3857:
3508:
1022:. We do this in a vector format; for example, the prime-power factorization of 504 is 2357, it is therefore represented by the exponent vector (3,2,0,1). Multiplying two integers then corresponds to adding their exponent vectors. A number is a square when its exponent vector is even in every coordinate. For example, the vectors (3,2,0,1) + (1,0,0,1) = (4,2,0,2), so (504)(14) is a square. Searching for a square requires knowledge only of the
3852:{\displaystyle {\begin{aligned}X&\equiv {\sqrt {15347}}-124&\equiv 8-124&\equiv 3{\pmod {17}}\\&&\equiv 9-124&\equiv 4{\pmod {17}}\\X&\equiv {\sqrt {15347}}-124&\equiv 11-124&\equiv 2{\pmod {23}}\\&&\equiv 12-124&\equiv 3{\pmod {23}}\\X&\equiv {\sqrt {15347}}-124&\equiv 8-124&\equiv 0{\pmod {29}}\\&&\equiv 21-124&\equiv 13{\pmod {29}}\\\end{aligned}}}
3005:
4483:
5370:, then this cofactor must be prime. In effect, it can be added to the factor base, by sorting the list of relations into order by cofactor. If y(a) = 7*11*23*137 and y(b) = 3*5*7*137, then y(a)y(b) = 3*5*11*23 * 7 * 137. This works by reducing the threshold of entries in the sieving array above which a full factorization is performed.
5627:, a factorizer written entirely in C containing implementation of self-initialising Quadratic Sieve. The project is inspired by a William Hart's FLINT factorizer. The source released in 2022 by Michel Leonard does not rely on external library, it is capable of factoring 240-bit numbers in a minute and 300-bit numbers in 2 hours.
2794:
4676:
2447:
5567:
computer algebra package includes an implementation of the self-initialising multiple polynomial quadratic sieve implementing the large prime variant. It was adapted by Thomas
Papanikolaou and Xavier Roblot from a sieve written for the LiDIA project. The self-initialisation scheme is based on an idea
5378:
Reducing the threshold even further, and using an effective process for factoring y(x) values into products of even relatively large primes - ECM is superb for this - can find relations with most of their factors in the factor base, but with two or even three larger primes. Cycle finding then allows
42:
under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties. It was invented by
4278:
1569:
1651:
Another way to increase the chance of smoothness is by simply increasing the size of the factor base. However, it is necessary to find at least one smooth relation more than the number of primes in the factor base, to ensure the existence of a linear dependency.
3000:{\displaystyle {\begin{aligned}V&={\begin{bmatrix}Y(0)&Y(1)&Y(2)&Y(3)&Y(4)&Y(5)&\cdots &Y(99)\end{bmatrix}}\\&={\begin{bmatrix}29&278&529&782&1037&1294&\cdots &34382\end{bmatrix}}\end{aligned}}}
3399:
4527:
4061:
643:
2176:
89:
to many processors, but the data processing phase requires large amounts of memory, and is difficult to parallelize efficiently over many nodes or if the processing nodes do not each have enough memory to store the whole matrix. The
5480:
was completed using QS. It was a 129-digit number, the product of two large primes, one of 64 digits and the other of 65 digits. The factor base for this factorization contained 524339 primes. The data collection phase took 5000
3151:
1915:
463:
901:
2008:
2786:
5680:
Carl
Pomerance, Analysis and Comparison of Some Integer Factoring Algorithms, in Computational Methods in Number Theory, Part I, H.W. Lenstra, Jr. and R. Tijdeman, eds., Math. Centre Tract 154, Amsterdam, 1982, pp
1623:
998:
303:
3316:
4861:
4738:
5067:
4478:{\displaystyle {\begin{aligned}29&=2^{0}\cdot 17^{0}\cdot 23^{0}\cdot 29^{1}\\782&=2^{1}\cdot 17^{1}\cdot 23^{1}\cdot 29^{0}\\22678&=2^{1}\cdot 17^{1}\cdot 23^{1}\cdot 29^{1}\\\end{aligned}}}
4923:
2078:
1813:
1748:
3496:
165:
3258:
1481:
4283:
3513:
2799:
2181:
1026:
of the numbers in the vectors, so it is sufficient to compute these vectors mod 2: (1,0,0,1) + (1,0,0,1) = (0,0,0,0). So given a set of (0,1)-vectors, we need to find a subset which adds to the
798:
694:
2601:) values that work, so the factorization process at the end doesn't have to be entirely reliable; often the processes misbehave on say 5% of inputs, requiring a small amount of extra sieving.
4789:
5581:, an implementation of the multiple polynomial quadratic sieve with support for single and double large primes, written by Jason Papadopoulos. Source code and a Windows binary are available.
1115:). They are harder to find, but using only smooth numbers keeps the vectors and matrices smaller and more tractable. The quadratic sieve searches for smooth numbers using a technique called
5553:, or can be downloaded in source form. SIMPQS is optimized for use on Athlon and Opteron machines, but will operate on most common 32- and 64-bit architectures. It is written entirely in C.
5541:
is a fast implementation of the self-initialising multiple polynomial quadratic sieve written by
William Hart. It provides support for the large prime variant and uses Jason Papadopoulos'
1334:
5274:
3056:
1073:
6344:
4263:
3444:
3327:
4671:{\displaystyle S\cdot {\begin{bmatrix}0&0&0&1\\1&1&1&0\\1&1&1&1\end{bmatrix}}\equiv {\begin{bmatrix}0&0&0&0\end{bmatrix}}{\pmod {2}}}
2582:) has been lost, but it has only small factors, and there are many good algorithms for factoring a number known to have only small factors, such as trial division by small primes,
3902:
5137:
3989:
733:
534:
5332:
5512:
The current QS factorization record is the 140-digit (463-bit) RSA-140, which was factored by
Patrick Konsor in June 2020 using approximately 6,000 core hours over 6 days.
2168:
5349:, the factor base and a collection of polynomials, and it will have no need to communicate with the central processor until it has finished sieving with its polynomials.
4519:
5179:
495:
2442:{\displaystyle {\begin{aligned}f(x)&=x^{2}-n\\f(x+kp)&=(x+kp)^{2}-n\\&=x^{2}+2xkp+(kp)^{2}-n\\&=f(x)+2xkp+(kp)^{2}\equiv f(x){\pmod {p}}\end{aligned}}}
542:
4149:
4122:
4095:
3929:
2685:
5387:
To illustrate typical parameter choices for a realistic example on a real implementation including the multiple polynomial and large prime optimizations, the tool
3191:
6348:
5199:
5088:
921:
5856:
5602:
instructions for AMD or Intel processors. It uses Jason
Papadopoulos' block Lanczos code. Source code and binaries for Windows and Linux are available.
3064:
1824:
381:
806:
4988:
5889:
1926:
2690:
171:, but when it does find one, more often than not, the congruence is nontrivial and the factorization is complete. This is roughly the basis of
5501:(massively parallel) supercomputer. This was the largest published factorization by a general-purpose algorithm, until NFS was used to factor
5482:
1575:
929:
2609:
This example will demonstrate standard quadratic sieve without logarithm optimizations or prime powers. Let the number to be factored
6404:
6213:
5849:
192:
3263:
5799:
4800:
4691:
5575:
computer algebra package. It is based on an implementation of Arjen
Lenstra from 1995, used in his "factoring by email" program.
4872:
6081:
2014:
1754:
1689:
5788:
1564:{\displaystyle y(x)=\left(\left\lceil {\sqrt {n}}\right\rceil +x\right)^{2}-n{\hbox{ (where }}x{\hbox{ is a small integer)}}}
3449:
3321:
Thus, starting at X=1 and incrementing by 2, each entry will be divisible by 2. Dividing each of those entries by 2 yields
6023:
5842:
5656:
5591:
was factored in less than 15 minutes on four cores of a 2.5 GHz Xeon 6248 CPU. All of the critical subroutines make use of
5458:
2104:
1079:
of order 2, that is we can divide by all non-zero numbers (there is only one, namely 1) when calculating modulo 2. It is a
108:
3196:
5952:
2103:, although this increases the running time for the data collection phase. Another method that has some acceptance is the
744:
6129:
651:
4749:
5758:
1015:
6076:
6038:
6013:
5927:
1270:
329:
172:
1679:
s are products of the same prime(s) outside the factor base. For example, if the factor base is {2, 3, 5, 7} and
6218:
6109:
6028:
6018:
5957:
5920:
4266:
3983:, etc., for each prime in the basis finds the smooth numbers which are products of unique primes (first powers).
3394:{\displaystyle V={\begin{bmatrix}29&139&529&391&1037&647&\cdots &17191\end{bmatrix}}}
1014:
So the problem has now been reduced to: given a set of integers, find a subset whose product is a square. By the
179:
5634:
package by Joseph Wood, provides an efficient implementation of the multiple polynomial quadratic sieve for the
5621:, an open source Java project containing basic implementation of QS. Released at February 4, 2016 by Ilya Gazman
5587:, written by Ben Buhrow, is probably the fastest available implementation of the quadratic sieve. For example,
5204:
4928:
Testing the result yields GCD(3070860 - 22678, 15347) = 103, a nontrivial factor of 15347, the other being 149.
3017:
1043:
6046:
5894:
5550:
2788:
and begin the sieving process for each prime in the basis, choosing to sieve the first 0 †X < 100 of Y(X):
2087:. Sometimes, forming a cycle from two partial relations leads directly to a congruence of squares, but rarely.
4219:
3411:
1199:
Use linear algebra to find a subset of these vectors which add to the zero vector. Multiply the corresponding
6299:
5780:
5572:
6294:
6261:
6223:
6124:
5534:
4056:{\displaystyle V={\begin{bmatrix}1&139&23&1&61&647&\cdots &17191\end{bmatrix}}}
2485:
3865:
1099:
factors, and so very long vectors and a very large matrix. The trick is to look specifically for numbers
6175:
5093:
2484:. This is finding a square root modulo a prime, for which there exist efficient algorithms, such as the
699:
500:
6340:
6330:
6289:
6065:
6059:
6033:
5904:
5454:
97:
The naive approach to finding a congruence of squares is to pick a random number, square it, divide by
35:
2083:
producing a full relation. Such a full relation (obtained by combining partial relations) is called a
6325:
6266:
375:, then finding a subset of these whose product is a square. This will yield a congruence of squares.
91:
5283:
6228:
6101:
5947:
5899:
5635:
1080:
2125:
6243:
5542:
5342:
4682:
4491:
1143:
1008:
638:{\displaystyle 32\cdot 200\equiv 41^{2}\cdot 43^{2}=(41\cdot 43)^{2}\equiv 114^{2}{\pmod {1649}}}
6134:
5142:
468:
6354:
6304:
6284:
5768:
5545:
implementation for the linear algebra stage. SIMPQS is accessible as the qsieve command in the
5494:
4931:
This demonstration should also serve to show that the quadratic sieve is only appropriate when
3502:> 2, there will be 2 resulting linear equations due to there being 2 modular square roots.
6364:
6005:
5980:
5909:
2497:
736:
60:
31:
5432:
Smooth values found: 25952 by sieving directly, 24462 by combining numbers with large primes
4127:
6359:
6251:
6233:
6208:
6170:
5914:
5615:
contains probably the fastest quadratic sieve written in Java (the successor of PSIQS 4.0).
4100:
4073:
3907:
2663:
1352:
or 1). If the factor is trivial, try again with a different linear dependency or different
1088:
1023:
82:
5465:
has exactly two prime factors of equal size), but in practice, QS is faster since it uses
8:
6369:
6335:
6256:
6160:
6119:
6114:
6091:
5995:
5808:
5457:(NFS), QS was the asymptotically fastest known general-purpose factoring algorithm. Now,
3170:
1004:
85:
and solves it to obtain a congruence of squares. The data collection phase can be easily
5719:"Useless Accomplishment: RSA-140 Factorization with Quadratic Sieve - mersenneforum.org"
5337:
This approach, called
Multiple Polynomial Quadratic Sieve (MPQS), is ideally suited for
2574:) which splits over the factor base. The information about exactly which primes divide
1132:
6200:
6147:
6144:
5985:
5934:
5884:
5718:
5703:
5338:
5184:
5073:
1360:
The remainder of this article explains details and extensions of this basic algorithm.
1038:
906:
86:
77:
phase, where it collects information that may lead to a congruence of squares; and the
63:
48:
5941:
5624:
6320:
6276:
5990:
5967:
5784:
5754:
5631:
5612:
2654:
1084:
6165:
5742:
5695:
5466:
182:. The general running time required for the quadratic sieve (to factor an integer
6155:
5828:
5538:
5470:
5435:
Final matrix size: 50294 × 50414, reduced by filtering to 35750 × 35862
3146:{\displaystyle Y(X)\equiv (X+\lceil {\sqrt {N}}\rceil )^{2}-N\equiv 0{\pmod {p}}}
1910:{\displaystyle {(21\cdot 29)^{2}\equiv 2^{1}\cdot 7^{1}\cdot 11^{2}{\pmod {91}}}}
5618:
4265:, the remainder of the algorithm follows equivalently to any other variation of
2551:
in order to recognise numbers divisible by small powers of a factor-base prime.
6185:
6086:
6071:
5975:
5876:
5746:
5691:
5661:
4936:
2547:. It is also necessary to solve the quadratic equation modulo small powers of
2100:
1150:, will control both the length of the vectors and the number of vectors needed.
1034:
458:{\displaystyle 41^{2}\equiv 32,42^{2}\equiv 115,43^{2}\equiv 200{\pmod {1649}}}
44:
5834:
5642:. For example, a 300-bit semiprime (91 digits) was factored in 1 hour and the
6398:
6180:
5865:
5560:
by Dario Alpern, that uses the quadratic sieve if certain conditions are met.
5485:, done in distributed fashion over the Internet. The data collected totaled 2
1112:
896:{\displaystyle 114^{2}-80^{2}=(114+80)\cdot (114-80)=194\cdot 34=k\cdot 1649}
363:
are hard to find. The quadratic sieve consists of computing the remainder of
102:
94:
can be used in the case of a few systems each capable of holding the matrix.
5608:, a simple Java implementation of the quadratic sieve for didactic purposes.
5379:
combining a set of relations sharing several primes into a single relation.
6190:
5646:
was factored in under 10 hours on a MacBook Air with an Apple M2 processor.
5605:
2003:{\displaystyle (58\cdot 21\cdot 29)^{2}\equiv 2^{1}\cdot 7^{1}{\pmod {91}}}
1459:. The quadratic sieve speeds up the process of finding relations by taking
1405:
1116:
1096:
1076:
5578:
5521:
5388:
2781:{\displaystyle Y(X)=(X+\lceil {\sqrt {N}}\rceil )^{2}-N=(X+124)^{2}-15347}
2657:
modulo each of these primes). These primes will be the basis for sieving.
5638:. It is written in C++ and is capable of comfortably factoring 100-digit
4971:, this growth can be reset by switching polynomials. As usual, we choose
4940:
2587:
1027:
1019:
5773:
5526:
1127:
To summarize, the basic quadratic sieve algorithm has these main steps:
69:(the integer to be factorized), which often leads to a factorization of
5557:
5506:
4952:
2120:
309:
34:
algorithm and, in practice, the second-fastest method known (after the
4935:
is large. For a number as small as 15347, this algorithm is overkill.
5868:
5639:
5392:
378:
For example, consider attempting to factor the number 1649. We have:
23:
167:. This approach finds a congruence of squares only rarely for large
5546:
5490:
5486:
4272:
Writing the exponents of the product of a subset of the equations
1920:
and multiply both sides by (11) modulo 91. 11 modulo 91 is 58, so:
1475:) will be smaller, and thus have a greater chance of being smooth.
5584:
1618:{\displaystyle y(x)\approx 2x\left\lceil {\sqrt {n}}\right\rceil }
5749:(2001). "Section 6.1: The quadratic sieve factorization method".
5643:
5596:
5588:
5564:
5502:
5477:
5402:
Sieve interval (per polynomial): 393216 (12 blocks of size 32768)
4743:
Thus the product of all three equations yields a square (mod N).
2645:
is small, only four primes are necessary. The first four primes
993:{\displaystyle 1649=\gcd(194,1649)\cdot \gcd(34,1649)=97\cdot 17}
39:
5531:
5441:
Total time (on a 1.6 GHz UltraSPARC III): 35 min 39 seconds
5498:
5366:, the remaining part of the number (the cofactor) is less than
2583:
1451:) that splits over the factor base, together with the value of
1348:. This produces a factor, although it may be a trivial factor (
1018:, any positive integer can be written uniquely as a product of
298:{\displaystyle e^{(1+o(1)){\sqrt {\ln n\ln \ln n}}}=L_{n}\left}
5461:
has the same asymptotic running time as QS (in the case where
4967:) starts to become large, resulting in poor density of smooth
3311:{\displaystyle X\equiv {\sqrt {15347}}-124\equiv 1{\pmod {2}}}
5801:
Factoring
Integers with the Self-Initializing Quadratic Sieve
4856:{\displaystyle 124^{2}\cdot 127^{2}\cdot 195^{2}=3070860^{2}}
4733:{\displaystyle S={\begin{bmatrix}1&1&1\end{bmatrix}}}
5062:{\displaystyle y(x)=(Ax+B)^{2}-n\qquad A,B\in \mathbb {Z} .}
1119:, discussed later, from which the algorithm takes its name.
5599:
5592:
4918:{\displaystyle 22678^{2}\equiv 3070860^{2}{\pmod {15347}}}
1668:) is not smooth, it may be possible to merge two of these
81:
phase, where it puts all the data it has collected into a
2503:
The sieve starts by setting every entry in a large array
2073:{\displaystyle 14^{2}\equiv 2^{1}\cdot 7^{1}{\pmod {91}}}
1808:{\displaystyle {29^{2}\equiv 2^{1}\cdot 11{\pmod {91}}}}
1743:{\displaystyle {21^{2}\equiv 7^{1}\cdot 11{\pmod {91}}}}
6385:
indicate that algorithm is for numbers of special forms
2590:, and ECM, which are usually used in some combination.
1368:
The quadratic sieve attempts to find pairs of integers
1363:
1083:
that with more vectors than each vector has entries, a
4943:
could have found a factor with much less computation.
4706:
4623:
4542:
4004:
3491:{\displaystyle X\equiv {\sqrt {15347}}-124{\pmod {p}}}
3342:
2944:
2817:
2095:
There are several ways to check for smoothness of the
1555:
1545:
5571:
A variant of the quadratic sieve is available in the
5286:
5280:
is a square or a smooth number, then only the factor
5207:
5187:
5145:
5096:
5076:
4991:
4875:
4803:
4752:
4694:
4530:
4494:
4281:
4222:
4130:
4103:
4076:
4070:
that equals 1 corresponds to a smooth number. Since
3992:
3910:
3868:
3511:
3452:
3414:
3330:
3266:
3199:
3173:
3067:
3020:
2797:
2693:
2666:
2613:= 15347, therefore the ceiling of the square root of
2179:
2128:
2017:
1929:
1827:
1757:
1692:
1578:
1484:
1273:
1046:
932:
909:
809:
747:
702:
654:
545:
503:
471:
384:
195:
160:{\displaystyle 80^{2}\equiv 441=21^{2}{\pmod {5959}}}
111:
5807:(M.A. thesis). University of Georgia. Archived from
5382:
3253:{\displaystyle (X+124)^{2}-15347\equiv 0{\pmod {2}}}
2558:
containing a value above a threshold of roughly log(
2488:. (This is where the quadratic sieve gets its name:
1252:), one by taking the square root in the integers of
1091:. However, simply squaring many random numbers mod
793:{\displaystyle 114^{2}\equiv 80^{2}{\pmod {1649}}.}
5772:
5509:factored since then have been factored using NFS.
5326:
5268:
5193:
5173:
5131:
5082:
5061:
4917:
4855:
4783:
4732:
4670:
4513:
4477:
4257:
4143:
4116:
4089:
4055:
3923:
3896:
3851:
3490:
3438:
3393:
3310:
3252:
3185:
3145:
3050:
2999:
2780:
2679:
2653:are 2, 17, 23, and 29 (in other words, 15347 is a
2441:
2162:
2072:
2002:
1909:
1807:
1742:
1617:
1563:
1328:
1067:
992:
915:
895:
792:
727:
689:{\displaystyle 41\cdot 43\equiv 114{\pmod {1649}}}
688:
637:
528:
489:
457:
297:
159:
5741:
2090:
1432:factorizes completely over the factor base. Such
1196:and generate exponent vectors mod 2 for each one.
6396:
5816:Describes many practical implementation details.
5707:. Vol. 43, no. 12. pp. 1473â1485.
5362:If, after dividing by all the factors less than
4784:{\displaystyle 29\cdot 782\cdot 22678=22678^{2}}
3010:The next step is to perform the sieve. For each
960:
939:
5864:
1655:
101:and hope the least non-negative remainder is a
5473:operations used by the elliptic curve method.
5850:
5489:. The data processing phase took 45 hours on
5201:. The polynomial y(x) can then be written as
1329:{\displaystyle (a+b)(a-b)\equiv 0{\pmod {n}}}
5753:(1st ed.). Springer. pp. 227â244.
3433:
3415:
3102:
3092:
3045:
3021:
2728:
2718:
5345:involved in the factorization can be given
4681:A solution to the equation is given by the
5857:
5843:
5751:Prime Numbers: A Computational Perspective
5269:{\displaystyle y(x)=A\cdot (Ax^{2}+2Bx+C)}
3051:{\displaystyle \lbrace 2,17,23,29\rbrace }
2621:is small, the basic polynomial is enough:
1416:such that the least absolute remainder of
1392:) satisfying a much weaker condition than
1095:produces a very large number of different
1068:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
5829:"The Quadratic Sieve Factoring Algorithm"
5690:
5549:computer algebra package, is part of the
5415:
5052:
2604:
2496:, and the sieving process works like the
1061:
1048:
178:The quadratic sieve is a modification of
73:. The algorithm works in two phases: the
5767:
5405:Smoothness bound: 1300967 (50294 primes)
4258:{\displaystyle Y\equiv Z^{2}{\pmod {N}}}
3439:{\displaystyle \lbrace 17,23,29\rbrace }
2511:, solve the quadratic equation mod
1244:from which we get two square roots of (
5821:
5797:
5476:On April 2, 1994, the factorization of
5422:Large prime bound: 128795733 (26 bits)
4946:
2523:, and then add an approximation to log(
1111:has only small prime factors (they are
6397:
5395:, producing the following parameters:
2649:for which 15347 has a square root mod
2464:generates a whole sequence of numbers
316:is the base of the natural logarithm.
5838:
2107:(ECM). In practice, a process called
332:entails a search for a single number
5657:Lenstra elliptic curve factorization
5568:from the thesis of Thomas Sosnowski.
5459:Lenstra elliptic curve factorization
5448:
5373:
4207:
3897:{\displaystyle X\equiv a{\pmod {p}}}
1364:How QS optimizes finding congruences
1144:denoting the number of prime numbers
5121:
4907:
4660:
4503:
4247:
3886:
3837:
3793:
3726:
3682:
3615:
3571:
3480:
3404:Similarly for the remaining primes
3300:
3242:
3135:
2554:At the end of the factor base, any
2427:
2062:
1992:
1898:
1796:
1731:
1683:= 91, there are partial relations:
1318:
779:
678:
627:
447:
149:
59:The algorithm attempts to set up a
13:
5515:
5357:
5334:has to be checked for smoothness.
5132:{\displaystyle B^{2}=n{\pmod {A}}}
4216:have been found with the property
2636:
1267:We now have the desired identity:
1232:We are now left with the equality
1087:always exists. It can be found by
728:{\displaystyle 32\cdot 200=80^{2}}
529:{\displaystyle 32\cdot 200=80^{2}}
14:
6416:
6066:Special number field sieve (SNFS)
6060:General number field sieve (GNFS)
5438:Nontrivial dependencies found: 15
5425:
5408:Number of factors for polynomial
5383:Parameters from realistic example
3156:to find the entries in the array
2480:), all of which are divisible by
1440:with respect to the factor base.
1206:together and give the result mod
1016:fundamental theorem of arithmetic
6405:Integer factorization algorithms
5505:, completed April 10, 1996. All
4151:equal one, this corresponds to:
3947:th value beyond that. Dividing
3498:is solved. Note that for every
2566:) will correspond to a value of
1636:. However, it also implies that
1443:The factorization of a value of
1340:with the difference (or sum) of
1122:
5798:Contini, Scott Patrick (1997).
5399:Trial factoring cutoff: 27 bits
5352:
5114:
5038:
4900:
4653:
4496:
4240:
3879:
3830:
3786:
3719:
3675:
3608:
3564:
3473:
3293:
3235:
3128:
2420:
2055:
1985:
1891:
1789:
1724:
1672:to form a full one, if the two
1311:
772:
671:
620:
440:
319:
142:
38:). It is still the fastest for
16:Integer factorization algorithm
5711:
5684:
5674:
5551:Fast Library for Number Theory
5327:{\displaystyle (Ax^{2}+2Bx+C)}
5321:
5287:
5263:
5229:
5217:
5211:
5125:
5115:
5023:
5007:
5001:
4995:
4911:
4901:
4664:
4654:
4507:
4497:
4251:
4241:
3890:
3880:
3841:
3831:
3797:
3787:
3730:
3720:
3686:
3676:
3619:
3609:
3575:
3565:
3484:
3474:
3304:
3294:
3246:
3236:
3213:
3200:
3139:
3129:
3106:
3083:
3077:
3071:
2918:
2912:
2899:
2893:
2885:
2879:
2871:
2865:
2857:
2851:
2843:
2837:
2829:
2823:
2763:
2750:
2732:
2709:
2703:
2697:
2486:Shanks–Tonelli algorithm
2431:
2421:
2416:
2410:
2395:
2385:
2364:
2358:
2330:
2320:
2267:
2251:
2241:
2226:
2193:
2187:
2138:
2132:
2091:Checking smoothness by sieving
2066:
2056:
1996:
1986:
1949:
1930:
1902:
1892:
1842:
1829:
1800:
1790:
1735:
1725:
1588:
1582:
1494:
1488:
1322:
1312:
1301:
1289:
1286:
1274:
975:
963:
954:
942:
866:
854:
848:
836:
783:
773:
682:
672:
631:
621:
597:
584:
451:
441:
222:
219:
213:
201:
153:
143:
1:
5781:American Mathematical Society
5667:
2492:is a quadratic polynomial in
497:is a square, but the product
351:, such that the remainder of
173:Fermat's factorization method
47:in 1981 as an improvement to
6024:Lenstra elliptic curve (ECM)
4951:In practice, many different
4267:Dixon's factorization method
2163:{\displaystyle f(x)=x^{2}-n}
1656:Partial relations and cycles
1463:close to the square root of
180:Dixon's factorization method
54:
7:
5650:
5453:Until the discovery of the
4514:{\displaystyle {\pmod {2}}}
2660:Now we construct our sieve
2527:) to every entry for which
2507:of bytes to zero. For each
10:
6421:
6331:Exponentiation by squaring
6014:Continued fraction (CFRAC)
5469:operations instead of the
5174:{\displaystyle B^{2}-n=AC}
2099:s. The most obvious is by
1660:Even if for some relation
1214:; similarly, multiply the
536:is a square. We also had
490:{\displaystyle 32,115,200}
36:general number field sieve
6378:
6313:
6275:
6242:
6199:
6143:
6100:
6004:
5966:
5875:
5444:Maximum memory used: 8 MB
1818:Multiply these together:
1644:times the square root of
1557: is a small integer)
1081:theorem of linear algebra
324:To factorize the integer
92:block Wiedemann algorithm
4983:, but now with the form
4979:) to be a square modulo
4866:So the algorithm found
1221:together which yields a
1157:) + 1 numbers
1153:Use sieving to locate Ï(
6244:Greatest common divisor
5769:Wagstaff, Samuel S. Jr.
3160:which are divisible by
1412:, and attempts to find
1404:). It selects a set of
1075:can be regarded as the
1009:greatest common divisor
696:. The observation that
465:. None of the integers
359:is a square. But these
186:) is conjectured to be
6355:Modular exponentiation
5696:"A Tale of Two Sieves"
5636:R programming language
5495:Telcordia Technologies
5328:
5270:
5195:
5175:
5133:
5084:
5063:
4919:
4857:
4785:
4734:
4672:
4515:
4479:
4259:
4145:
4144:{\displaystyle V_{71}}
4118:
4091:
4057:
3925:
3898:
3853:
3492:
3440:
3395:
3312:
3254:
3187:
3147:
3052:
3001:
2782:
2681:
2605:Example of basic sieve
2443:
2164:
2111:is typically used. If
2074:
2004:
1911:
1809:
1744:
1619:
1565:
1436:values are said to be
1330:
1069:
994:
923:. We can then factor
917:
897:
794:
729:
690:
639:
530:
491:
459:
299:
161:
6082:Shanks's square forms
6006:Integer factorization
5981:Sieve of Eratosthenes
5723:www.mersenneforum.org
5391:was run on a 267-bit
5329:
5271:
5196:
5176:
5134:
5085:
5064:
4920:
4858:
4786:
4735:
4673:
4516:
4480:
4260:
4212:Since smooth numbers
4146:
4119:
4117:{\displaystyle V_{3}}
4092:
4090:{\displaystyle V_{0}}
4058:
3926:
3924:{\displaystyle V_{x}}
3899:
3854:
3493:
3441:
3396:
3313:
3255:
3188:
3148:
3053:
3002:
2783:
2682:
2680:{\displaystyle V_{X}}
2498:Sieve of Eratosthenes
2444:
2165:
2105:elliptic curve method
2075:
2005:
1912:
1810:
1745:
1620:
1566:
1336:. Compute the GCD of
1331:
1070:
995:
918:
898:
795:
737:congruence of squares
730:
691:
640:
531:
492:
460:
300:
162:
61:congruence of squares
32:integer factorization
6360:Montgomery reduction
6234:Function field sieve
6209:Baby-step giant-step
6055:Quadratic sieve (QS)
5822:Other external links
5783:. pp. 195â202.
5775:The Joy of Factoring
5416:Multiple polynomials
5284:
5205:
5185:
5143:
5094:
5090:is chosen such that
5074:
4989:
4947:Multiple polynomials
4873:
4801:
4750:
4692:
4528:
4492:
4279:
4220:
4128:
4101:
4074:
3990:
3908:
3866:
3509:
3450:
3412:
3328:
3264:
3260:to get the solution
3197:
3171:
3065:
3018:
2795:
2691:
2664:
2177:
2126:
2015:
1927:
1825:
1755:
1690:
1640:grows linearly with
1632:is on the order of 2
1576:
1482:
1467:. This ensures that
1271:
1260:, and the other the
1089:Gaussian elimination
1044:
930:
907:
807:
745:
700:
652:
543:
501:
469:
382:
193:
109:
6370:Trachtenberg system
6336:Integer square root
6277:Modular square root
5996:Wheel factorization
5948:Quadratic Frobenius
5928:LucasâLehmerâRiesel
3931:being divisible by
3186:{\displaystyle p=2}
3058:solve the equation
3014:in our factor base
1388:) is a function of
1264:computed in step 4.
1005:Euclidean algorithm
6262:Extended Euclidean
6201:Discrete logarithm
6130:SchönhageâStrassen
5986:Sieve of Pritchard
5779:. Providence, RI:
5704:Notices of the AMS
5537:2020-05-06 at the
5455:number field sieve
5324:
5266:
5191:
5171:
5129:
5080:
5059:
4915:
4853:
4781:
4730:
4724:
4668:
4646:
4609:
4511:
4475:
4473:
4255:
4141:
4114:
4087:
4053:
4047:
3921:
3894:
3849:
3847:
3488:
3436:
3391:
3385:
3308:
3250:
3183:
3143:
3048:
2997:
2995:
2987:
2923:
2778:
2677:
2439:
2437:
2160:
2070:
2000:
1907:
1805:
1740:
1628:This implies that
1615:
1561:
1559:
1549:
1547: (where
1326:
1065:
1037:problem since the
990:
913:
893:
790:
725:
686:
635:
526:
487:
455:
295:
157:
6392:
6391:
5991:Sieve of Sundaram
5831:by Eric Landquist
5790:978-1-4704-1048-3
5743:Crandall, Richard
5694:(December 1996).
5625:C Quadratic Sieve
5613:java-math-library
5522:PPMPQS and PPSIQS
5449:Factoring records
5412:coefficients: 10
5374:More large primes
5194:{\displaystyle C}
5083:{\displaystyle B}
4208:Matrix processing
4205:
4204:
4201:2 âą 17 âą 23 âą 29
4190:2 âą 17 âą 23 âą 29
4179:2 âą 17 âą 23 âą 29
3753:
3642:
3531:
3464:
3278:
3100:
2726:
2655:quadratic residue
2515:to get two roots
1670:partial relations
1609:
1558:
1548:
1515:
1085:linear dependency
1007:to calculate the
916:{\displaystyle k}
903:for some integer
251:
51:'s linear sieve.
6412:
6341:Integer relation
6314:Other algorithms
6219:Pollard kangaroo
6110:Ancient Egyptian
5968:Prime-generating
5953:SolovayâStrassen
5866:Number-theoretic
5859:
5852:
5845:
5836:
5835:
5827:Reference paper
5815:
5813:
5806:
5794:
5778:
5764:
5733:
5732:
5730:
5729:
5715:
5709:
5708:
5700:
5688:
5682:
5678:
5558:factoring applet
5467:single-precision
5333:
5331:
5330:
5325:
5302:
5301:
5275:
5273:
5272:
5267:
5244:
5243:
5200:
5198:
5197:
5192:
5180:
5178:
5177:
5172:
5155:
5154:
5138:
5136:
5135:
5130:
5128:
5106:
5105:
5089:
5087:
5086:
5081:
5068:
5066:
5065:
5060:
5055:
5031:
5030:
4924:
4922:
4921:
4916:
4914:
4898:
4897:
4885:
4884:
4862:
4860:
4859:
4854:
4852:
4851:
4839:
4838:
4826:
4825:
4813:
4812:
4790:
4788:
4787:
4782:
4780:
4779:
4739:
4737:
4736:
4731:
4729:
4728:
4677:
4675:
4674:
4669:
4667:
4651:
4650:
4614:
4613:
4520:
4518:
4517:
4512:
4510:
4484:
4482:
4481:
4476:
4474:
4470:
4469:
4457:
4456:
4444:
4443:
4431:
4430:
4407:
4406:
4394:
4393:
4381:
4380:
4368:
4367:
4344:
4343:
4331:
4330:
4318:
4317:
4305:
4304:
4264:
4262:
4261:
4256:
4254:
4238:
4237:
4154:
4153:
4150:
4148:
4147:
4142:
4140:
4139:
4123:
4121:
4120:
4115:
4113:
4112:
4096:
4094:
4093:
4088:
4086:
4085:
4062:
4060:
4059:
4054:
4052:
4051:
3930:
3928:
3927:
3922:
3920:
3919:
3903:
3901:
3900:
3895:
3893:
3858:
3856:
3855:
3850:
3848:
3844:
3805:
3804:
3800:
3754:
3749:
3733:
3694:
3693:
3689:
3643:
3638:
3622:
3583:
3582:
3578:
3532:
3527:
3497:
3495:
3494:
3489:
3487:
3465:
3460:
3445:
3443:
3442:
3437:
3400:
3398:
3397:
3392:
3390:
3389:
3317:
3315:
3314:
3309:
3307:
3279:
3274:
3259:
3257:
3256:
3251:
3249:
3221:
3220:
3192:
3190:
3189:
3184:
3152:
3150:
3149:
3144:
3142:
3114:
3113:
3101:
3096:
3057:
3055:
3054:
3049:
3006:
3004:
3003:
2998:
2996:
2992:
2991:
2932:
2928:
2927:
2787:
2785:
2784:
2779:
2771:
2770:
2740:
2739:
2727:
2722:
2686:
2684:
2683:
2678:
2676:
2675:
2633:+ 124) â 15347.
2448:
2446:
2445:
2440:
2438:
2434:
2403:
2402:
2348:
2338:
2337:
2301:
2300:
2285:
2275:
2274:
2212:
2211:
2169:
2167:
2166:
2161:
2153:
2152:
2079:
2077:
2076:
2071:
2069:
2053:
2052:
2040:
2039:
2027:
2026:
2009:
2007:
2006:
2001:
1999:
1983:
1982:
1970:
1969:
1957:
1956:
1916:
1914:
1913:
1908:
1906:
1905:
1889:
1888:
1876:
1875:
1863:
1862:
1850:
1849:
1814:
1812:
1811:
1806:
1804:
1803:
1781:
1780:
1768:
1767:
1749:
1747:
1746:
1741:
1739:
1738:
1716:
1715:
1703:
1702:
1678:
1624:
1622:
1621:
1616:
1614:
1610:
1605:
1570:
1568:
1567:
1562:
1560:
1556:
1550:
1546:
1537:
1536:
1531:
1527:
1520:
1516:
1511:
1455:, is known as a
1335:
1333:
1332:
1327:
1325:
1225:-smooth square
1133:smoothness bound
1074:
1072:
1071:
1066:
1064:
1056:
1051:
999:
997:
996:
991:
922:
920:
919:
914:
902:
900:
899:
894:
832:
831:
819:
818:
799:
797:
796:
791:
786:
770:
769:
757:
756:
734:
732:
731:
726:
724:
723:
695:
693:
692:
687:
685:
644:
642:
641:
636:
634:
618:
617:
605:
604:
580:
579:
567:
566:
535:
533:
532:
527:
525:
524:
496:
494:
493:
488:
464:
462:
461:
456:
454:
432:
431:
413:
412:
394:
393:
350:
304:
302:
301:
296:
294:
290:
280:
267:
266:
254:
253:
252:
226:
166:
164:
163:
158:
156:
140:
139:
121:
120:
6420:
6419:
6415:
6414:
6413:
6411:
6410:
6409:
6395:
6394:
6393:
6388:
6374:
6309:
6271:
6238:
6195:
6139:
6096:
6000:
5962:
5935:Proth's theorem
5877:Primality tests
5871:
5863:
5824:
5819:
5811:
5804:
5791:
5761:
5747:Pomerance, Carl
5737:
5736:
5727:
5725:
5717:
5716:
5712:
5698:
5692:Pomerance, Carl
5689:
5685:
5679:
5675:
5670:
5653:
5632:RcppBigIntAlgos
5539:Wayback Machine
5518:
5516:Implementations
5471:multi-precision
5451:
5385:
5376:
5360:
5358:One large prime
5355:
5339:parallelization
5297:
5293:
5285:
5282:
5281:
5239:
5235:
5206:
5203:
5202:
5186:
5183:
5182:
5150:
5146:
5144:
5141:
5140:
5113:
5101:
5097:
5095:
5092:
5091:
5075:
5072:
5071:
5051:
5026:
5022:
4990:
4987:
4986:
4949:
4899:
4893:
4889:
4880:
4876:
4874:
4871:
4870:
4847:
4843:
4834:
4830:
4821:
4817:
4808:
4804:
4802:
4799:
4798:
4775:
4771:
4751:
4748:
4747:
4723:
4722:
4717:
4712:
4702:
4701:
4693:
4690:
4689:
4683:left null space
4652:
4645:
4644:
4639:
4634:
4629:
4619:
4618:
4608:
4607:
4602:
4597:
4592:
4586:
4585:
4580:
4575:
4570:
4564:
4563:
4558:
4553:
4548:
4538:
4537:
4529:
4526:
4525:
4495:
4493:
4490:
4489:
4472:
4471:
4465:
4461:
4452:
4448:
4439:
4435:
4426:
4422:
4415:
4409:
4408:
4402:
4398:
4389:
4385:
4376:
4372:
4363:
4359:
4352:
4346:
4345:
4339:
4335:
4326:
4322:
4313:
4309:
4300:
4296:
4289:
4282:
4280:
4277:
4276:
4239:
4233:
4229:
4221:
4218:
4217:
4210:
4135:
4131:
4129:
4126:
4125:
4108:
4104:
4102:
4099:
4098:
4081:
4077:
4075:
4072:
4071:
4046:
4045:
4040:
4035:
4030:
4025:
4020:
4015:
4010:
4000:
3999:
3991:
3988:
3987:
3915:
3911:
3909:
3906:
3905:
3878:
3867:
3864:
3863:
3846:
3845:
3829:
3819:
3802:
3801:
3785:
3775:
3761:
3748:
3741:
3735:
3734:
3718:
3708:
3691:
3690:
3674:
3664:
3650:
3637:
3630:
3624:
3623:
3607:
3597:
3580:
3579:
3563:
3553:
3539:
3526:
3519:
3512:
3510:
3507:
3506:
3472:
3459:
3451:
3448:
3447:
3413:
3410:
3409:
3384:
3383:
3378:
3373:
3368:
3363:
3358:
3353:
3348:
3338:
3337:
3329:
3326:
3325:
3292:
3273:
3265:
3262:
3261:
3234:
3216:
3212:
3198:
3195:
3194:
3172:
3169:
3168:
3127:
3109:
3105:
3095:
3066:
3063:
3062:
3019:
3016:
3015:
2994:
2993:
2986:
2985:
2980:
2975:
2970:
2965:
2960:
2955:
2950:
2940:
2939:
2930:
2929:
2922:
2921:
2907:
2902:
2888:
2874:
2860:
2846:
2832:
2813:
2812:
2805:
2798:
2796:
2793:
2792:
2766:
2762:
2735:
2731:
2721:
2692:
2689:
2688:
2671:
2667:
2665:
2662:
2661:
2639:
2637:Data collection
2607:
2593:There are many
2436:
2435:
2419:
2398:
2394:
2346:
2345:
2333:
2329:
2296:
2292:
2283:
2282:
2270:
2266:
2244:
2220:
2219:
2207:
2203:
2196:
2180:
2178:
2175:
2174:
2148:
2144:
2127:
2124:
2123:
2093:
2054:
2048:
2044:
2035:
2031:
2022:
2018:
2016:
2013:
2012:
1984:
1978:
1974:
1965:
1961:
1952:
1948:
1928:
1925:
1924:
1890:
1884:
1880:
1871:
1867:
1858:
1854:
1845:
1841:
1828:
1826:
1823:
1822:
1788:
1776:
1772:
1763:
1759:
1758:
1756:
1753:
1752:
1723:
1711:
1707:
1698:
1694:
1693:
1691:
1688:
1687:
1676:
1658:
1604:
1600:
1577:
1574:
1573:
1554:
1544:
1532:
1510:
1506:
1505:
1501:
1500:
1483:
1480:
1479:
1366:
1310:
1272:
1269:
1268:
1219:
1204:
1194:
1176:
1169:
1162:
1138:. The number Ï(
1125:
1060:
1052:
1047:
1045:
1042:
1041:
931:
928:
927:
908:
905:
904:
827:
823:
814:
810:
808:
805:
804:
771:
765:
761:
752:
748:
746:
743:
742:
719:
715:
701:
698:
697:
670:
653:
650:
649:
619:
613:
609:
600:
596:
575:
571:
562:
558:
544:
541:
540:
520:
516:
502:
499:
498:
470:
467:
466:
439:
427:
423:
408:
404:
389:
385:
383:
380:
379:
337:
330:Fermat's method
322:
312:. The constant
276:
272:
268:
262:
258:
225:
200:
196:
194:
191:
190:
141:
135:
131:
116:
112:
110:
107:
106:
105:. For example,
79:data processing
75:data collection
57:
21:quadratic sieve
17:
12:
11:
5:
6418:
6408:
6407:
6390:
6389:
6387:
6386:
6379:
6376:
6375:
6373:
6372:
6367:
6362:
6357:
6352:
6338:
6333:
6328:
6323:
6317:
6315:
6311:
6310:
6308:
6307:
6302:
6297:
6295:TonelliâShanks
6292:
6287:
6281:
6279:
6273:
6272:
6270:
6269:
6264:
6259:
6254:
6248:
6246:
6240:
6239:
6237:
6236:
6231:
6229:Index calculus
6226:
6224:PohligâHellman
6221:
6216:
6211:
6205:
6203:
6197:
6196:
6194:
6193:
6188:
6183:
6178:
6176:Newton-Raphson
6173:
6168:
6163:
6158:
6152:
6150:
6141:
6140:
6138:
6137:
6132:
6127:
6122:
6117:
6112:
6106:
6104:
6102:Multiplication
6098:
6097:
6095:
6094:
6089:
6087:Trial division
6084:
6079:
6074:
6072:Rational sieve
6069:
6062:
6057:
6052:
6044:
6036:
6031:
6026:
6021:
6016:
6010:
6008:
6002:
6001:
5999:
5998:
5993:
5988:
5983:
5978:
5976:Sieve of Atkin
5972:
5970:
5964:
5963:
5961:
5960:
5955:
5950:
5945:
5938:
5931:
5924:
5917:
5912:
5907:
5902:
5900:Elliptic curve
5897:
5892:
5887:
5881:
5879:
5873:
5872:
5862:
5861:
5854:
5847:
5839:
5833:
5832:
5823:
5820:
5818:
5817:
5814:on 2015-04-05.
5795:
5789:
5765:
5759:
5738:
5735:
5734:
5710:
5683:
5672:
5671:
5669:
5666:
5665:
5664:
5662:primality test
5659:
5652:
5649:
5648:
5647:
5628:
5622:
5616:
5609:
5603:
5582:
5576:
5569:
5561:
5554:
5529:
5524:
5517:
5514:
5450:
5447:
5446:
5445:
5442:
5439:
5436:
5433:
5430:
5420:
5406:
5403:
5400:
5384:
5381:
5375:
5372:
5359:
5356:
5354:
5351:
5323:
5320:
5317:
5314:
5311:
5308:
5305:
5300:
5296:
5292:
5289:
5265:
5262:
5259:
5256:
5253:
5250:
5247:
5242:
5238:
5234:
5231:
5228:
5225:
5222:
5219:
5216:
5213:
5210:
5190:
5170:
5167:
5164:
5161:
5158:
5153:
5149:
5127:
5124:
5120:
5117:
5112:
5109:
5104:
5100:
5079:
5058:
5054:
5050:
5047:
5044:
5041:
5037:
5034:
5029:
5025:
5021:
5018:
5015:
5012:
5009:
5006:
5003:
5000:
4997:
4994:
4948:
4945:
4937:Trial division
4926:
4925:
4913:
4910:
4906:
4903:
4896:
4892:
4888:
4883:
4879:
4864:
4863:
4850:
4846:
4842:
4837:
4833:
4829:
4824:
4820:
4816:
4811:
4807:
4792:
4791:
4778:
4774:
4770:
4767:
4764:
4761:
4758:
4755:
4741:
4740:
4727:
4721:
4718:
4716:
4713:
4711:
4708:
4707:
4705:
4700:
4697:
4679:
4678:
4666:
4663:
4659:
4656:
4649:
4643:
4640:
4638:
4635:
4633:
4630:
4628:
4625:
4624:
4622:
4617:
4612:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4587:
4584:
4581:
4579:
4576:
4574:
4571:
4569:
4566:
4565:
4562:
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4543:
4541:
4536:
4533:
4509:
4506:
4502:
4499:
4486:
4485:
4468:
4464:
4460:
4455:
4451:
4447:
4442:
4438:
4434:
4429:
4425:
4421:
4418:
4416:
4414:
4411:
4410:
4405:
4401:
4397:
4392:
4388:
4384:
4379:
4375:
4371:
4366:
4362:
4358:
4355:
4353:
4351:
4348:
4347:
4342:
4338:
4334:
4329:
4325:
4321:
4316:
4312:
4308:
4303:
4299:
4295:
4292:
4290:
4288:
4285:
4284:
4253:
4250:
4246:
4243:
4236:
4232:
4228:
4225:
4209:
4206:
4203:
4202:
4199:
4196:
4192:
4191:
4188:
4185:
4181:
4180:
4177:
4174:
4170:
4169:
4166:
4161:
4138:
4134:
4111:
4107:
4084:
4080:
4064:
4063:
4050:
4044:
4041:
4039:
4036:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4009:
4006:
4005:
4003:
3998:
3995:
3918:
3914:
3892:
3889:
3885:
3882:
3877:
3874:
3871:
3862:Each equation
3860:
3859:
3843:
3840:
3836:
3833:
3828:
3825:
3822:
3820:
3818:
3815:
3812:
3809:
3806:
3803:
3799:
3796:
3792:
3789:
3784:
3781:
3778:
3776:
3774:
3771:
3768:
3765:
3762:
3760:
3757:
3752:
3747:
3744:
3742:
3740:
3737:
3736:
3732:
3729:
3725:
3722:
3717:
3714:
3711:
3709:
3707:
3704:
3701:
3698:
3695:
3692:
3688:
3685:
3681:
3678:
3673:
3670:
3667:
3665:
3663:
3660:
3657:
3654:
3651:
3649:
3646:
3641:
3636:
3633:
3631:
3629:
3626:
3625:
3621:
3618:
3614:
3611:
3606:
3603:
3600:
3598:
3596:
3593:
3590:
3587:
3584:
3581:
3577:
3574:
3570:
3567:
3562:
3559:
3556:
3554:
3552:
3549:
3546:
3543:
3540:
3538:
3535:
3530:
3525:
3522:
3520:
3518:
3515:
3514:
3486:
3483:
3479:
3476:
3471:
3468:
3463:
3458:
3455:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3402:
3401:
3388:
3382:
3379:
3377:
3374:
3372:
3369:
3367:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3347:
3344:
3343:
3341:
3336:
3333:
3306:
3303:
3299:
3296:
3291:
3288:
3285:
3282:
3277:
3272:
3269:
3248:
3245:
3241:
3238:
3233:
3230:
3227:
3224:
3219:
3215:
3211:
3208:
3205:
3202:
3182:
3179:
3176:
3154:
3153:
3141:
3138:
3134:
3131:
3126:
3123:
3120:
3117:
3112:
3108:
3104:
3099:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3008:
3007:
2990:
2984:
2981:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2945:
2943:
2938:
2935:
2933:
2931:
2926:
2920:
2917:
2914:
2911:
2908:
2906:
2903:
2901:
2898:
2895:
2892:
2889:
2887:
2884:
2881:
2878:
2875:
2873:
2870:
2867:
2864:
2861:
2859:
2856:
2853:
2850:
2847:
2845:
2842:
2839:
2836:
2833:
2831:
2828:
2825:
2822:
2819:
2818:
2816:
2811:
2808:
2806:
2804:
2801:
2800:
2777:
2774:
2769:
2765:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2738:
2734:
2730:
2725:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2674:
2670:
2638:
2635:
2617:is 124. Since
2606:
2603:
2450:
2449:
2433:
2430:
2426:
2423:
2418:
2415:
2412:
2409:
2406:
2401:
2397:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2349:
2347:
2344:
2341:
2336:
2332:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2299:
2295:
2291:
2288:
2286:
2284:
2281:
2278:
2273:
2269:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2245:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2221:
2218:
2215:
2210:
2206:
2202:
2199:
2197:
2195:
2192:
2189:
2186:
2183:
2182:
2159:
2156:
2151:
2147:
2143:
2140:
2137:
2134:
2131:
2101:trial division
2092:
2089:
2081:
2080:
2068:
2065:
2061:
2058:
2051:
2047:
2043:
2038:
2034:
2030:
2025:
2021:
2010:
1998:
1995:
1991:
1988:
1981:
1977:
1973:
1968:
1964:
1960:
1955:
1951:
1947:
1944:
1941:
1938:
1935:
1932:
1918:
1917:
1904:
1901:
1897:
1894:
1887:
1883:
1879:
1874:
1870:
1866:
1861:
1857:
1853:
1848:
1844:
1840:
1837:
1834:
1831:
1816:
1815:
1802:
1799:
1795:
1792:
1787:
1784:
1779:
1775:
1771:
1766:
1762:
1750:
1737:
1734:
1730:
1727:
1722:
1719:
1714:
1710:
1706:
1701:
1697:
1657:
1654:
1626:
1625:
1613:
1608:
1603:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1571:
1553:
1543:
1540:
1535:
1530:
1526:
1523:
1519:
1514:
1509:
1504:
1499:
1496:
1493:
1490:
1487:
1365:
1362:
1358:
1357:
1324:
1321:
1317:
1314:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1265:
1230:
1217:
1202:
1197:
1192:
1187:
1174:
1167:
1160:
1151:
1124:
1121:
1113:smooth numbers
1063:
1059:
1055:
1050:
1035:linear algebra
1001:
1000:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
912:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
830:
826:
822:
817:
813:
801:
800:
789:
785:
782:
778:
775:
768:
764:
760:
755:
751:
722:
718:
714:
711:
708:
705:
684:
681:
677:
674:
669:
666:
663:
660:
657:
646:
645:
633:
630:
626:
623:
616:
612:
608:
603:
599:
595:
592:
589:
586:
583:
578:
574:
570:
565:
561:
557:
554:
551:
548:
523:
519:
515:
512:
509:
506:
486:
483:
480:
477:
474:
453:
450:
446:
443:
438:
435:
430:
426:
422:
419:
416:
411:
407:
403:
400:
397:
392:
388:
321:
318:
306:
305:
293:
289:
286:
283:
279:
275:
271:
265:
261:
257:
250:
247:
244:
241:
238:
235:
232:
229:
224:
221:
218:
215:
212:
209:
206:
203:
199:
155:
152:
148:
145:
138:
134:
130:
127:
124:
119:
115:
103:perfect square
56:
53:
45:Carl Pomerance
15:
9:
6:
4:
3:
2:
6417:
6406:
6403:
6402:
6400:
6384:
6381:
6380:
6377:
6371:
6368:
6366:
6363:
6361:
6358:
6356:
6353:
6350:
6346:
6342:
6339:
6337:
6334:
6332:
6329:
6327:
6324:
6322:
6319:
6318:
6316:
6312:
6306:
6303:
6301:
6298:
6296:
6293:
6291:
6290:Pocklington's
6288:
6286:
6283:
6282:
6280:
6278:
6274:
6268:
6265:
6263:
6260:
6258:
6255:
6253:
6250:
6249:
6247:
6245:
6241:
6235:
6232:
6230:
6227:
6225:
6222:
6220:
6217:
6215:
6212:
6210:
6207:
6206:
6204:
6202:
6198:
6192:
6189:
6187:
6184:
6182:
6179:
6177:
6174:
6172:
6169:
6167:
6164:
6162:
6159:
6157:
6154:
6153:
6151:
6149:
6146:
6142:
6136:
6133:
6131:
6128:
6126:
6123:
6121:
6118:
6116:
6113:
6111:
6108:
6107:
6105:
6103:
6099:
6093:
6090:
6088:
6085:
6083:
6080:
6078:
6075:
6073:
6070:
6068:
6067:
6063:
6061:
6058:
6056:
6053:
6051:
6049:
6045:
6043:
6041:
6037:
6035:
6034:Pollard's rho
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6011:
6009:
6007:
6003:
5997:
5994:
5992:
5989:
5987:
5984:
5982:
5979:
5977:
5974:
5973:
5971:
5969:
5965:
5959:
5956:
5954:
5951:
5949:
5946:
5944:
5943:
5939:
5937:
5936:
5932:
5930:
5929:
5925:
5923:
5922:
5918:
5916:
5913:
5911:
5908:
5906:
5903:
5901:
5898:
5896:
5893:
5891:
5888:
5886:
5883:
5882:
5880:
5878:
5874:
5870:
5867:
5860:
5855:
5853:
5848:
5846:
5841:
5840:
5837:
5830:
5826:
5825:
5810:
5803:
5802:
5796:
5792:
5786:
5782:
5777:
5776:
5770:
5766:
5762:
5760:0-387-94777-9
5756:
5752:
5748:
5744:
5740:
5739:
5724:
5720:
5714:
5706:
5705:
5697:
5693:
5687:
5677:
5673:
5663:
5660:
5658:
5655:
5654:
5645:
5641:
5637:
5633:
5629:
5626:
5623:
5620:
5617:
5614:
5610:
5607:
5604:
5601:
5598:
5594:
5590:
5586:
5583:
5580:
5577:
5574:
5570:
5566:
5562:
5559:
5555:
5552:
5548:
5544:
5543:block Lanczos
5540:
5536:
5533:
5530:
5528:
5525:
5523:
5520:
5519:
5513:
5510:
5508:
5504:
5500:
5496:
5492:
5488:
5484:
5479:
5474:
5472:
5468:
5464:
5460:
5456:
5443:
5440:
5437:
5434:
5431:
5429:
5427:
5421:
5419:
5417:
5411:
5407:
5404:
5401:
5398:
5397:
5396:
5394:
5390:
5380:
5371:
5369:
5365:
5350:
5348:
5344:
5341:, since each
5340:
5335:
5318:
5315:
5312:
5309:
5306:
5303:
5298:
5294:
5290:
5279:
5260:
5257:
5254:
5251:
5248:
5245:
5240:
5236:
5232:
5226:
5223:
5220:
5214:
5208:
5188:
5168:
5165:
5162:
5159:
5156:
5151:
5147:
5122:
5118:
5110:
5107:
5102:
5098:
5077:
5069:
5056:
5048:
5045:
5042:
5039:
5035:
5032:
5027:
5019:
5016:
5013:
5010:
5004:
4998:
4992:
4984:
4982:
4978:
4974:
4970:
4966:
4962:
4959:so that when
4958:
4955:are used for
4954:
4944:
4942:
4938:
4934:
4929:
4908:
4904:
4894:
4890:
4886:
4881:
4877:
4869:
4868:
4867:
4848:
4844:
4840:
4835:
4831:
4827:
4822:
4818:
4814:
4809:
4805:
4797:
4796:
4795:
4776:
4772:
4768:
4765:
4762:
4759:
4756:
4753:
4746:
4745:
4744:
4725:
4719:
4714:
4709:
4703:
4698:
4695:
4688:
4687:
4686:
4684:
4661:
4657:
4647:
4641:
4636:
4631:
4626:
4620:
4615:
4610:
4604:
4599:
4594:
4589:
4582:
4577:
4572:
4567:
4560:
4555:
4550:
4545:
4539:
4534:
4531:
4524:
4523:
4522:
4504:
4500:
4466:
4462:
4458:
4453:
4449:
4445:
4440:
4436:
4432:
4427:
4423:
4419:
4417:
4412:
4403:
4399:
4395:
4390:
4386:
4382:
4377:
4373:
4369:
4364:
4360:
4356:
4354:
4349:
4340:
4336:
4332:
4327:
4323:
4319:
4314:
4310:
4306:
4301:
4297:
4293:
4291:
4286:
4275:
4274:
4273:
4270:
4268:
4248:
4244:
4234:
4230:
4226:
4223:
4215:
4200:
4197:
4194:
4193:
4189:
4186:
4183:
4182:
4178:
4175:
4172:
4171:
4167:
4165:
4162:
4159:
4156:
4155:
4152:
4136:
4132:
4109:
4105:
4082:
4078:
4069:
4066:Any entry of
4048:
4042:
4037:
4032:
4027:
4022:
4017:
4012:
4007:
4001:
3996:
3993:
3986:
3985:
3984:
3982:
3978:
3974:
3970:
3966:
3962:
3958:
3954:
3950:
3946:
3942:
3938:
3934:
3916:
3912:
3887:
3883:
3875:
3872:
3869:
3838:
3834:
3826:
3823:
3821:
3816:
3813:
3810:
3807:
3794:
3790:
3782:
3779:
3777:
3772:
3769:
3766:
3763:
3758:
3755:
3750:
3745:
3743:
3738:
3727:
3723:
3715:
3712:
3710:
3705:
3702:
3699:
3696:
3683:
3679:
3671:
3668:
3666:
3661:
3658:
3655:
3652:
3647:
3644:
3639:
3634:
3632:
3627:
3616:
3612:
3604:
3601:
3599:
3594:
3591:
3588:
3585:
3572:
3568:
3560:
3557:
3555:
3550:
3547:
3544:
3541:
3536:
3533:
3528:
3523:
3521:
3516:
3505:
3504:
3503:
3501:
3481:
3477:
3469:
3466:
3461:
3456:
3453:
3430:
3427:
3424:
3421:
3418:
3407:
3386:
3380:
3375:
3370:
3365:
3360:
3355:
3350:
3345:
3339:
3334:
3331:
3324:
3323:
3322:
3319:
3301:
3297:
3289:
3286:
3283:
3280:
3275:
3270:
3267:
3243:
3239:
3231:
3228:
3225:
3222:
3217:
3209:
3206:
3203:
3180:
3177:
3174:
3165:
3163:
3159:
3136:
3132:
3124:
3121:
3118:
3115:
3110:
3097:
3089:
3086:
3080:
3074:
3068:
3061:
3060:
3059:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3013:
2988:
2982:
2977:
2972:
2967:
2962:
2957:
2952:
2947:
2941:
2936:
2934:
2924:
2915:
2909:
2904:
2896:
2890:
2882:
2876:
2868:
2862:
2854:
2848:
2840:
2834:
2826:
2820:
2814:
2809:
2807:
2802:
2791:
2790:
2789:
2775:
2772:
2767:
2759:
2756:
2753:
2747:
2744:
2741:
2736:
2723:
2715:
2712:
2706:
2700:
2694:
2672:
2668:
2658:
2656:
2652:
2648:
2644:
2634:
2632:
2628:
2624:
2620:
2616:
2612:
2602:
2600:
2596:
2591:
2589:
2585:
2581:
2577:
2573:
2569:
2565:
2561:
2557:
2552:
2550:
2546:
2542:
2539:... that is,
2538:
2534:
2530:
2526:
2522:
2518:
2514:
2510:
2506:
2501:
2499:
2495:
2491:
2487:
2483:
2479:
2475:
2471:
2467:
2463:
2459:
2455:
2452:Thus solving
2428:
2424:
2413:
2407:
2404:
2399:
2391:
2388:
2382:
2379:
2376:
2373:
2370:
2367:
2361:
2355:
2352:
2350:
2342:
2339:
2334:
2326:
2323:
2317:
2314:
2311:
2308:
2305:
2302:
2297:
2293:
2289:
2287:
2279:
2276:
2271:
2263:
2260:
2257:
2254:
2248:
2246:
2238:
2235:
2232:
2229:
2223:
2216:
2213:
2208:
2204:
2200:
2198:
2190:
2184:
2173:
2172:
2171:
2157:
2154:
2149:
2145:
2141:
2135:
2129:
2122:
2118:
2114:
2110:
2106:
2102:
2098:
2088:
2086:
2063:
2059:
2049:
2045:
2041:
2036:
2032:
2028:
2023:
2019:
2011:
1993:
1989:
1979:
1975:
1971:
1966:
1962:
1958:
1953:
1945:
1942:
1939:
1936:
1933:
1923:
1922:
1921:
1899:
1895:
1885:
1881:
1877:
1872:
1868:
1864:
1859:
1855:
1851:
1846:
1838:
1835:
1832:
1821:
1820:
1819:
1797:
1793:
1785:
1782:
1777:
1773:
1769:
1764:
1760:
1751:
1732:
1728:
1720:
1717:
1712:
1708:
1704:
1699:
1695:
1686:
1685:
1684:
1682:
1675:
1671:
1667:
1663:
1653:
1649:
1647:
1643:
1639:
1635:
1631:
1611:
1606:
1601:
1597:
1594:
1591:
1585:
1579:
1572:
1551:
1541:
1538:
1533:
1528:
1524:
1521:
1517:
1512:
1507:
1502:
1497:
1491:
1485:
1478:
1477:
1476:
1474:
1470:
1466:
1462:
1458:
1454:
1450:
1446:
1441:
1439:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1361:
1355:
1351:
1347:
1343:
1339:
1319:
1315:
1307:
1304:
1298:
1295:
1292:
1283:
1280:
1277:
1266:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1228:
1224:
1220:
1213:
1209:
1205:
1198:
1195:
1188:
1185:
1181:
1177:
1170:
1163:
1156:
1152:
1149:
1145:
1141:
1137:
1134:
1130:
1129:
1128:
1123:The algorithm
1120:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1057:
1053:
1040:
1036:
1031:
1029:
1025:
1021:
1017:
1012:
1010:
1006:
987:
984:
981:
978:
972:
969:
966:
957:
951:
948:
945:
936:
933:
926:
925:
924:
910:
890:
887:
884:
881:
878:
875:
872:
869:
863:
860:
857:
851:
845:
842:
839:
833:
828:
824:
820:
815:
811:
787:
780:
776:
766:
762:
758:
753:
749:
741:
740:
739:
738:
735:thus gives a
720:
716:
712:
709:
706:
703:
679:
675:
667:
664:
661:
658:
655:
628:
624:
614:
610:
606:
601:
593:
590:
587:
581:
576:
572:
568:
563:
559:
555:
552:
549:
546:
539:
538:
537:
521:
517:
513:
510:
507:
504:
484:
481:
478:
475:
472:
448:
444:
436:
433:
428:
424:
420:
417:
414:
409:
405:
401:
398:
395:
390:
386:
376:
374:
370:
366:
362:
358:
354:
348:
344:
340:
335:
331:
327:
317:
315:
311:
291:
287:
284:
281:
277:
273:
269:
263:
259:
255:
248:
245:
242:
239:
236:
233:
230:
227:
216:
210:
207:
204:
197:
189:
188:
187:
185:
181:
176:
174:
170:
150:
146:
136:
132:
128:
125:
122:
117:
113:
104:
100:
95:
93:
88:
84:
80:
76:
72:
68:
65:
62:
52:
50:
46:
41:
37:
33:
29:
25:
22:
6382:
6064:
6054:
6047:
6039:
5958:MillerâRabin
5940:
5933:
5926:
5921:LucasâLehmer
5919:
5809:the original
5800:
5774:
5750:
5726:. Retrieved
5722:
5713:
5702:
5686:
5676:
5511:
5475:
5462:
5452:
5426:Large primes
5423:
5413:
5409:
5386:
5377:
5367:
5363:
5361:
5353:Large primes
5346:
5336:
5277:
5070:
4985:
4980:
4976:
4972:
4968:
4964:
4960:
4956:
4950:
4932:
4930:
4927:
4865:
4793:
4742:
4680:
4487:
4271:
4213:
4211:
4163:
4157:
4067:
4065:
3980:
3976:
3972:
3968:
3964:
3960:
3956:
3952:
3948:
3944:
3940:
3936:
3932:
3861:
3499:
3446:the equation
3405:
3403:
3320:
3166:
3161:
3157:
3155:
3011:
3009:
2659:
2650:
2646:
2642:
2640:
2630:
2626:
2622:
2618:
2614:
2610:
2608:
2598:
2594:
2592:
2579:
2575:
2571:
2567:
2563:
2559:
2555:
2553:
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2520:
2516:
2512:
2508:
2504:
2502:
2493:
2489:
2481:
2477:
2473:
2469:
2465:
2461:
2457:
2453:
2451:
2116:
2112:
2108:
2096:
2094:
2084:
2082:
1919:
1817:
1680:
1673:
1669:
1665:
1661:
1659:
1650:
1645:
1641:
1637:
1633:
1629:
1627:
1472:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1442:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1401:
1397:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1367:
1359:
1353:
1349:
1345:
1341:
1337:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1233:
1226:
1222:
1215:
1211:
1207:
1200:
1190:
1183:
1179:
1172:
1165:
1158:
1154:
1147:
1139:
1135:
1126:
1108:
1104:
1100:
1092:
1077:Galois field
1032:
1020:prime powers
1013:
1002:
802:
647:
377:
372:
371:for several
368:
364:
360:
356:
352:
346:
342:
338:
333:
325:
323:
320:The approach
313:
307:
183:
177:
168:
98:
96:
87:parallelized
78:
74:
70:
66:
58:
27:
20:
18:
6214:Pollard rho
6171:Goldschmidt
5905:Pocklington
5895:BaillieâPSW
5507:RSA numbers
4953:polynomials
4941:Pollard rho
4488:as a matrix
3904:results in
2588:Pollard rho
1410:factor base
1408:called the
1189:Factor the
1028:zero vector
355:divided by
6326:Cornacchia
6321:Chakravala
5869:algorithms
5728:2020-07-07
5668:References
5640:semiprimes
5483:MIPS-years
4685:, simply
2535:) = 0 mod
2468:for which
2121:polynomial
1164:such that
1146:less than
1103:such that
1033:This is a
1003:using the
310:L-notation
49:Schroeppel
6300:Berlekamp
6257:Euclidean
6145:Euclidean
6125:ToomâCook
6120:Karatsuba
5393:semiprime
5343:processor
5227:⋅
5181:for some
5157:−
5049:∈
5033:−
4887:≡
4828:⋅
4815:⋅
4763:⋅
4757:⋅
4616:≡
4535:⋅
4459:⋅
4446:⋅
4433:⋅
4396:⋅
4383:⋅
4370:⋅
4333:⋅
4320:⋅
4307:⋅
4227:≡
4038:⋯
3943:and each
3873:≡
3824:≡
3814:−
3808:≡
3780:≡
3770:−
3764:≡
3756:−
3746:≡
3713:≡
3703:−
3697:≡
3669:≡
3659:−
3653:≡
3645:−
3635:≡
3602:≡
3592:−
3586:≡
3558:≡
3548:−
3542:≡
3534:−
3524:≡
3467:−
3457:≡
3376:⋯
3287:≡
3281:−
3271:≡
3229:≡
3223:−
3122:≡
3116:−
3103:⌉
3093:⌈
3081:≡
2978:⋯
2905:⋯
2773:−
2742:−
2729:⌉
2719:⌈
2456:⥠0 (mod
2405:≡
2340:−
2277:−
2214:−
2155:−
2119:) is the
2042:⋅
2029:≡
1972:⋅
1959:≡
1943:⋅
1937:⋅
1878:⋅
1865:⋅
1852:≡
1836:⋅
1783:⋅
1770:≡
1718:⋅
1705:≡
1592:≈
1539:−
1380:) (where
1305:≡
1296:−
1210:the name
1131:Choose a
985:⋅
958:⋅
888:⋅
876:⋅
861:−
852:⋅
821:−
759:≡
707:⋅
665:≡
659:⋅
607:≡
591:⋅
569:⋅
556:≡
550:⋅
508:⋅
434:≡
415:≡
396:≡
246:
240:
231:
123:≡
55:Basic aim
24:algorithm
6399:Category
6267:Lehmer's
6161:Chunking
6148:division
6077:Fermat's
5771:(2013).
5651:See also
5547:SageMath
5535:Archived
5493:'s (now
5491:Bellcore
4521:yields:
4168:factors
2170:we have
1612:⌉
1602:⌈
1518:⌉
1508:⌈
1457:relation
1186:-smooth.
1030:mod 2.
308:in the
40:integers
30:) is an
6383:Italics
6305:Kunerth
6285:Cipolla
6166:Fourier
6135:FĂŒrer's
6029:Euler's
6019:Dixon's
5942:PĂ©pin's
5681:89-139.
5644:RSA-100
5619:Java QS
5597:AVX-512
5589:RSA-100
5565:PARI/GP
5503:RSA-130
5478:RSA-129
4891:3070860
4845:3070860
2109:sieving
1256:namely
1117:sieving
6365:Schoof
6252:Binary
6156:Binary
6092:Shor's
5910:Fermat
5787:
5757:
5579:msieve
5532:SIMPQS
5499:MasPar
5428:above)
5418:above)
5389:msieve
4198:22678
4124:, and
3193:solve
2641:Since
2584:SQUFOF
2460:) for
1438:smooth
1406:primes
1024:parity
803:Hence
648:since
83:matrix
64:modulo
6186:Short
5915:Lucas
5812:(PDF)
5805:(PDF)
5699:(PDF)
5606:Ariel
5573:MAGMA
5424:(see
5414:(see
5276:. If
5139:, so
4909:15347
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4413:22678
4160:+ 124
4043:17191
3751:15347
3640:15347
3529:15347
3462:15347
3381:17191
3276:15347
3226:15347
2983:34382
2776:15347
2629:) = (
2085:cycle
1677:'
1400:(mod
1182:) is
1097:prime
345:<
341:<
6181:Long
6115:Long
5785:ISBN
5755:ISBN
5630:The
5611:The
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5593:AVX2
5585:YAFU
5563:The
5527:mpqs
4794:and
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4187:782
4184:127
4173:124
3366:1037
3167:For
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2968:1037
2543:and
2519:and
2454:f(x)
1428:mod
1424:) =
1372:and
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1248:mod
1240:mod
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1107:mod
1039:ring
973:1649
952:1649
934:1649
891:1649
781:1649
680:1649
629:1649
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151:5959
19:The
6345:LLL
6191:SRT
6050:+ 1
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5119:mod
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2760:124
2687:of
2500:.)
2425:mod
2060:mod
1990:mod
1896:mod
1794:mod
1729:mod
1316:mod
1171:= (
1142:),
961:gcd
946:194
940:gcd
873:194
858:114
840:114
812:114
777:mod
750:114
710:200
676:mod
668:114
625:mod
611:114
553:200
511:200
485:200
479:115
445:mod
437:200
418:115
147:mod
126:441
6401::
6349:KZ
6347:;
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5701:.
5556:a
5497:)
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4754:29
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4400:29
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4337:29
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4137:71
4097:,
4028:61
4018:23
3979:+3
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3971:+2
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3827:13
3811:21
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3700:12
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3431:29
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3318:.
3164:.
3043:29
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2948:29
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2586:,
2064:91
2020:14
1994:91
1946:29
1940:21
1934:58
1900:91
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1761:29
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662:43
656:41
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573:43
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399:32
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5160:n
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5111:n
5108:=
5103:2
5099:B
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