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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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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 \Gamma 1. AMS Subject Classification (1991): 11Y05, 11Y40 Keywords & Phrases: number field sieve, factorization Note: This report has been submitted for publication elsewhere. Note: This research is funded by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for Scientific Research, NWO) through the Stichting Mathematisch Centrum (SMC), under grant number 611307022. 1. Introduction The Number Field Sieve (NFS)  introduced in 1988 by John Pollard [19]  is the asymptotically fastest known algorithm for factoring inte...
Computational Number Theory at CWI in 19701994
, 1994
"... this paper we present a concise survey of the research in Computational ..."