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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 ..."
Abstract

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...
Pippenger's Exponentiation Algorithm
, 2002
"... Pippenger's exponentiation algorithm computes a power, or a product of powers, or a sequence of powers, or a sequence of products of powers, with very few multiplications. Pippenger's algorithm was published twentyve years ago, but it is still not widely understood or appreciated, althoug ..."
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Cited by 8 (0 self)
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Pippenger's exponentiation algorithm computes a power, or a product of powers, or a sequence of powers, or a sequence of products of powers, with very few multiplications. Pippenger's algorithm was published twentyve years ago, but it is still not widely understood or appreciated, although certain parts of it have recently been reinvented, republished, and popularized. This paper is an exposition of the state of the art in generic exponentiation algorithmsin particular, Pippenger's algorithm. 1.