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 Rivest-Shamir-Adelman (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 60-decimal 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

1.4 Simple XOR

We compare implementations of two integer factorisation algorithms, the elliptic curve method (ECM) and a variant of the Pollard "rho " method, on three machines (the Fujitsu AP1000, VP2200 and VPP500) with parallel and/or vector architectures. ECM is scalable and well suited for both vector and parallel architectures.

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In an earlier Report (NM-R9212, June 1992), two of us gave tables of factorizations of a n \Sigma 1 for 13 a ! 100. The exponents n satisfied a n ! 10 255 if a ! 30, and n 100 if a 30. The factorizations were complete for n 46, and the tables contained no composite numbers smaller than 10 80 . In this Report we update the original tables. The factorizations are now complete for n 58, and there are no composite numbers smaller than 10 86 . 1991 Mathematics Subject Classification: Primary 11A25; Secondary 11-04. Keywords and Phrases: Factor Tables. Appeared as Report NM-R94??, Centrum voor Wiskunde en Informatica, Amsterdam, September 1994, 46 pp. Only the front matter is given here. Copyright c fl 1994, the authors. rpb134u1 typeset using T E X 1. Introduction For many years there has been an interest in the prime factors of numbers of the form a n \Sigma 1, where a is a small integer (the base) and n is a positive exponent. Such numbers often arise. For example, i...

In an earlier Report (NM-R9212, June 1992), two of us gave tables of factorizations of a n \Sigma 1 for 13 a ! 100. The exponents n satisfied a n ! 10 255 if a ! 30, and n 100 if a 30. The factorizations were complete for n 46, and the tables contained no composite numbers smaller than 10 80 . In this Report we update the original tables. The factorizations are now complete for n 58, and there are no composite numbers smaller than 10 86 . 1991 Mathematics Subject Classification: Primary 11A25; Secondary 11-04. Keywords and Phrases: Factor Tables. 1. Introduction For many years there has been an interest in the prime factors of numbers of the form a n \Sigma 1, where a is a small integer (the base) and n is a positive exponent. Such numbers often arise. For example, if a is prime then there is a finite field F with a n elements, and the multiplicative group of F has a n \Gamma 1 elements. Also, for prime a the sum of divisors of a n is oe(a n ) = (a n+1 ...

As an extension of the "Cunningham" tables, we present tables of factorizations of a n \Sigma 1 for 13 a ! 100. The exponents n satisfy a n ! 10 255 if a ! 30, and n 100 if a 30. The factorizations are complete for n 46, and the tables contain no composite numbers smaller than 10 80 . 1991 Mathematics Subject Classification: Primary 11A25; Secondary 11-04. Keywords and Phrases: Factor Tables. Appeared as Report NM-R9212, Centrum voor Wiskunde en Informatica, Amsterdam, June 1992, 368 pp. Only the front matter is given here. Copyright c fl 1992, the authors. rpb134 typeset using T E X 1. Introduction For many years there has been an interest in the prime factors of numbers of the form a n \Sigma 1, where a is a moderately small integer (the base) and n is a positive exponent. Such numbers often arise. For example, if a is prime then there is a finite field F with a n elements, and the multiplicative group of F has a n \Gamma 1 elements. Also, for prime a the sum ...

. Two systematic search methods are employed to find multipliers for linear congruential pseudo-random number generation which are optimal with respect to an upper bound for the discrepancy of pairs of successive pseudo-random numbers. The efficiency of these search procedures when executed on parallel systems is assessed by experimental results of a MIMD parallel implementation on a Meiko CS-2 and a workstation cluster. Furthermore the quality of the computed multipliers is evaluated by using the spectral--test in dimensions 2 -- 8 and by calculating the actual discrepancy of pairs of the resulting full--period sequences. AMS subject classification: 65C10, 65Y05, 68Q22, 11A55. Key words: Random number generation, parallel processing, continued fractions. 1 Introduction. The statistical properties of the most commonly used pseudo-random numbers (PRN), namely of those generated by the linear congruential generator (LCG), depend strongly on the choice of parameters in the method. In ...

The first practical public key cryptosystem to be published, the Diffie-Hellman key exchange algorithm, was based on the assumption that discrete logarithms are hard to compute. This intractability hypothesis is also the foundation for the presumed security of a variety of other public key schemes. While there have been substantial advances in discrete log algorithms in the last two decades, in general the discrete log still appears to be hard, especially for some groups, such as those from elliptic curves. Unfortunately no proofs of hardness are available in this area, so it is necessary to rely on experience and intuition in judging what parameters to use for cryptosystems. This paper presents a brief survey of the current state of the art in discrete logs. 1. Introduction Many of the popular public key cryptosystems are based on discrete exponentiation. If G is a group, such as the multiplicative group of a finite field or the group of points on an elliptic curve, and g is an elem...

In an earlier Report (NM-R9212, June 1992), two of us gave tables of factorizations of a 1 for 13 a < 100. The exponents n satis ed a if a < 30, and n 100 if a 30. The factorizations were complete for n 46, and the tables contained no composite numbers smaller than 10 80 . In this Report we update the original tables. The factorizations are now complete for n 58, and there are no composite numbers smaller than 10 86 .