1104:
334:
388:. 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
948:
749:
89:
1702:
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.
1623:
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.
1478:
1477:, 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
2537:
572:
94:
798:
2541:
1022:
567:
2049:
329:{\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}}}
1088:
1053:
786:
473:
2082:
1552:, 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
1612:
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
2597:
2406:
2042:
1710:
software written by
Kleinjung, or by msieve, and lattice sieving by GPL software written by Franke and Kleinjung; these are distributed in GGNFS.
2274:
1806:
384:(see below). The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of
2216:
2035:
2145:
2322:
351:: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from
2120:
2015:
1737:(which contains final-processing code, a polynomial selection optimized for smaller numbers and an implementation of the line sieve)
1144:
1695:
2231:
2269:
2206:
2150:
2113:
2411:
2302:
2221:
2211:
1126:
65:
2087:
2239:
2492:
1392:, and we have a better chance for them to be smooth at the same time. The current best-known approach for this search is
2487:
2416:
2317:
2454:
2368:
1865:
358:
The principle of the number field sieve (both special and general) can be understood as an improvement to the simpler
953:
The product of any two such values can be computed by taking the product as polynomials, then reducing any powers of
2533:
2523:
2482:
2258:
2226:
2097:
1752:
1672:
943:{\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} .}
348:
2518:
2459:
1482:
1966:
2421:
2294:
2140:
2092:
1122:
392:. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve.
1679:. A project called NFSNET ran from 2002 through at least 2007. It used volunteer distributed computing on the
744:{\displaystyle {\begin{aligned}(a+bi)(c+di)&=ac+(ad+bc)i+(bd)i^{2}\\&=(ac-bd)+(ad+bc)i.\end{aligned}}}
987:
2436:
2327:
1589:
2547:
2497:
2477:
1188:
1177:
978:
755:
43:
2003:(eds.). "The development of the number field sieve". Lecture Notes in Math. (1993) 1554. Springer-Verlag.
1209:. An optimal strategy for choosing these polynomials is not known; one simple method is to pick a degree
2198:
2173:
2102:
1699:
1628:
51:
2557:
1620:
shown above is suboptimal in many practical situations, leading to the development of better methods.
2552:
2444:
2426:
2401:
2363:
2107:
1880:
1400:
1940:
1846:
2562:
2528:
2449:
2353:
2312:
2307:
2284:
2188:
1114:
1694:
Until 2007, the gold-standard implementation was a suite of software developed and distributed by
1564:
can be obtained as a square in two waysâone for each homomorphism. Thus, one can find two numbers
2393:
2340:
2337:
2178:
2077:
1676:
1528:
1203:
1196:
2134:
2127:
2513:
2469:
2183:
2160:
2011:
1802:
1698:
in the
Netherlands, which was available only under a relatively restrictive license. In 2007,
1415:
of squares in our number fields, but that condition can be achieved by this method too. Each
1062:
1027:
760:
456:
969:
as described above, yielding a value in the same form. To ensure that this field is actually
2358:
1888:
1456:
1056:
982:
2348:
2247:
2000:
1393:
1192:
1059:
with integer coefficients. In some cases, this ring of integers is equivalent to the ring
363:
1884:
1823:
2378:
2279:
2264:
2168:
2069:
2007:
1772:
1688:
789:
359:
340:
1411:
to be squares at the same time. A slightly stronger condition is neededâthat they are
2591:
2373:
2058:
1996:
1671:
Some implementations focus on a certain smaller class of numbers. These are known as
1369:
371:
31:
2383:
1776:
1724:
1719:
1684:
1505:
973:-dimensional and does not collapse to an even smaller field, it is sufficient that
440:
389:
1893:
1451:, with a "square root" which can be determined (as a product of known factors in
352:
2027:
1915:
1412:
1158:
344:
2061:
2010:. Prime Numbers: A Computational Perspective (2001). 2nd edition, Springer.
61:
47:
2024:
Matthew E. Briggs: An
Introduction to the General Number Field Sieve, 1998
1734:
1680:
1396:; to get acceptable yields, it is necessary to use a large factor base.
1714:
17:
1970:
55:
1218:
1584:
and again with probability at least one half we get a factor of
2031:
1707:
1097:
1920:
Presentation at EIDMA-CWI Workshop on
Factoring Large Numbers
1866:"On polynomial selection for the general number field sieve"
1729:
1944:
1627:
The best reported results were achieved by the method of
508:. This equation can be used to reduce away any powers of
380:. The size of these values is exponential in the size of
1433:
and hence that the product of the corresponding factors
2578:
indicate that algorithm is for numbers of special forms
1740:
1653:
composed of small prime factors congruent to 1 modulo 2
1353:) is an integer. The goal is to find integer values of
1248:) as the polynomial with the smallest coefficients and
1187:
are chosen, which have integer coefficients, which are
1065:
1030:
990:
763:
459:
406:
or the number of bits in the binary representation of
366:. When using such algorithms to factor a large number
1824:"On quadratic polynomials for the number field sieve"
1119:. In particular, there are no examples or pseudocode.
801:
570:
92:
2018:. Section 6.2: Number field sieve, pp. 278â301.
1455:)âit will typically be represented as an irrational
2506:
2468:
2435:
2392:
2336:
2293:
2197:
2159:
2068:
1082:
1047:
1016:
981:over the rationals. Similarly, one may define the
942:
780:
743:
467:
328:
427:. The running time of the number field sieve is
374:(i.e. numbers with small prime factors) of order
1786:. Vol. 43, no. 12. pp. 1473â1485.
1691:and Richard Wackerbarth of Texas were involved.
561:. This allows us to define the complex product:
355:(which are trivial to factor by taking roots).
1706:Polynomial selection is normally performed by
2043:
8:
1213:for a polynomial, consider the expansion of
1125:. There might be a discussion about this on
1372:relative to the chosen basis of primes. If
2050:
2036:
2028:
1828:Australian Computer Science Communications
395:The size of the input to the algorithm is
1892:
1459:. Similarly, the product of the factors
1145:Learn how and when to remove this message
1067:
1066:
1064:
1032:
1031:
1029:
999:
998:
997:
993:
992:
989:
933:
932:
917:
898:
889:
880:
870:
857:
847:
822:
806:
800:
765:
764:
762:
670:
571:
569:
461:
460:
458:
307:
303:
291:
274:
260:
252:
230:
226:
190:
186:
136:
132:
120:
93:
91:
1017:{\textstyle \mathbb {O} _{\mathbb {Q} }}
1764:
754:In general, this leads directly to the
1090:. However, there are many exceptions.
1388:will be small too, about the size of
788:, which can be defined as the set of
500:as a linear combination of powers of
494:, which can be rearranged to express
7:
1864:Kleinjung, Thorsten (October 2006).
429:super-polynomial but sub-exponential
1914:Paul Leyland (December 12, 2003).
1403:, one can get products of certain
25:
2259:Special number field sieve (SNFS)
2253:General number field sieve (GNFS)
1822:Murphy, B.; Brent, R. P. (1998),
1659:and over leading coefficients of
2598:Integer factorization algorithms
1943:. April 23, 2007. Archived from
1675:techniques, such as used in the
1399:Having enough such pairs, using
1267:Consider the number field rings
1102:
370:, it is necessary to search for
347:. It is a generalization of the
1848:On RSA 200 and larger projects
1313:are integers, then so will be
1305:with integer coefficients, if
1195:, and which, when interpreted
1077:
1071:
1042:
1036:
1009:
1003:
775:
769:
728:
710:
704:
686:
663:
654:
645:
627:
608:
593:
590:
575:
300:
285:
158:
152:
129:
114:
1:
1894:10.1090/S0025-5718-06-01870-9
1289:are roots of the polynomials
2217:Lenstra elliptic curve (ECM)
1232:) for a number of different
552: + 1 = 0
475:(the rational numbers), and
1663:which are divisible by 60.
1608:Improving polynomial choice
410:. Any element of the order
2614:
2524:Exponentiation by squaring
2207:Continued fraction (CFRAC)
1873:Mathematics of Computation
1753:Special number field sieve
1673:special number field sieve
1224:(allowing digits between â
438:
431:in the size of the input.
349:special number field sieve
36:general number field sieve
2571:
1797:Ribenboim, Paulo (1972).
1407:and of the corresponding
1361:that simultaneously make
1083:{\textstyle \mathbb {Z} }
1048:{\textstyle \mathbb {Q} }
781:{\textstyle \mathbb {Q} }
468:{\textstyle \mathbb {Q} }
68:for factoring an integer
1916:"NFSNET: the first year"
1202:, have a common integer
453:-degree polynomial over
2437:Greatest common divisor
1590:greatest common divisor
27:Factorization algorithm
2548:Modular exponentiation
1801:. Wiley-Interscience.
1777:"A Tale of Two Sieves"
1084:
1049:
1018:
979:irreducible polynomial
944:
782:
756:algebraic number field
745:
543:is the imaginary unit
469:
330:
2275:Shanks's square forms
2199:Integer factorization
2174:Sieve of Eratosthenes
2006:Richard Crandall and
1845:Franke, Jens (2006),
1085:
1050:
1019:
945:
783:
746:
479:is a complex root of
470:
331:
83:bits) is of the form
2553:Montgomery reduction
2427:Function field sieve
2402:Baby-step giant-step
2248:Quadratic sieve (QS)
1649:, and searches over
1401:Gaussian elimination
1115:confusing or unclear
1063:
1028:
988:
799:
761:
568:
457:
90:
2563:Trachtenberg system
2529:Integer square root
2470:Modular square root
2189:Wheel factorization
2141:Quadratic Frobenius
2121:LucasâLehmerâRiesel
1947:on October 22, 2007
1941:"Welcome to NFSNET"
1885:2006MaCom..75.2037K
1123:clarify the section
1055:which are roots of
2455:Extended Euclidean
2394:Discrete logarithm
2323:SchönhageâStrassen
2179:Sieve of Pritchard
2001:H. W. Lenstra, Jr.
1879:(256): 2037â2047.
1784:Notices of the AMS
1700:Jason Papadopoulos
1677:Cunningham project
1629:Thorsten Kleinjung
1492:is a root of both
1080:
1045:
1014:
940:
778:
741:
739:
524:. For example, if
465:
420:is exponential in
326:
324:
52:factoring integers
2585:
2584:
2184:Sieve of Sundaram
1808:978-0-471-71804-8
1799:Algebraic Numbers
1775:(December 1996).
1329:), which we call
1155:
1154:
1147:
1057:monic polynomials
1024:as the subset of
892:
891: where
16:(Redirected from
2605:
2534:Integer relation
2507:Other algorithms
2412:Pollard kangaroo
2303:Ancient Egyptian
2161:Prime-generating
2146:SolovayâStrassen
2059:Number-theoretic
2052:
2045:
2038:
2029:
1997:Arjen K. Lenstra
1983:
1982:
1980:
1978:
1969:. Archived from
1963:
1957:
1956:
1954:
1952:
1937:
1931:
1930:
1928:
1926:
1911:
1905:
1904:
1902:
1901:
1896:
1870:
1861:
1855:
1854:
1853:
1842:
1836:
1835:
1819:
1813:
1812:
1794:
1788:
1787:
1781:
1769:
1662:
1658:
1652:
1648:
1619:
1615:
1457:algebraic number
1380:are small, then
1150:
1143:
1139:
1136:
1130:
1106:
1105:
1098:
1089:
1087:
1086:
1081:
1070:
1054:
1052:
1051:
1046:
1035:
1023:
1021:
1020:
1015:
1013:
1012:
1002:
996:
983:ring of integers
976:
972:
968:
958:
949:
947:
946:
941:
936:
928:
927:
903:
902:
893:
890:
885:
884:
875:
874:
862:
861:
852:
851:
833:
832:
817:
816:
787:
785:
784:
779:
768:
750:
748:
747:
742:
740:
679:
675:
674:
560:
553:
546:
542:
538:
523:
513:
507:
503:
499:
493:
482:
478:
474:
472:
471:
466:
464:
452:
448:
426:
419:
415:
409:
405:
387:
383:
379:
369:
335:
333:
332:
327:
325:
321:
317:
316:
315:
311:
295:
278:
265:
264:
253:
244:
240:
239:
238:
234:
225:
221:
199:
198:
194:
185:
181:
165:
161:
145:
144:
140:
124:
96:
82:
80:
71:
58:
21:
2613:
2612:
2608:
2607:
2606:
2604:
2603:
2602:
2588:
2587:
2586:
2581:
2567:
2502:
2464:
2431:
2388:
2332:
2289:
2193:
2155:
2128:Proth's theorem
2070:Primality tests
2064:
2056:
2021:
1992:
1987:
1986:
1976:
1974:
1965:
1964:
1960:
1950:
1948:
1939:
1938:
1934:
1924:
1922:
1913:
1912:
1908:
1899:
1897:
1868:
1863:
1862:
1858:
1851:
1844:
1843:
1839:
1821:
1820:
1816:
1809:
1796:
1795:
1791:
1779:
1773:Pomerance, Carl
1771:
1770:
1766:
1761:
1749:
1669:
1667:Implementations
1660:
1654:
1650:
1632:
1631:, which allows
1617:
1613:
1610:
1588:by finding the
1547:
1540:
1508:from the rings
1483:Block Wiedemann
1473:is a square in
1469:
1447:is a square in
1443:
1429:
1394:lattice sieving
1288:
1281:
1151:
1140:
1134:
1131:
1120:
1107:
1103:
1096:
1061:
1060:
1026:
1025:
991:
986:
985:
974:
970:
960:
954:
913:
894:
876:
866:
853:
843:
818:
802:
797:
796:
790:complex numbers
759:
758:
738:
737:
677:
676:
666:
611:
566:
565:
559: = â1
555:
548:
544:
540:
525:
515:
509:
505:
501:
495:
492:) = 0
484:
480:
476:
455:
454:
450:
446:
443:
437:
421:
417:
416:for a constant
411:
407:
400:
396:
385:
381:
375:
367:
364:quadratic sieve
323:
322:
299:
270:
266:
256:
254:
246:
245:
205:
201:
200:
171:
167:
166:
128:
113:
109:
108:
104:
88:
87:
78:
77:
73:
72:(consisting of
69:
56:
28:
23:
22:
15:
12:
11:
5:
2611:
2609:
2601:
2600:
2590:
2589:
2583:
2582:
2580:
2579:
2572:
2569:
2568:
2566:
2565:
2560:
2555:
2550:
2545:
2531:
2526:
2521:
2516:
2510:
2508:
2504:
2503:
2501:
2500:
2495:
2490:
2488:TonelliâShanks
2485:
2480:
2474:
2472:
2466:
2465:
2463:
2462:
2457:
2452:
2447:
2441:
2439:
2433:
2432:
2430:
2429:
2424:
2422:Index calculus
2419:
2417:PohligâHellman
2414:
2409:
2404:
2398:
2396:
2390:
2389:
2387:
2386:
2381:
2376:
2371:
2369:Newton-Raphson
2366:
2361:
2356:
2351:
2345:
2343:
2334:
2333:
2331:
2330:
2325:
2320:
2315:
2310:
2305:
2299:
2297:
2295:Multiplication
2291:
2290:
2288:
2287:
2282:
2280:Trial division
2277:
2272:
2267:
2265:Rational sieve
2262:
2255:
2250:
2245:
2237:
2229:
2224:
2219:
2214:
2209:
2203:
2201:
2195:
2194:
2192:
2191:
2186:
2181:
2176:
2171:
2169:Sieve of Atkin
2165:
2163:
2157:
2156:
2154:
2153:
2148:
2143:
2138:
2131:
2124:
2117:
2110:
2105:
2100:
2095:
2093:Elliptic curve
2090:
2085:
2080:
2074:
2072:
2066:
2065:
2057:
2055:
2054:
2047:
2040:
2032:
2026:
2025:
2020:
2019:
2008:Carl Pomerance
2004:
1993:
1991:
1988:
1985:
1984:
1973:on May 9, 2008
1967:"About NFSNET"
1958:
1932:
1906:
1856:
1837:
1814:
1807:
1789:
1763:
1762:
1760:
1757:
1756:
1755:
1748:
1745:
1744:
1743:
1738:
1732:
1727:
1725:factor by gnfs
1722:
1717:
1689:United Kingdom
1668:
1665:
1609:
1606:
1545:
1538:
1527:(the integers
1467:
1441:
1427:
1286:
1279:
1153:
1152:
1110:
1108:
1101:
1095:
1092:
1079:
1076:
1073:
1069:
1044:
1041:
1038:
1034:
1011:
1008:
1005:
1001:
995:
959:with exponent
951:
950:
939:
935:
931:
926:
923:
920:
916:
912:
909:
906:
901:
897:
888:
883:
879:
873:
869:
865:
860:
856:
850:
846:
842:
839:
836:
831:
828:
825:
821:
815:
812:
809:
805:
777:
774:
771:
767:
752:
751:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
688:
685:
682:
680:
678:
673:
669:
665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
617:
614:
612:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
573:
537: + 1
533:) =
514:with exponent
463:
439:Main article:
436:
433:
398:
372:smooth numbers
360:rational sieve
337:
336:
320:
314:
310:
306:
302:
298:
294:
290:
287:
284:
281:
277:
273:
269:
263:
259:
255:
251:
248:
247:
243:
237:
233:
229:
224:
220:
217:
214:
211:
208:
204:
197:
193:
189:
184:
180:
177:
174:
170:
164:
160:
157:
154:
151:
148:
143:
139:
135:
131:
127:
123:
119:
116:
112:
107:
103:
100:
97:
95:
75:
42:) is the most
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2610:
2599:
2596:
2595:
2593:
2577:
2574:
2573:
2570:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2543:
2539:
2535:
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2515:
2512:
2511:
2509:
2505:
2499:
2496:
2494:
2491:
2489:
2486:
2484:
2483:Pocklington's
2481:
2479:
2476:
2475:
2473:
2471:
2467:
2461:
2458:
2456:
2453:
2451:
2448:
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2443:
2442:
2440:
2438:
2434:
2428:
2425:
2423:
2420:
2418:
2415:
2413:
2410:
2408:
2405:
2403:
2400:
2399:
2397:
2395:
2391:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
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2350:
2347:
2346:
2344:
2342:
2339:
2335:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2300:
2298:
2296:
2292:
2286:
2283:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2260:
2256:
2254:
2251:
2249:
2246:
2244:
2242:
2238:
2236:
2234:
2230:
2228:
2227:Pollard's rho
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2204:
2202:
2200:
2196:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2170:
2167:
2166:
2164:
2162:
2158:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2136:
2132:
2130:
2129:
2125:
2123:
2122:
2118:
2116:
2115:
2111:
2109:
2106:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2086:
2084:
2081:
2079:
2076:
2075:
2073:
2071:
2067:
2063:
2060:
2053:
2048:
2046:
2041:
2039:
2034:
2033:
2030:
2023:
2022:
2017:
2016:0-387-25282-7
2013:
2009:
2005:
2002:
1998:
1995:
1994:
1989:
1972:
1968:
1962:
1959:
1946:
1942:
1936:
1933:
1921:
1917:
1910:
1907:
1895:
1890:
1886:
1882:
1878:
1874:
1867:
1860:
1857:
1850:
1849:
1841:
1838:
1833:
1829:
1825:
1818:
1815:
1810:
1804:
1800:
1793:
1790:
1785:
1778:
1774:
1768:
1765:
1758:
1754:
1751:
1750:
1746:
1742:
1739:
1736:
1733:
1731:
1728:
1726:
1723:
1721:
1718:
1716:
1713:
1712:
1711:
1709:
1704:
1701:
1697:
1692:
1690:
1686:
1682:
1678:
1674:
1666:
1664:
1657:
1647:
1644: +
1643:
1639:
1635:
1630:
1625:
1621:
1607:
1605:
1603:
1600: â
1599:
1595:
1591:
1587:
1583:
1580:divisible by
1579:
1576: â
1575:
1571:
1567:
1563:
1559:
1556: â
1555:
1551:
1544:
1537:
1534:), which map
1533:
1532:
1526:
1523:
1519:
1515:
1511:
1507:
1506:homomorphisms
1503:
1499:
1495:
1491:
1486:
1484:
1480:
1479:Block Lanczos
1476:
1472:
1466:
1463: â
1462:
1458:
1454:
1450:
1446:
1440:
1437: â
1436:
1432:
1426:
1423: â
1422:
1419:is a norm of
1418:
1414:
1410:
1406:
1402:
1397:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1333:. Similarly,
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1301:is of degree
1300:
1296:
1292:
1285:
1278:
1274:
1270:
1265:
1263:
1260: â
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1222:
1216:
1212:
1208:
1205:
1201:
1200:
1194:
1190:
1186:
1182:
1179:
1175:
1171:
1167:
1163:
1160:
1149:
1146:
1138:
1128:
1127:the talk page
1124:
1118:
1116:
1111:This section
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1100:
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1006:
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967:
963:
957:
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907:
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863:
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854:
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840:
837:
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819:
813:
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807:
803:
795:
794:
793:
791:
772:
757:
734:
731:
725:
722:
719:
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713:
707:
701:
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692:
689:
683:
681:
671:
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563:
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512:
498:
491:
487:
442:
435:Number fields
434:
432:
430:
425:
414:
404:
393:
391:
390:number fields
378:
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361:
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137:
133:
125:
121:
117:
110:
105:
101:
98:
86:
85:
84:
67:
63:
62:Heuristically
59:
53:
49:
45:
41:
37:
33:
32:number theory
19:
2575:
2257:
2252:
2240:
2232:
2151:MillerâRabin
2133:
2126:
2119:
2114:LucasâLehmer
2112:
1975:. Retrieved
1971:the original
1961:
1949:. Retrieved
1945:the original
1935:
1923:. Retrieved
1919:
1909:
1898:. Retrieved
1876:
1872:
1859:
1847:
1840:
1831:
1827:
1817:
1798:
1792:
1783:
1767:
1705:
1693:
1685:Paul Leyland
1670:
1655:
1645:
1641:
1637:
1633:
1626:
1622:
1611:
1601:
1597:
1593:
1585:
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1573:
1569:
1565:
1561:
1557:
1553:
1549:
1542:
1535:
1530:
1524:
1521:
1517:
1516:to the ring
1513:
1509:
1504:, there are
1501:
1497:
1493:
1489:
1487:
1474:
1470:
1464:
1460:
1452:
1448:
1444:
1438:
1434:
1430:
1424:
1420:
1416:
1408:
1404:
1398:
1389:
1385:
1381:
1377:
1373:
1366:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1283:
1276:
1272:
1268:
1266:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1233:
1229:
1225:
1220:
1214:
1210:
1206:
1198:
1184:
1180:
1173:
1169:
1165:
1161:
1156:
1141:
1132:
1121:Please help
1112:
965:
961:
955:
952:
753:
556:
549:
534:
530:
526:
520:
516:
510:
496:
489:
485:
444:
441:Number field
428:
423:
412:
402:
394:
376:
357:
353:prime powers
338:
54:larger than
39:
35:
29:
2407:Pollard rho
2364:Goldschmidt
2098:Pocklington
2088:BaillieâPSW
1485:are used.
1240:, and pick
1189:irreducible
1176:) of small
1159:polynomials
345:L-notations
2519:Cornacchia
2514:Chakravala
2062:algorithms
1990:References
1900:2007-12-13
1117:to readers
792:given by:
504:less than
66:complexity
50:known for
46:classical
2493:Berlekamp
2450:Euclidean
2338:Euclidean
2318:ToomâCook
2313:Karatsuba
1977:August 9,
1951:August 9,
1925:August 9,
1834:: 199â213
1236:of order
1193:rationals
1191:over the
930:∈
922:−
908:…
838:⋯
827:−
811:−
696:−
422:log
216:
210:
176:
102:
48:algorithm
44:efficient
2592:Category
2460:Lehmer's
2354:Chunking
2341:division
2270:Fermat's
1747:See also
1730:CADO-NFS
1715:NFS@Home
1681:Internet
1616:in base
1297:. Since
1275:, where
1135:May 2021
483:. Then,
445:Suppose
2576:Italics
2498:Kunerth
2478:Cipolla
2359:Fourier
2328:FĂŒrer's
2222:Euler's
2212:Dixon's
2135:PĂ©pin's
1881:Bibcode
1687:of the
1572:, with
1529:modulo
1178:degrees
1113:may be
547:, then
2558:Schoof
2445:Binary
2349:Binary
2285:Shor's
2103:Fermat
2014:
1805:
1741:kmGNFS
1735:msieve
1488:Since
1370:smooth
1168:) and
1094:Method
977:is an
401:
64:, its
34:, the
18:NFSNet
2379:Short
2108:Lucas
1869:(PDF)
1852:(PDF)
1780:(PDF)
1759:Notes
1720:GGNFS
1413:norms
1256:) as
1219:base
554:, or
449:is a
81:â + 1
2374:Long
2308:Long
2012:ISBN
1999:and
1979:2011
1953:2011
1927:2011
1803:ISBN
1640:) =
1596:and
1568:and
1560:mod
1541:and
1512:and
1500:mod
1496:and
1384:and
1376:and
1365:and
1357:and
1309:and
1293:and
1282:and
1271:and
1228:and
1204:root
1197:mod
1183:and
1157:Two
539:and
343:and
74:âlog
40:GNFS
2538:LLL
2384:SRT
2243:+ 1
2235:â 1
2083:APR
2078:AKS
1889:doi
1708:GPL
1696:CWI
1592:of
1548:to
1481:or
1217:in
397:log
362:or
339:in
213:log
207:log
173:log
99:exp
30:In
2594::
2542:KZ
2540:;
1918:.
1887:.
1877:75
1875:.
1871:.
1832:20
1830:,
1826:,
1782:.
1683:.
1642:ax
1604:.
1558:mb
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1264:.
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2233:p
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1981:.
1955:.
1929:.
1903:.
1891::
1883::
1811:.
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1656:d
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1520:/
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