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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 100 (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...
Efficient generation of minimal length addition chains
 SIAM Journal on Computing
, 1999
"... Abstract. An addition chain for a positive integer n is a set 1 = a0
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Abstract. An addition chain for a positive integer n is a set 1 = a0 <a1 < ·· · <ar = n of integers such that for each i ≥ 1, ai = aj + ak for some k ≤ j<i. This paper is concerned with some of the computational aspects of generating minimal length addition chains for an integer n. Particular attention is paid to various pruning techniques that cut down the search time for such chains. Certain of these techniques are influenced by the multiplicative structure of n. Later sections of the paper present some results that have been uncovered by searching for minimal length addition chains.
Redundant trinomials for finite fields of characteristic 2
 Proceedings of ACISP 05, LNCS 3574
, 2005
"... Abstract. In this paper we introduce socalled redundant trinomials to represent elements of nite elds of characteristic 2. The concept is in fact similar to almost irreducible trinomials introduced by Brent and Zimmermann in the context of random numbers generators in [BZ 2003]. See also [BZ]. In f ..."
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Cited by 7 (0 self)
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Abstract. In this paper we introduce socalled redundant trinomials to represent elements of nite elds of characteristic 2. The concept is in fact similar to almost irreducible trinomials introduced by Brent and Zimmermann in the context of random numbers generators in [BZ 2003]. See also [BZ]. In fact, Blake et al. [BGL 1994, BGL 1996] and Tromp et al. [TZZ 1997] explored also similar ideas some years ago. However redundant trinomials have been discovered independently and this paper develops applications to cryptography, especially based on elliptic curves. After recalling well known techniques to perform e cient arithmetic in extensions of F2, we describe redundant trinomial bases and discuss how to implement them e ciently. They are well suited to build F2n when no irreducible trinomial of degree n exists. Depending on n ∈ [2, 10, 000] tests with NTL show that improvements for squaring and exponentiation are respectively up to 45 % and 25%. More attention is given to relevant extension degrees for doing elliptic and hyperelliptic curve cryptography. For this range, a scalar multiplication can be speeded up by a factor up to 15%. 1.
Computing special powers in finite fields
 e7 ← −e2 + yq; (e7 = −ypr0 + yq) 7: e8 ← −e0 + e4; (e8 = −r 2 0 + ypyq) 8: e9 ← e7e8; (e9 = (−ypr0 + yq)(−r 2 0 + ypyq)) 9: a1 ← e9 − e3 − e5; a0 ← e3 − e5 − yp; 10: a3 ← −e1 + e6; a2 ← −yp; a4 ← 0; a5 ← −yq; B Techniques for Reducing Partial Products in
, 2003
"... Abstract. We study exponentiation in nonprime finite fields with very special exponents such as they occur, for example, in inversion, primitivity tests, and polynomial factorization. Our algorithmic approach improves the corresponding exponentiation problem from about quadratic to about linear time ..."
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Abstract. We study exponentiation in nonprime finite fields with very special exponents such as they occur, for example, in inversion, primitivity tests, and polynomial factorization. Our algorithmic approach improves the corresponding exponentiation problem from about quadratic to about linear time. 1.
Methods for Regular VLSI Implementations of Wavelet Filters
, 1996
"... We investigate three approaches to VLSI implementation of wavelet filters. The direct form structure, the lattice form structure, and an algebraic structure are used to derive different architectures for wavelet filters. The algebraic structure exploits conjugacy properties in number fields. All app ..."
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We investigate three approaches to VLSI implementation of wavelet filters. The direct form structure, the lattice form structure, and an algebraic structure are used to derive different architectures for wavelet filters. The algebraic structure exploits conjugacy properties in number fields. All approaches are explained in detail for the Daubechies 4tab filters. We outline the philosophy of a design method for integrated circuits. Keywords: Wavelet filter, Daubechies wavelets, integrated circuits, VLSI. 1 INTRODUCTION We investigate different methods to implement orthonormal wavelet filters as integrated circuits. Many applications of these filters, e. g. in video coding, require high performance and cost effective implementations, which can be achieved by full custom VLSI implementations. Our main interest is to investigate the relation between the mathematical structure of the filters and their physical implementations. Wavelet filters can be realized in various ways. Basically,...