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15
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 109 (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...
Boltzmann Samplers For The Random Generation Of Combinatorial Structures
 Combinatorics, Probability and Computing
, 2004
"... This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combina ..."
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Cited by 107 (3 self)
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This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class  an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on realarithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.
An asymptotic theory for CauchyEuler differential equations with applications to the analysis of algorithms
, 2002
"... CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We ..."
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Cited by 27 (12 self)
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CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We study in this paper the most general framework for CauchyEuler equations and propose an asymptotic theory that covers almost all applications where CauchyEuler equations appear. Our approach is very general and requires almost no background on differential equations. Indeed the whole theory can be stated in terms of recurrences instead of functions. Old and new applications of the theory are given. New phase changes of limit laws of new variations of quicksort are systematically derived. We apply our theory to about a dozen of diverse examples in quicksort, binary search trees, urn models, increasing trees, etc.
COMBINATORIAL MODELS OF CREATION–ANNIHILATION
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 65 (2011), ARTICLE B65C
, 2011
"... Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB − BA = 1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator ..."
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Cited by 18 (3 self)
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Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB − BA = 1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator polynomials in such an algebra. The connection is achieved through suitable labelled graphs, or “diagrams”, that are composed of elementary “gates”. In this way, many normal form evaluations can be systematically obtained, thanks to models that involve set partitions, permutations, increasing trees, as well as weighted lattice paths. Extensions to qanalogues, multivariate frameworks, and urn models are also briefly discussed.
LambdaUpsilonOmega  The 1989 Cookbook
, 1989
"... LambdaUpsilonOmega ( \Upsilon\Omega ) is a research tool designed to assist the average case analysis of some well defined classes of algorithms and data structures. This cookbook consists of an informal introduction to the system together with eighteen examples of programmes that are automatica ..."
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Cited by 18 (6 self)
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LambdaUpsilonOmega ( \Upsilon\Omega ) is a research tool designed to assist the average case analysis of some well defined classes of algorithms and data structures. This cookbook consists of an informal introduction to the system together with eighteen examples of programmes that are automatically analyzed. Amongst the applications treated here, we find: addition chains, quantitative concurrency analysis of simple systems, symbolic manipulation algorithms such as formal differentiation, simplification and rewriting systems, as well as combinatorial models including various tree and permutation statistics and functional graphs with applications to integer factorisation.
A Calculus for the Random Generation of Combinatorial Structures
, 1993
"... A systematic approach to the random generation of labelled combinatorial objects is presented. It applies to structures that are decomposable, i.e., formally specifiable by grammars involving set, sequence, and cycle constructions. A general strategy is developed for solving the random generation pr ..."
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Cited by 12 (2 self)
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A systematic approach to the random generation of labelled combinatorial objects is presented. It applies to structures that are decomposable, i.e., formally specifiable by grammars involving set, sequence, and cycle constructions. A general strategy is developed for solving the random generation problem with two closely related types of methods: for structures of size n, the boustrophedonic algorithms exhibit a worstcase behaviour of the form O(n log n); the sequential algorithms haveworst case O(n²), while offering good potential for optimizations in the average case. (Both methods appeal to precomputed numerical tables of linear size.) A companion calculus permits to systematically compute the average case cost of the sequential generation algorithm associated to a given specification. Using optimizations dictated by the cost calculus, several random generation algorithms are developed, based on the sequential principle; most of them have expected complexity 1/2 n log n,thu...
Object Grammars and Random Generation
, 1997
"... This article presents a new systematic approach for the uniform random generation of combinatorial objects. The method is based on the notion of object grammars which are recursive descriptions of objects generalizing contextfree grammars. The application of particular valuations to these grammars ..."
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Cited by 12 (2 self)
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This article presents a new systematic approach for the uniform random generation of combinatorial objects. The method is based on the notion of object grammars which are recursive descriptions of objects generalizing contextfree grammars. The application of particular valuations to these grammars allows to reach to enumeration and random generation of objects according to non algebraic parameters.
A Calculus of Random Generation
, 1993
"... A systematic approach to the random generation of labelled combinatorial objects is presented. It applies to structures that are decomposable, i.e., formally specifiable by grammars involving union, product, set, sequence, and cycle constructions. ..."
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Cited by 7 (0 self)
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A systematic approach to the random generation of labelled combinatorial objects is presented. It applies to structures that are decomposable, i.e., formally specifiable by grammars involving union, product, set, sequence, and cycle constructions.
A generic approach for the unranking of labeled combinatorial classes
 Random Structures & Algorithms 19 (2001), no. 34, 472–497, Analysis of algorithms (Krynica Morska
, 2000
"... ABSTRACT: In this article, we design and analyze algorithms that solve the unranking problem (i.e., generating a combinatorial structure of size, n given its rank) for a large collection of labeled combinatorial classes, those that can be built using operators like unions (+), products (⋆), sequence ..."
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Cited by 7 (0 self)
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ABSTRACT: In this article, we design and analyze algorithms that solve the unranking problem (i.e., generating a combinatorial structure of size, n given its rank) for a large collection of labeled combinatorial classes, those that can be built using operators like unions (+), products (⋆), sequences, sets, cycles, and substitutions. We also analyze the performance of these algorithms and show that the worstcase is ��n 2 � (��n log n � if the socalled boustrophedonic order is used), and provide an algebra for the analysis of the average performance and higherorder moments together with a few examples of its application. © 2001 John Wiley
Analysis of DPA Countermeasures Based on Randomizing the Binary Algorithm
, 2003
"... One of the major threats to the security of cryptosystems nowadays is the information leaked through side channels. For instance, power analysis attacks have been successfully mounted on cryptosystems embedded into small devices such as smart cards. ..."
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Cited by 5 (2 self)
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One of the major threats to the security of cryptosystems nowadays is the information leaked through side channels. For instance, power analysis attacks have been successfully mounted on cryptosystems embedded into small devices such as smart cards.