1793:
1789:
1777:
1771:Pomerance, Carl
1769:
1768:
1764:
1759:
1747:
1667:
1665:Implementations
1658:
1652:
1648:
1630:
1629:, which allows
1615:
1611:
1608:
1586:by finding the
1545:
1538:
1506:from the rings
1481:Block Wiedemann
1471:is a square in
1467:
1445:is a square in
1441:
1427:
1392:lattice sieving
1286:
1279:
1149:
1138:
1132:
1129:
1118:
1105:
1101:
1094:
1059:
1058:
1024:
1023:
989:
984:
983:
972:
968:
958:
952:
911:
892:
874:
864:
851:
841:
816:
800:
795:
794:
788:complex numbers
757:
756:
736:
735:
675:
674:
664:
609:
564:
563:
557: = â1
553:
546:
542:
538:
523:
513:
507:
503:
499:
493:
490:) = 0
482:
478:
474:
453:
452:
448:
444:
441:
435:
419:
415:
414:for a constant
409:
405:
398:
394:
383:
379:
373:
365:
362:quadratic sieve
321:
320:
297:
268:
264:
254:
252:
244:
243:
203:
199:
198:
169:
165:
164:
126:
111:
107:
106:
102:
86:
85:
76:
75:
71:
70:(consisting of
67:
55:
28:
23:
22:
15:
12:
11:
5:
2609:
2607:
2599:
2598:
2588:
2587:
2581:
2580:
2578:
2577:
2570:
2567:
2566:
2564:
2563:
2558:
2553:
2548:
2543:
2529:
2524:
2519:
2514:
2508:
2506:
2502:
2501:
2499:
2498:
2493:
2488:
2486:TonelliâShanks
2483:
2478:
2472:
2470:
2464:
2463:
2461:
2460:
2455:
2450:
2445:
2439:
2437:
2431:
2430:
2428:
2427:
2422:
2420:Index calculus
2417:
2415:PohligâHellman
2412:
2407:
2402:
2396:
2394:
2388:
2387:
2385:
2384:
2379:
2374:
2369:
2367:Newton-Raphson
2364:
2359:
2354:
2349:
2343:
2341:
2332:
2331:
2329:
2328:
2323:
2318:
2313:
2308:
2303:
2297:
2295:
2293:Multiplication
2289:
2288:
2286:
2285:
2280:
2278:Trial division
2275:
2270:
2265:
2263:Rational sieve
2260:
2253:
2248:
2243:
2235:
2227:
2222:
2217:
2212:
2207:
2201:
2199:
2193:
2192:
2190:
2189:
2184:
2179:
2174:
2169:
2167:Sieve of Atkin
2163:
2161:
2155:
2154:
2152:
2151:
2146:
2141:
2136:
2129:
2122:
2115:
2108:
2103:
2098:
2093:
2091:Elliptic curve
2088:
2083:
2078:
2072:
2070:
2064:
2063:
2055:
2053:
2052:
2045:
2038:
2030:
2024:
2023:
2018:
2017:
2006:Carl Pomerance
2002:
1991:
1989:
1986:
1983:
1982:
1971:on May 9, 2008
1965:"About NFSNET"
1956:
1930:
1904:
1854:
1835:
1812:
1805:
1787:
1761:
1760:
1758:
1755:
1754:
1753:
1746:
1743:
1742:
1741:
1736:
1730:
1725:
1723:factor by gnfs
1720:
1715:
1687:United Kingdom
1666:
1663:
1607:
1604:
1543:
1536:
1525:(the integers
1465:
1439:
1425:
1284:
1277:
1151:
1150:
1108:
1106:
1099:
1093:
1090:
1077:
1074:
1071:
1067:
1042:
1039:
1036:
1032:
1009:
1006:
1003:
999:
993:
957:with exponent
949:
948:
937:
933:
929:
924:
921:
918:
914:
910:
907:
904:
899:
895:
886:
881:
877:
871:
867:
863:
858:
854:
848:
844:
840:
837:
834:
829:
826:
823:
819:
813:
810:
807:
803:
775:
772:
769:
765:
750:
749:
734:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
678:
676:
671:
667:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
610:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
571:
535: + 1
531:) =
512:with exponent
461:
437:Main article:
434:
431:
396:
370:smooth numbers
358:rational sieve
335:
334:
318:
312:
308:
304:
300:
296:
292:
288:
285:
282:
279:
275:
271:
267:
261:
257:
253:
249:
246:
245:
241:
235:
231:
227:
222:
218:
215:
212:
209:
206:
202:
195:
191:
187:
182:
178:
175:
172:
168:
162:
158:
155:
152:
149:
146:
141:
137:
133:
129:
125:
121:
117:
114:
110:
105:
101:
98:
95:
93:
73:
42:) is the most
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2608:
2597:
2594:
2593:
2591:
2575:
2572:
2571:
2568:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2541:
2537:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2513:
2510:
2509:
2507:
2503:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2481:Pocklington's
2479:
2477:
2474:
2473:
2471:
2469:
2465:
2459:
2456:
2454:
2451:
2449:
2446:
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2436:
2432:
2426:
2423:
2421:
2418:
2416:
2413:
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2408:
2406:
2403:
2401:
2398:
2397:
2395:
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2389:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
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2337:
2333:
2327:
2324:
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2319:
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2314:
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2309:
2307:
2304:
2302:
2299:
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2290:
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2279:
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2274:
2271:
2269:
2266:
2264:
2261:
2259:
2258:
2254:
2252:
2249:
2247:
2244:
2242:
2240:
2236:
2234:
2232:
2228:
2226:
2225:Pollard's rho
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2202:
2200:
2198:
2194:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2164:
2162:
2160:
2156:
2150:
2147:
2145:
2142:
2140:
2137:
2135:
2134:
2130:
2128:
2127:
2123:
2121:
2120:
2116:
2114:
2113:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2073:
2071:
2069:
2065:
2061:
2058:
2051:
2046:
2044:
2039:
2037:
2032:
2031:
2028:
2021:
2020:
2015:
2014:0-387-25282-7
2011:
2007:
2003:
2000:
1996:
1993:
1992:
1987:
1970:
1966:
1960:
1957:
1944:
1940:
1934:
1931:
1919:
1915:
1908:
1905:
1893:
1888:
1884:
1880:
1876:
1872:
1865:
1858:
1855:
1848:
1847:
1839:
1836:
1831:
1827:
1823:
1816:
1813:
1808:
1802:
1798:
1791:
1788:
1783:
1776:
1772:
1766:
1763:
1756:
1752:
1749:
1748:
1744:
1740:
1737:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1710:
1709:
1707:
1702:
1699:
1695:
1690:
1688:
1684:
1680:
1676:
1672:
1664:
1662:
1655:
1645:
1642: +
1641:
1637:
1633:
1628:
1623:
1619:
1605:
1603:
1601:
1598: â
1597:
1593:
1589:
1585:
1581:
1578:divisible by
1577:
1574: â
1573:
1569:
1565:
1561:
1557:
1554: â
1553:
1549:
1542:
1535:
1532:), which map
1531:
1530:
1524:
1521:
1517:
1513:
1509:
1505:
1504:homomorphisms
1501:
1497:
1493:
1489:
1484:
1482:
1478:
1477:Block Lanczos
1474:
1470:
1464:
1461: â
1460:
1456:
1452:
1448:
1444:
1438:
1435: â
1434:
1430:
1424:
1421: â
1420:
1417:is a norm of
1416:
1412:
1408:
1404:
1400:
1395:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1366:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1331:. Similarly,
1330:
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1299:is of degree
1298:
1294:
1290:
1283:
1276:
1272:
1268:
1263:
1261:
1258: â
1257:
1253:
1249:
1245:
1241:
1237:
1233:
1229:
1225:
1221:
1220:
1214:
1210:
1206:
1203:
1199:
1198:
1192:
1188:
1184:
1180:
1177:
1173:
1169:
1165:
1161:
1158:
1147:
1144:
1136:
1126:
1125:the talk page
1122:
1116:
1114:
1109:This section
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1004:
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978:
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961:
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908:
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893:
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861:
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838:
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832:
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824:
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817:
811:
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801:
793:
792:
791:
789:
770:
755:
732:
729:
723:
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690:
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496:
489:
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440:
433:Number fields
432:
430:
428:
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402:
391:
389:
388:number fields
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123:
119:
115:
108:
103:
99:
96:
84:
83:
82:
65:
61:
60:Heuristically
53:
49:
45:
41:
37:
33:
32:number theory
19:
2573:
2255:
2250:
2238:
2230:
2149:MillerâRabin
2131:
2124:
2117:
2112:LucasâLehmer
2110:
1973:. Retrieved
1969:the original
1959:
1947:. Retrieved
1943:the original
1933:
1921:. Retrieved
1917:
1907:
1896:. Retrieved
1874:
1870:
1857:
1845:
1838:
1829:
1825:
1815:
1796:
1790:
1781:
1765:
1703:
1691:
1683:Paul Leyland
1668:
1653:
1643:
1639:
1635:
1631:
1624:
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1609:
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1591:
1583:
1579:
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1567:
1563:
1559:
1555:
1551:
1547:
1540:
1533:
1528:
1522:
1519:
1515:
1514:to the ring
1511:
1507:
1502:, there are
1499:
1495:
1491:
1487:
1485:
1472:
1468:
1462:
1458:
1450:
1446:
1442:
1436:
1432:
1428:
1422:
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1414:
1406:
1402:
1396:
1387:
1383:
1379:
1375:
1371:
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1281:
1274:
1270:
1266:
1264:
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1218:
1212:
1208:
1204:
1196:
1182:
1178:
1171:
1167:
1163:
1159:
1154:
1139:
1130:
1119:Please help
1110:
963:
959:
953:
950:
751:
554:
547:
532:
528:
524:
518:
514:
508:
494:
487:
483:
442:
439:Number field
426:
421:
410:
400:
392:
374:
355:
351:prime powers
336:
54:larger than
39:
35:
29:
2405:Pollard rho
2362:Goldschmidt
2096:Pocklington
2086:BaillieâPSW
1483:are used.
1238:, and pick
1187:irreducible
1174:) of small
1157:polynomials
343:L-notations
2517:Cornacchia
2512:Chakravala
2060:algorithms
1988:References
1898:2007-12-13
1115:to readers
790:given by:
502:less than
64:complexity
50:known for
46:classical
2491:Berlekamp
2448:Euclidean
2336:Euclidean
2316:ToomâCook
2311:Karatsuba
1975:August 9,
1949:August 9,
1923:August 9,
1832:: 199â213
1234:of order
1191:rationals
1189:over the
928:∈
920:−
906:…
836:⋯
825:−
809:−
694:−
420:log
214:
208:
174:
100:
48:algorithm
44:efficient
2590:Category
2458:Lehmer's
2352:Chunking
2339:division
2268:Fermat's
1745:See also
1728:CADO-NFS
1713:NFS@Home
1679:Internet
1614:in base
1295:. Since
1273:, where
1133:May 2021
481:. Then,
443:Suppose
2574:Italics
2496:Kunerth
2476:Cipolla
2357:Fourier
2326:FĂŒrer's
2220:Euler's
2210:Dixon's
2133:PĂ©pin's
1879:Bibcode
1685:of the
1570:, with
1527:modulo
1176:degrees
1111:may be
545:, then
2556:Schoof
2443:Binary
2347:Binary
2283:Shor's
2101:Fermat
2012:
1803:
1739:kmGNFS
1733:msieve
1486:Since
1368:smooth
1166:) and
1092:Method
975:is an
399:
62:, its
34:, the
2377:Short
2106:Lucas
1867:(PDF)
1850:(PDF)
1778:(PDF)
1757:Notes
1718:GGNFS
1411:norms
1254:) as
1217:base
552:, or
447:is a
79:â + 1
2372:Long
2306:Long
2010:ISBN
1997:and
1977:2011
1951:2011
1925:2011
1801:ISBN
1638:) =
1594:and
1566:and
1558:mod
1539:and
1510:and
1498:mod
1494:and
1382:and
1374:and
1363:and
1355:and
1307:and
1291:and
1280:and
1269:and
1226:and
1202:root
1195:mod
1181:and
1155:Two
537:and
341:and
72:âlog
40:GNFS
2536:LLL
2382:SRT
2241:+ 1
2233:â 1
2081:APR
2076:AKS
1887:doi
1706:GPL
1694:CWI
1590:of
1546:to
1479:or
1215:in
395:log
360:or
337:in
211:log
205:log
171:log
97:exp
30:In
2592::
2540:KZ
2538:;
1916:.
1885:.
1875:75
1873:.
1869:.
1830:20
1828:,
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1780:.
1681:.
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1602:.
1556:mb
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1262:.
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1881::
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