A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Aw = 0, w 6= 0. Throughout the paper A will be a singular N Theta N matrix over the Galois field with q elements K =GF(q) and w a vector of N unknowns. One fundamental application of this problem is integer and polynomial factorization, where such linear systems arise with N over 200; 000 [23, 25, 19]. This has motivated several authors to develop fast finite field counterpart to numerical iterative methods. The conjugate gradient method has been used in [23] the Lanczos method in [23, 12] and the block Lanczos method in [8, 29] But up to now, only the probabilistic analysis of Wiedemann ....

A.K. Lenstra, H.W. Lenstra, M.S. Manasse, and J.M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349.

.... d ailleurs tr es rapidement, et ce n est qu en 1990 que le neuvi eme d entre eux, 2 2 9 1, un nombre de 154 chi res, fut factoris e au terme d un calcul auquel contribu erent plus de 700 ordinateurs a travers le monde travaillant en parall ele et sans arr et pendant pr es de quatre mois [LLMP]. Aujourd hui encore, les algorithmes les plus sophistiqu es arrivent dicilement a bout des nombres de plus de 150 ou 200 chi res, m eme sur les ordinateurs les plus puissants. Absorb es dans la contemplation de nombres dont la taille d epasse de loin ce qui peut se rencontrer en physique, les ....

Lenstra, A. K.; Lenstra, H. W., Jr.; Manasse, M. S.; Pollard, J. M. The factorization of the ninth Fermat number. Math. Comp. 61 (1993), no. 203, 319-349.

....above. Furthermore, a reduction in the sieve array size allows a reduction in the size of n; see, e.g. 52] Pollard s number eld sieve, as generalized by Buhler, Lenstra, and Pomerance, seems to always succeed in time exp( 64=9 o(1) 1=3 (log D) 1=3 (log log D) 2=3 ) See [138] [104], 105] 3] 44] 139] 60] 24] 41] 145] 121] 82] 19] 69] 148] 72] 73] 74] 76] 61] 75] 123] 126] 130] and [127] The algorithm in this paper can again be used to indirectly speed up sieving and reduce the size of n. Coppersmith s number eld sieve variant in ....

Arjen K. Lenstra, Hendrik W. Lenstra, Jr., Mark S. Manasse, John M. Pollard, The factorization of the ninth Fermat number, Mathematics of Computation 61 (1993), 319-349. MR 93k:11116.

....two integers u mv and u 5 8v 5 . Note that, for moderate u and v, both these integers are much smaller than N , in fact they are O(N 1 d ) where d =5isthedegreeof the algebraic number field. The optimal choice of d is discussed in 6. Using these and related ideas, Lenstra et al. [31] factored F 9 in June 1990, obtaining F 9 = 2424833 7455602825647884208337395736200454918783366342657 p 99 , where p 99 is an 99 digit prime, and the 7 digit factor was already known (although SNFS was unable to take advantage of this) The collection of relations took less than two months on ....

A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349.

....powerful known factoring method. It was first introduced in 1988 by John Pollard [17] to factor numbers of form x 3 k. Then it was modified to handle numbers of the form r e Gamma s for small positive r and jsj: this was successfully applied to the Fermat number F 9 = 2 512 1 (see [11]) This version of the algorithm is now called the special number field sieve (SNFS) 10] in contrast with the general number field sieve (GNFS) 3] which can handle arbitrary integers. GNFS factors integers n in heuristic time exp i (c g o(1) ln 1=3 n ln 2=3 ln n j with c g = 64=9) ....

....root of fl. Unfortunately, the ring Z[ff] is not necessarily a UFD for the arbitrary number fields GNFS encounters. And even though Z[ff] is a UFD, computing a system of fundamental units is not an obvious task (see [4] The method was nevertheless applied with success to the factorization of F 9 [11]. 3 Brute force method. One factorizes the polynomial P (X) X 2 Gamma fl over K [X ] To do so, one has to explicitly write the algebraic number fl, for instance by expanding the product: one thus gets the (rational) coefficients of fl as a polynomial of degree at most d Gamma 1 in ff. ....

Lenstra, A. K., Lenstra, Jr., H. W., Manasse, M. S., and Pollard, J. M. The factorization of the ninth fermat number. Math. Comp. 61 (1993), 319--349.

....two integers u mv and ju 5 8v 5 j. Note that, for moderate u and v, both these integers are much smaller than N , in fact they are O(N 1=d ) where d = 5 is the degree of the algebraic number eld. The optimal choice of d is discussed in x6. Using these and related ideas, Lenstra et al. [31] factored F 9 in June 1990, obtaining F 9 = 2424833 7455602825647884208337395736200454918783366342657 p 99 ; 6 where p 99 is an 99 digit prime, and the 7 digit factor was already known (although SNFS was unable to take advantage of this) The collection of relations took less than two ....

A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319-349.

....L x [u; v] is an increasing function of the more important parameter u. As L x [u; v] interpolates between powers of ln x and powers of exp(ln x) x, the order of L x [u; v] is said to be subexponential in the size ln x of x if 0 u 1. 2 K. NAKAMULA A good introduction to the NFS is given in [10]. Basic facts, improvements, related topics, references and a history of the NFS can be found in the lecture notes [9] A historical description of the NFS including its recent developments can be found in [23] A variation of the GNFS with four large primes is introduced in [5] A multiple ....

A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, J. M. Pollard, The factorization of the ninth Fermat number, Math. Comput. 61 (1993), 319-349.

....which can be factored, to 130 or 140 digits. A general number is one with no special form that might make it easier to factor; an RSA modulus is a general number. Note that a 512 bit number has about 155 digits. Numbers that have a special form can already be factored up to 155 digits or more [48]. The Cunningham Project [14] keeps track of the factorizations of numbers with these special forms and maintains a 10 Most Wanted list of desired factorizations. Also, a good way to survey current factoring capability is to look at recent results of the RSA Factoring Challenge (see Question ....

A.K. Lenstra, H.W. Lenstra Jr., M.S. Manasse, and J.M. Pollard. The factorization of the ninth Fermat number. 1991. To appear.

....premiers. 4.2.1 Factorisation Malgr e ces complications d ordre th eorique, on peut mettre au point des algorithmes de factorisation d entiers utilisant des factorisations interm ediaires dans des corps de nombres. C est le cas de l algorithme NFS (Number Field Sieve) du a Pollard [LLMP90, LLMP91] Cet al..gorithme appartient egalement a la famille des algorithmes a combinaisons de congruences. A l heure actuelle, il est tr es bien adapt e a la factorisation des nombres de la forme r e Sigma s avec r et s petits. Pour un nombre N de cette forme, on montre que le temps de calcul est ....

.... montre que le temps de calcul est exp(c(log N) 1=3 (log log N) 2=3 ) avec c une constante positive (effective) Cet al..gorithme a ainsi permis de factoriser le nombre F 9 = 2 2 9 1, un nombre de 155 chiffres d ecimaux grace a la collaboration d une centaine de chercheurs de par le monde [LLMP91] Cet al..gorithme ne permet pas, a l heure actuelle, de factoriser des grands nombres de forme quelconque [BLP92] 4.2.2 Logarithme discret Notons egalement que par un transfert courant, NFS peut s adapter a la r esolution du probl eme de logarithme discret dans Z=pZ (cf. Gor92] donnant un ....

Lenstra (A. K.), Lenstra, Jr. (H. W.), Manasse (M. S.) et Pollard (J. M.). -- The factorization of the ninth Fermat number. -- 1991. To appear.

....Fn (perhaps only F 0 ; F 4 ) are prime. The complete factorization of the Fermat numbers F 6 ; F 7 ; has been a challenge since Euler s time. Because the Fn grow rapidly in size, a method which factors Fn may be inadequate for Fn 1 . Historical details and references can be found in [21, 35, 36, 44, 74], and some recent results are given in [17, 26, 27, 34] In the following, pn denotes a prime number with n decimal digits (not necessarily the same at each occurrence) Similarly, c n denotes a composite number with n decimal digits. Landry [41] factored F 6 = 274177 Delta p 14 in 1880, but ....

.... overcome by the invention of the (special) number field sieve (SNFS) based on a new idea of Pollard [43, 61] In 1990, Lenstra, Lenstra, Manasse and Pollard, with the assistance of many collaborators and approximately 700 workstations scattered around the world, completely factored F 9 by SNFS [44, 45]. The factorization is F 9 = 2424833 Delta 7455602825647884208337395736200454918783366342657 Delta p 99 : In x8 we show that it would have been possible (though more expensive) to complete the factorization of F 9 by ECM. F 10 was the most wanted number in various lists of composite numbers ....

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A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349. MR 93k:11116

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A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349.

....was quickly generalized to a factoring method for numbers of the form x d k, a method that is currently referred to as the special number eld sieve. It proved to be practical by factoring the ninth Fermat number F 9 = 2 2 9 1. This happened in 1990, long before F 9 was expected to fall [66]. The heuristic expected runtime of the special number eld sieve is Ln [1=3; 1:526] where 1:526 # (32=9) 1=3 . It was the rst factoring algorithm with runtime substantially below Ln [1=2; c] for constant c) and as such an enormous breakthrough. It was also an unpleasant surprise for ....

....of more than two large primes, so that relatively small factor bases can be used during sieving. The relation collection stage can be distributed over almost any number of loosely coupled processors, similar to quadratic sieve. For an introductory description of the number eld sieve, refer to [63, 66, 93]. For complete details see [65] and the references given there. The latest developments are described in [21, 81] The largest special number factored using the special number eld sieve is 2 773 1 (see [33] This was done by the same group that achieved the current general number eld ....

A.K. Lenstra, H.W. Lenstra, Jr., M.S. Manasse, J.M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993) 319-349.

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A.K. Lenstra, H.W. Lenstra, Jr., M.S. Manasse, and J.M. Pollard. The factorization of the Ninth Fermat number. Mathematics of Computation, 61(203):319--349, July 1993.

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A.K. Lenstra, H.W. Lenstra, Jr., M.S. Manasse, and J.M. Pollard. The factorization of the Ninth Fermat number. Mathematics of Computation, 61(203):319--349, July 1993.

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A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349.

.... factoring algorithms are the quadratic sieve (QS) and the number field sieve (NFS) cf. 12] and [2] Throughout this paper, NFS is the generalized version (from [2] of the algorithm from [8] the latter algorithm is much faster, but can only be applied to composites of a very special form, cf. [9]. Let L x [a; b] exp Gamma (b o(1) log x) a (log log x) 1 Gammaa Delta for real a, b, x, and x 1. To factor an odd integer n 1 which is not a prime power, QS runs in time (1) Ln [1=2; 1] and NFS in (2) Ln [1=3; 1:923] It follows that NFS is asymptotically superior to QS, but ....

....seven percent. We did not attempt to analyse the effects of a factor base that is even larger than 245810. Matrix reduction. As a result of the sieving stage we got a 252222 Theta 245810 bit matrix. To find dependencies modulo two among its rows the third author used the technique described in [9] with the extension from [1] structured Gaussian elimination (cf. 5; 13] followed by the incremental version from [1] of the MasPar dense matrix eliminator described in [6] Structured Gauss managed to reduce the size of the matrix to 89304 Theta 89088. This took 15 hours on a Sparc10 ....

Lenstra, A.K., Lenstra, Jr., H.W., Manasse, M. S., Pollard, J. M.: The factorization of the ninth Fermat number. Math. Comp. 61 (1993) 319--349

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A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349.

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A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349. MR 93k:11116

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A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349.

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A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349.

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A.K. Lenstra, H.W. Lenstra Jr., M.S. Manasse, and J.M. Pollard. The factorization of the ninth Fermat number. 1991. To appear.

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A.K. Lenstra, H.W. Lenstra Jr., M.S. Manasse, and J.M. Pollard. The factorization of the ninth Fermat number. Antonia J. Jones:18 December

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A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319--349.

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A.K. Lenstra, H.W. Lenstra, Jr., M.S. Manasse, and J.M. Pollard. The factorization of the ninth Fermat number. Mathematics of Computation, 61:319--349, 1993.

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A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse and J. M. Pollard "The factorization of the ninth Fermat number", Mathematics of Computation 61 (1993), 319--349.

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