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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 44 (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.
Multidigit Multiplication For Mathematicians
, 2001
"... This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStr ..."
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Cited by 35 (8 self)
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This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStrassen trick, Schönhage's trick, Nussbaumer's trick, the cyclic SchönhageStrassen trick, and the CantorKaltofen theorem. It emphasizes the underlying ring homomorphisms.
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 28 (12 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 .
Recent progress and prospects for integer factorisation algorithms
 In Proc. of COCOON 2000
, 2000
"... Abstract. The integer factorisation and discrete logarithm problems are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed difficulty of solving these problems. This paper considers primarily the integer factorisation problem. In ..."
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Cited by 24 (1 self)
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Abstract. The integer factorisation and discrete logarithm problems are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed difficulty of solving these problems. This paper considers primarily the integer factorisation problem. In recent years the limits of the best integer factorisation algorithms have been extended greatly, due in part to Moore’s law and in part to algorithmic improvements. It is now routine to factor 100decimal digit numbers, and feasible to factor numbers of 155 decimal digits (512 bits). We outline several integer factorisation algorithms, consider their suitability for implementation on parallel machines, and give examples of their current capabilities. In particular, we consider the problem of parallel solution of the large, sparse linear systems which arise with the MPQS and NFS methods. 1
Irregular primes and cyclotomic invariants to four million
 Math. Comp
, 1993
"... Dedicated to the computational genius of Derrick Lehmer Abstract. Recent computations of irregular primes, and associated cyclotomic invariants, were extended to all primes below four million using an enhanced multisectioning/convolution method. Fermat's "Last Theorem " and Vandiver&a ..."
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Cited by 22 (2 self)
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Dedicated to the computational genius of Derrick Lehmer Abstract. Recent computations of irregular primes, and associated cyclotomic invariants, were extended to all primes below four million using an enhanced multisectioning/convolution method. Fermat's "Last Theorem " and Vandiver's conjecture were found to be true for those primes, and the cyclotomic invariants behaved as expected. There is exactly one prime less than four million whose index of irregularity is equal to seven. An irregular pair (p, t) consists of an odd prime p and an even integer t such that 0 < t < p 1 and p divides (the numerator of) the Bernoulli number Bt. The index of irregularity rp for a prime p is the number of irregular pairs for p. Kummer computed the irregular pairs for odd primes p less than 165 by 1874. In the 1920s and 1930s, H. S. Vandiver used desk calculators and graduate students to find the irregular primes for p < 620, and used these computations to verify Fermat's "Last Theorem " (FLT) for those primes. Derrick and Emma
Performing outofcore FFTs on parallel disk systems
 PARALLEL COMPUTING
, 1998
"... The Fast Fourier Transform (FFT) plays a key role in many areas of computational science and engineering. Although most onedimensional FFT problems can be solved entirely in main memory, some important classes of applications require outofcore techniques. For these, use of parallel I/O systems ca ..."
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Cited by 20 (7 self)
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The Fast Fourier Transform (FFT) plays a key role in many areas of computational science and engineering. Although most onedimensional FFT problems can be solved entirely in main memory, some important classes of applications require outofcore techniques. For these, use of parallel I/O systems can improve performance considerably. This paper shows how to perform onedimensional FFTs using a parallel disk system with independent disk accesses. We present both analytical and experimental results for performing outofcore FFTs in two ways: using traditional virtual memory with demand paging, and using a provably asymptotically optimal algorithm for the Parallel Disk Model (PDM) of Vitter and Shriver. When run on a DEC 2100 server with a large memory and eight parallel disks, the optimal algorithm for the PDM runs up to 144.7 times faster than incore methods under demand paging. Moreover, even including I/O costs, the normalized times for the optimal PDM algorithm are competitive, or better than, those for incore methods even when they run entirely in memory.
Portable HighPerformance Programs
, 1999
"... right notice and this permission notice are preserved on all copies. ..."
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Cited by 19 (0 self)
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right notice and this permission notice are preserved on all copies.
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...
A GMPbased implementation of SchönhageStrassen’s large integer multiplication algorithm
 In Proceedings of ISSAC’07
, 2007
"... Abstract. SchönhageStrassen’s algorithm is one of the best known algorithms for multiplying large integers. Implementing it efficiently is of utmost importance, since many other algorithms rely on it as a subroutine. We present here an improved implementation, based on the one distributed within th ..."
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Cited by 17 (5 self)
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Abstract. SchönhageStrassen’s algorithm is one of the best known algorithms for multiplying large integers. Implementing it efficiently is of utmost importance, since many other algorithms rely on it as a subroutine. We present here an improved implementation, based on the one distributed within the GMP library. The following ideas and techniques were used or tried: faster arithmetic modulo 2 n + 1, improved cache locality, Mersenne transforms, Chinese Remainder Reconstruction, the √ 2 trick, Harley’s and Granlund’s tricks, improved tuning. We also discuss some ideas we plan to try in the future.
The twentyfourth Fermat number is composite
, 2002
"... We have shown by machine proof that F24 = 2^2^24 + 1 is composite. The rigorous Pépin primality test was performed using independently developed programs running simultaneously on two different, physically separated processors. Each program employed a floatingpoint, FFTbased discrete weighted tran ..."
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Cited by 10 (2 self)
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We have shown by machine proof that F24 = 2^2^24 + 1 is composite. The rigorous Pépin primality test was performed using independently developed programs running simultaneously on two different, physically separated processors. Each program employed a floatingpoint, FFTbased discrete weighted transform (DWT) to effect multiplication modulo F24. The final, respective Pépin residues obtained by these two machines were in complete agreement. Using intermediate residues stored periodically during one of the floatingpoint runs, a separate algorithm for pureinteger negacyclic convolution verified the result in a “wavefront” paradigm, by running simultaneously on numerous additional machines, to effect piecewise verification of a saturating set of deterministic links for the Pépin chain. We deposited a final Pépin residue for possible use by future investigators in the event that a proper factor of F24 should be discovered; herein we report the more compact, traditional SelfridgeHurwitz residues. For the sake of completeness, we also generated a Pépin residue for F23, and via the Suyama test determined that the known cofactor of this number is composite.