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Four-Variable Expanders Over The Prime Fields

Publications associées (40)

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The Security of DSA and ECDSA

Serge Vaudenay

DSA and ECDSA are well established standards for digital signature based on the discrete logarithm problem. In this paper we survey known properties, certification issues regarding the public parameters, and security proofs. ECDSA also includes a standard ...

2003

Irregular primes and Cyclotomic Invariants to Twelve Million

Mohammad Amin Shokrollahi

Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to twelve million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes (Shokrollahi (1996)). ...

2001

Selecting cryptographic key sizes

Arjen Lenstra

In this article we offer guidelines for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm-based cryptosystems both over finite fields and over groups of elliptic curves over prime fields. Our recommendations are based ...

Springer Verlag2001

Deciding properties of number fields without factoring

Mohammad Amin Shokrollahi

The polynomial time algorithm of Lenstra, Lenstra, and Lovász [15] for factoring integer polynomials and variants thereof have been widely used to show that various computational problems in number theory have polynomial time solutions. Among them is the p ...

1997

Let

p

be an odd prime and

zeta_p

be a primitive pth root of unity over

smallBbbQ

. The Galois group

G

K:=smallBbbQ(zeta_p)

over

smallBbbQ

is a cyclic group of order

p-1

. The integral group ring

smallBbbZ[G]

contains the Stickelberger idea ...

1996

The Factorization of the ninth Fermat Number

Arjen Lenstra

In this paper we exhibit the full prime factorization of the ninth Fermat number F9 = 2(512) + 1. It is the product of three prime factors that have 7, 49, and 99 decimal digits. We found the two largest prime factors by means of the number field sieve, wh ...

American Mathematical Society1993

Factoring multivariate polynomials over algebraic number fields

Arjen Lenstra

We present an algorithm to factor multivariate polynomials over algebraic number fields that is polynomial-time in the degrees of the polynomial to be factored. The algorithm is an immediate generalization of the polynomial-time algorithm to factor univari ...

1987

Factoring multivariate polynomials over finite fields

Arjen Lenstra

This paper describes an algorithm for the factorization of multivariate polynomials with coefficients in a finite field that is polynomial-time in the degrees of the polynomial to be factored. The algorithm makes use of a new basis reduction algorithm for ...

1983

Factoring multivariate integral polynomials

Arjen Lenstra

An algorithm is presented to factorize polynomials in several variables with integral coefficients that is polynomial-time in the degrees of the polynomial to be factored, for any fixed number of variables. The algorithm generalizes the algorithm presented ...

1983

Lattices and factorization of polynomials

Arjen Lenstra

Discusses an algorithm to factorize univariate polynomials over an algebraic number field which has been implemented in Algol-68 on a CDC-Cyber 170-750 computer ...

Association for Computing Machinery1981

Let

p

be an odd prime and

zeta_p

be a primitive pth root of unity over

smallBbbQ

. The Galois group

G

K:=smallBbbQ(zeta_p)

over

smallBbbQ

is a cyclic group of order

p-1

. The integral group ring

smallBbbZ[G]

contains the Stickelberger idea ...

1996