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Parallel Algorithms for Integer Factorisation
"... The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends o ..."
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Cited by 41 (17 self)
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The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorisation algorithms have improved greatly, to the point where it is now easy to factor a 60decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. We describe several algorithms, including the elliptic curve method (ECM), and the multiplepolynomial quadratic sieve (MPQS) algorithm, and discuss their parallel implementation. It turns out that some of the algorithms are very well suited to parallel implementation. Doubling the degree of parallelism (i.e. the amount of hardware devoted to the problem) roughly increases the size of a number which can be factored in a fixed time by 3 decimal digits. Some recent computational results are mentioned – for example, the complete factorisation of the 617decimal digit Fermat number F11 = 2211 + 1 which was accomplished using ECM.
Factorization Of The Tenth Fermat Number
 MATH. COMP
, 1999
"... We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factor ..."
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Cited by 22 (10 self)
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We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 ; : : : ; F 11 .
Factorization of the tenth and eleventh Fermat numbers
, 1996
"... . We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a ..."
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Cited by 17 (8 self)
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. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40digit factor of the tenth Fermat number was found after about 140 Mflopyears of computation. We discuss aspects of the practical implementation of ECM, including the use of specialpurpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the nth Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...
Two new factors of Fermat numbers
, 1997
"... Abstract. We report the discovery of new 27decimal digit factors of the thirteenth and sixteenth Fermat numbers. Each of the new factors was found by the elliptic curve method. After division by the new factors and other known factors, the quotients are seen to be composite numbers with 2391 and 19 ..."
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Cited by 5 (2 self)
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Abstract. We report the discovery of new 27decimal digit factors of the thirteenth and sixteenth Fermat numbers. Each of the new factors was found by the elliptic curve method. After division by the new factors and other known factors, the quotients are seen to be composite numbers with 2391 and 19694 decimal digits respectively. 1.
Three New Factors of Fermat Numbers
 Math. Comp
, 2000
"... We report the discovery of a new factor for each of the Fermat numbers F 13 ,F 15 ,F 16 . These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are ..."
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Cited by 4 (0 self)
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We report the discovery of a new factor for each of the Fermat numbers F 13 ,F 15 ,F 16 . These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are seen to be composite numbers with 2391, 9808 and 19694 decimal digits respectively. 1.
Integer Factorization
, 1994
"... 6.19> public key cryptosystems (also known as asymmetric cryptosystems and open encryption key cryptosystems) [12, 13]. The security of such systems depends on the (assumed) difficulty of factoring the product of two large primes. This is a practical motivation for the current interest in integer f ..."
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Cited by 1 (0 self)
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6.19> public key cryptosystems (also known as asymmetric cryptosystems and open encryption key cryptosystems) [12, 13]. The security of such systems depends on the (assumed) difficulty of factoring the product of two large primes. This is a practical motivation for the current interest in integer factorisation algorithms. Parallel algorithms We would hope that an algorithm which required time T 1 on a computer with one processor could be implemented to run in time T P ¸ T 1 =P on a computer with P independent processors. This is not always the case, since it may be impossible to use all P processors effectively. However, it is true for many integer factorisation algorithms, provided that P is not too large. Integer factorization algorithms There are many algorithms for finding a nontrivial fac
Integer Factorization Summary
, 1994
"... The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends o ..."
Abstract
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The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorization algorithms have improved greatly, to the point where it is now easy to factor a 60decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. However, the problem of integer factorization still appears difficult, both in a practical sense (for numbers of more than about 80 decimal digits), and in a theoretical sense (because none of the algorithms run in polynomial time). We outline several recent integer factorization algorithms, including the elliptic curve algorithm (ECM), the multiple polynomial quadratic sieve (MPQS), and the special/general number field sieve (NFS), give examples of their use, and mention some applications. Public key cryptography Large primes have at least one practical application – they can be used to construct public key