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Speeding Up The Computations On An Elliptic Curve Using AdditionSubtraction Chains
 Theoretical Informatics and Applications
, 1990
"... We show how to compute x k using multiplications and divisions. We use this method in the context of elliptic curves for which a law exists with the property that division has the same cost as multiplication. Our best algorithm is 11.11% faster than the ordinary binary algorithm and speeds up acco ..."
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Cited by 97 (4 self)
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We show how to compute x k using multiplications and divisions. We use this method in the context of elliptic curves for which a law exists with the property that division has the same cost as multiplication. Our best algorithm is 11.11% faster than the ordinary binary algorithm and speeds up accordingly the factorization and primality testing algorithms using elliptic curves. 1. Introduction. Recent algorithms used in primality testing and integer factorization make use of elliptic curves defined over finite fields or Artinian rings (cf. Section 2). One can define over these sets an abelian law. As a consequence, one can transpose over the corresponding groups all the classical algorithms that were designed over Z/NZ. In particular, one has the analogue of the p \Gamma 1 factorization algorithm of Pollard [29, 5, 20, 22], the Fermatlike primality testing algorithms [1, 14, 21, 26] and the public key cryptosystems based on RSA [30, 17, 19]. The basic operation performed on an elli...
An Implementation of the Number Field Sieve
 EXPERIMENTAL MATHEMATICS
, 1995
"... The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical and lattice sieving, the block Lanczos method and a new square root algorith ..."
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The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical and lattice sieving, the block Lanczos method and a new square root algorithm. Finally some data on factorizations obtained with this implementation are listed, including the record factorization of 12^151  1.
Interrelation Among RSA Security, Strong Primes, and Factoring
"... Abstract The security of RSA depends critically on the inability of an adversary to compute private key from the public key. The problem of computing private key from public key is equivalent to the problem of factoring n into its prime factors. Therefore it is important for the RSA user to select ..."
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Abstract The security of RSA depends critically on the inability of an adversary to compute private key from the public key. The problem of computing private key from public key is equivalent to the problem of factoring n into its prime factors. Therefore it is important for the RSA user to select prime numbers in such a way that the problem of factoring n is computationally infeasible for an adversary. Choosing the factors as “strong primes ” has been recommended as a way of maximizing the difficulty of factoring n. There are such arguments that say that strong primes are needed to protect against factoring attacks. The purpose of this paper is to extend this scrutiny by carefully examining the common recommendation that one should use strong primes when constructing keys for an RSA. We review the RSA security and describe that this problem can be solved using more key size, more factors or choosing strong primes numbers.