Richard P. Brent. Factorization of the tenth and eleventh Fermat Numbers. Computer Science Laboratory, Australian National Univ., Canberra, Report TR-CS-96-02:1--25, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....p. The heuristics are, however, con rmed by experiments and the elliptic curve method so far behaves as heuristically predicted. Remarkable successes of the elliptic curve method were the factorizations of the tenth and eleventh Fermat numbers, F 10 = 2 2 10 1 and F 11 = 2 2 11 1; see [16]. Factors of over 50 decimal digits have occasionally been found using the elliptic curve method. The method is not considered to be a threat against RSA, but its existence implies that care should be taken when using RSA moduli consisting of more than two prime factors. A variant of the elliptic ....

R.P. Brent, Factorization of the tenth and eleventh Fermat numbers, manuscript, 1996.

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R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS-9602, Computer Sciences Laboratory, Australian National Univ., Canberra, February 1996, 25 pp. ftp://nimbus.anu.edu.au/pub/Brent/rpb161tr.dvi.Z .

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R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS96 -02, Computer Sciences Laboratory, Australian National Univ., Canberra, Feb. 1996. ftp://nimbus.anu.edu.au/pub/Brent/rpb161tr.dvi.gz .

....with 2391, 9808 and 19694 decimal digits respectively. 1. Introduction For a nonnegative integer n,then th Fermat number is Fn =2 2 n 1. Itis known [12] that Fn is prime for 0 # n # 4, and composite for 5 # n # 23. For a brief history of attempts to factor Fermat numbers, we refer to [3, 1] and [5] In recent years several factors of Fermat numbers have been found by the elliptic curve method (ECM) Brent [2, 3, 4] completed the factorization of F 10 (by finding a 40 digit factor) and F 11 . He also rediscovered the 49 digit factor of F 9 and the five known prime factors of F 12 ....

....Fn =2 2 n 1. Itis known [12] that Fn is prime for 0 # n # 4, and composite for 5 # n # 23. For a brief history of attempts to factor Fermat numbers, we refer to [3, 1] and [5] In recent years several factors of Fermat numbers have been found by the elliptic curve method (ECM) Brent [2, 3, 4] completed the factorization of F 10 (by finding a 40 digit factor) and F 11 . He also rediscovered the 49 digit factor of F 9 and the five known prime factors of F 12 . Crandall [10] discovered two 19 digit factors of F 13 . This paper reports the discovery of 27 digit factors of F 13 and F 16 ....

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R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS96 -02, Computer Sciences Laboratory, Australian National Univ., Canberra, Feb. 1997. ftp://nimbus.anu.edu.au/pub/Brent/rpb161tr.dvi.gz .

....2 2 n 1. It is known [12] that F n is prime for 0 n 4, and composite for 5 n 23. For a brief history of attempts to factor Fermat numbers, we refer to [3, x1] and [5] In recent years several factors of Fermat numbers have been found by the elliptic curve method (ECM) Brent [2, 3, 4] completed the factorization of F 10 (by nding a 40 digit factor) and F 11 . He also rediscovered the 49 digit factor of F 9 and the ve known prime factors of F 12 . Crandall [10] discovered two 19 digit factors of F 13 . This paper reports the discovery of 27 digit factors of F 13 and F 16 ....

....factor has been found by ECM. Factors of larger numbers are customarily found by trial division [16, 18] 2. The Elliptic Curve Method ECM was invented by H. W. Lenstra, Jr. 22] Various practical re nements were suggested by Brent [1] Montgomery [23, 24] and Suyama [31] We refer to [3, 14, 21, 25, 30] for a description of ECM and some of its implementations. In the following, we assume that ECM is used to nd a prime factor p 3 of a composite number N , not a prime power [20, x2.5] The rst phase limit for ECM is denoted by B 1 . Although p is unknown, it is convenient to describe ECM in ....

R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS96 -02, Computer Sciences Laboratory, Australian National Univ., Canberra, Feb. 1997. ftp://nimbus.anu.edu.au/pub/Brent/rpb161tr.dvi.gz .

....is Fn = 2 2 n 1. It is known [12] that Fn is prime for 0 n 4, and composite for 5 n 23. For a brief history of attempts to factor Fermat numbers, we refer to [3, x1] and [5] In recent years several factors of Fermat numbers have been found by the elliptic curve method (ECM) Brent [2, 3, 4] completed the factorization of F 10 (by finding a 40 digit factor) and F 11 . He also rediscovered the 49 digit factor of F 9 and the five known prime factors of F 12 . Crandall [10] discovered two 19 digit factors of F 13 . This paper reports the discovery of 27 digit factors of F 13 and F 16 ....

....factor has been found by ECM. Factors of larger numbers are customarily found by trial division [16, 18] 2. The Elliptic Curve Method ECM was invented by H. W. Lenstra, Jr. 23] Various practical refinements were suggested by Brent [1] Montgomery [24, 25] and Suyama [32] We refer to [3, 14, 22, 26, 31] for a description of ECM and some of its implementations. In the following, we assume that ECM is used to find a prime factor p 3 of a composite number N , not a prime power [21, x2.5] The first phase limit for ECM is denoted by B 1 . 1991 Mathematics Subject Classification. 11Y05, 11B83, ....

R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS96 -02, Computer Sciences Laboratory, Australian National Univ., Canberra, Feb. 1997. ftp://nimbus.anu.edu.au/pub/Brent/rpb161tr.dvi.gz .

....a number as large as the 564 digit factor of F 11 is a nontrivial task. In x9 we discuss primality proofs and certificates of primality for the factors of Fn , n 11. Attempts to factor Fermat numbers by ECM are continuing. For example, 27digit factors of F 13 and F 16 have recently been found [13, 17]. The smallest Fermat number which is not yet completely factored is F 12 . It is known that F 12 has at least seven prime factors, and F 12 = 114689 Delta 26017793 Delta 63766529 Delta 190274191361 Delta 1256132134125569 Delta c 1187 : The prospects for further progress in factoring F 12 ....

.... Sigma 1 using binary shift and add subtract operations, which are much faster (for large n) than multiply or divide operations, a significant speedup may be possible. This idea was not implemented in programs B D, but was used successfully in programs which found factors of F 13 and F 16 , see [13, 17]. 6. Factorization of F 10 When ECM was implemented on the Fujitsu VP100 in March 1988, some of the first numbers which we attempted to factor were the Fermat numbers F 9 ; F 10 ; F 11 and F 12 , using variants of program C. We were soon successful with F 11 (see x7) but not with the other ....

R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS96 -02, Computer Sciences Laboratory, Australian National Univ., Canberra, Feb. 1996. ftp://nimbus.anu.edu.au/pub/Brent/rpb161tr.dvi.gz .

....number is Fn = 2 2 n 1. It is known [12] that Fn is prime for 0 n 4, and composite for 5 n 23. For a brief history of attempts to factor Fermat numbers, we refer to [3, x1] and [5] In recent years several factors of Fermat numbers have been found by the elliptic curve method (ECM) Brent [2, 3, 4] completed the factorization of F 10 (by finding a 40 digit factor) and F 11 . He also rediscovered the 49 digit factor of F 9 and the five known prime factors of F 12 . Crandall [10] discovered two 19 digit factors of F 13 . This paper reports the discovery of 27 digit factors of F 13 and F 16 ....

....factor has been found by ECM. Factors of larger numbers are customarily found by trial division [16, 18] 2. The Elliptic Curve Method ECM was invented by H. W. Lenstra, Jr. 22] Various practical refinements were suggested by Brent [1] Montgomery [23, 24] and Suyama [31] We refer to [3, 14, 21, 25, 30] for a description of ECM and some of its implementations. In the following, we assume that ECM is used to find a prime factor p 3 of a composite number N , not a prime power [20, x2.5] The first phase limit for ECM is denoted by B 1 . Although p is unknown, it is convenient to describe ECM in ....

R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS96 -02, Computer Sciences Laboratory, Australian National Univ., Canberra, Feb. 1997. ftp://nimbus.anu.edu.au/pub/Brent/rpb161tr.dvi.gz .

.... field sieve (NFS) For a brief survey and historical introduction, see [40] A more comprehensive introduction is Riesel [104] ECM was introduced by Hendrik Lenstra [79] and practical improvements were suggested and implemented by several people [23, 84] The state of the art is summarized in [30, 31]. The quadratic sieve was popularized by Carl Pomerance, although it is based on earlier ideas of Fermat, Kraitchik, Brillhart, Morrison and others, and improvements have been contributed by Davis, Montgomery, Silverman and others [40] The best current version is the two large primes ....

Richard P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS-96-02, Computer Sciences Laboratory, Australian National Univ., Canberra, Feb. 1996. ftp:// nimbus.anu.edu.au/pub/Brent/rpb161tr.dvi.gz. A preliminary version of [30], including more detail on different implementations of ECM.

....the known non algebraic factors of the number a n Sigma 1) We do not describe ECM, MPQS or SNFS here. The reader should refer to [17, 18, 20] for a general description of ECM, to [24] for MPQS, and to [16] for SNFS. The particular implementations of ECM by Brent and Montgomery are described in [7, 19]. Computational details regarding the factorizations of various entries in Update 1 and in this Update 2, obtained with an implementation of the two large primes variation (PPMPQS) of MPQS on SGI workstations and on a Cray C90 vector computer, are given in [2] The implementation of SNFS used by ....

.... Elkenbracht Huizing and Peter Montgomery is described in [13] In the following we do not distinguish between different versions of the basic methods (e.g. PMPQS and PPMPQS, ECM and ECM FFT) ECM is useful for finding factors of up to about 35 digits, although it occasionally finds larger factors [6, 7]. The largest factor found by ECM in the course of preparation of this update is a 47 digit factor p 47 = 28207978317787299519881883345010831781124600233 of 30 109 Gamma 1 (found by Peter Montgomery on 25 February 1996) We knew that 30 109 Gamma 1 = 29 Delta 134507 Delta ....

R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS-9602, Computer Sciences Laboratory, Australian National Univ., Canberra, February 1996, 25 pp. ftp://nimbus.anu.edu.au/pub/Brent/rpb161tr.dvi.Z .

No context found.

Richard P. Brent. Factorization of the tenth and eleventh Fermat Numbers. Computer Science Laboratory, Australian National Univ., Canberra, Report TR-CS-96-02:1--25, 1996.

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