Results 1  10
of
23
Solving Large Sparse Linear Systems Over Finite Fields
, 1991
"... Many of the fast methods for factoring integers and computing discrete logarithms require the solution of large sparse linear systems of equations over finite fields. This paper presents the results of implementations of several linear algebra algorithms. It shows that very large sparse systems can ..."
Abstract

Cited by 75 (2 self)
 Add to MetaCart
Many of the fast methods for factoring integers and computing discrete logarithms require the solution of large sparse linear systems of equations over finite fields. This paper presents the results of implementations of several linear algebra algorithms. It shows that very large sparse systems can be solved efficiently by using combinations of structured Gaussian elimination and the conjugate gradient, Lanczos, and Wiedemann methods. 1. Introduction Factoring integers and computing discrete logarithms often requires solving large systems of linear equations over finite fields. General surveys of these areas are presented in [14, 17, 19]. So far there have been few implementations of discrete logarithm algorithms, but many of integer factoring methods. Some of the published results have involved solving systems of over 6 \Theta 10 4 equations in more than 6 \Theta 10 4 variables [12]. In factoring, equations have had to be solved over the field GF (2). In that situation, ordinary...
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 ..."
Abstract

Cited by 41 (17 self)
 Add to MetaCart
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.
Computation of Discrete Logarithms in Prime Fields
 Design, Codes and Cryptography
, 1991
"... The presumed difficulty of computing discrete logarithms in finite fields is the basis of several popular public key cryptosystems. The secure identification option of the Sun Network File System, for example, uses discrete logarithms in a field GF (p) with p a prime of 192 bits. This paper describe ..."
Abstract

Cited by 38 (1 self)
 Add to MetaCart
The presumed difficulty of computing discrete logarithms in finite fields is the basis of several popular public key cryptosystems. The secure identification option of the Sun Network File System, for example, uses discrete logarithms in a field GF (p) with p a prime of 192 bits. This paper describes an implementation of a discrete logarithm algorithm which shows that primes of under 200 bits, such as that in the Sun system, are very insecure. Some enhancements to this system are suggested. 1. Introduction If p is a prime and g and x integers, then computation of y such that y j g x mod p; 0 y p \Gamma 1 (1.1) is referred to as discrete exponentiation. Using the successive squaring method, it is very fast (polynomial in the number of bits of jpj + jgj + jxj). On the other hand, the inverse problem, namely, given p; g, and y, to compute some x such that Equation 1.1 holds, which is referred to as the discrete logarithm problem, appears to be quite hard in general. Many of the mos...
Discrete Logarithms and Smooth Polynomials, in Finite
 Contemporary Mathematics, Volume 168, American Mathematical Society
, 1994
"... ..."
Improvements to the general number field sieve for discrete logarithms in prime fields
 Mathematics of Computation
, 2003
"... Abstract. In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Abstract. In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number field sieve outperforms the gaussian integer method in the hundred digit range. We also illustrate our results by successfully computing discrete logarithms with GNFS in a large prime field. 1.
Reducing Logarithms in Totally NonMaximal Imaginary Quadratic Orders to Logarithms in Finite Fields (Extended Abstract)
, 1999
"... Since nobody can guarantee that the computation of discrete logarithms in elliptic curves or IF p remains intractible for the future it is important to study cryptosystems based on alternative groups. A promising candidate, which was proposed by Buchmann and Williams [8], is the class group Cl(\D ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
Since nobody can guarantee that the computation of discrete logarithms in elliptic curves or IF p remains intractible for the future it is important to study cryptosystems based on alternative groups. A promising candidate, which was proposed by Buchmann and Williams [8], is the class group Cl(\Delta) of an imaginary quadratic order O \Delta . This ring is isomorphic to the endomorphism ring of a nonsupersingular elliptic curve over a finite field. While in the meantime there was found a subexponential algorithm for the computation of discrete logarithms in Cl(\Delta) [16], this algorithm only has running time L \Delta [ 1 2 ; c] and is far less efficient than the number field sieve with L p [ 1 3 ; c] to compute logarithms in IF p . Thus one may choose the parameters smaller to obtain the same level of security. It is an open question whether there is an L \Delta [ 1 3 ; c] algorithm to compute discrete logarithms in arbitrary Cl(\Delta). Recently there were proposed cry...
A Note on Cyclic Groups, Finite Fields, and the Discrete Logarithm Problem
 Applicable Algebra in Engineering, Communication and Computing
, 1992
"... We show how the discrete logarithm problem in some finite cyclic groups can easily be reduced to the discrete logarithm problem in a finite field. The cyclic groups that we consider are the set of points on a singular elliptic curve over a finite field, the set of points on a genus 0 curve over a fi ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
We show how the discrete logarithm problem in some finite cyclic groups can easily be reduced to the discrete logarithm problem in a finite field. The cyclic groups that we consider are the set of points on a singular elliptic curve over a finite field, the set of points on a genus 0 curve over a finite field given by the Pell equation, and certain subgroups of the general linear group.
On the complexity of computing discrete logarithms and factoring integers
 Algorithmic Number Theory Symposium (ANTS VII
, 1987
"... ..."
Discrete Logarithms in Finite Fields
, 1996
"... Given a finite field F q of order q, and g a primitive element of F q , the discrete logarithm base g of an arbitrary, nonzero y 2 F q is that integer x, 0 x q \Gamma 2, such that g x = y in F q . The security of many realworld cryptographic schemes depends on the difficulty of computing discr ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Given a finite field F q of order q, and g a primitive element of F q , the discrete logarithm base g of an arbitrary, nonzero y 2 F q is that integer x, 0 x q \Gamma 2, such that g x = y in F q . The security of many realworld cryptographic schemes depends on the difficulty of computing discrete logarithms in large finite fields. This thesis is a survey of the discrete logarithm problem in finite fields, including: some cryptographic applications (password authentication, the DiffieHellman key exchange, and the ElGamal publickey cryptosystem and digital signature scheme); Niederreiter's proof of explicit formulas for the discrete logarithm; and algorithms for computing discrete logarithms (especially Shank's algorithm, Pollard's aemethod, the PohligHellman algorithm, Coppersmith's algorithm in fields of order 2 n , and the Gaussian integers method for fields of prime order).