Philippe DUMAS | Institut National de Recherche en Informatique et Automatique (INRIA) (original) (raw)
Papers by Philippe DUMAS
Gazette Des Mathematiciens, 2011
Dmtcs, 2007
We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which origi... more We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which originates from differential cryptanalysis, using elementary tools from analysis and linear algebra, and more sophisticated tools from analytic number theory. We show that a probability distribution function describes the asymptotic behaviour of the rational series according to the length of words. As a result, the non-classical rational sequence, obtained by interpreting this rational series via the octal numeration system, admits an oscillating asymptotic behaviour for its first-order summation function. The distribution function and the periodic function are differentiable almost everywhere and not differentiable on an everywhere dense set. We compute the moments of the distribution function using a functional equation, which brings to light a self-similarity phenomenon, and we derive a Fourier representation of the periodic function using a Dirichlet series with vector coefficients. The method is applicable to a wide class of sequences rational with respect to a numeration system essentially under the condition that they admit a linear representation with nonnegative coefficients.
We give a randomized algorithm for estimating the score vector of matches between a text string o... more We give a randomized algorithm for estimating the score vector of matches between a text string of length N and a pattern string of length M; this is the vector obtained when the pattern is slid along the text, and the number of matches is counted for each position. The randomized algorithm takes deterministic time O«NjM)CODV(M)) where Conv(M) is the time for performing a convolution of two vectors of size M each. In particular, using the fast Fourier transform for convolut.ions and thus assuming that all arit.hmetic operations take constant time yields a time complexity of O(N log M). The algorithm finds an unbiased estimator of the scores, whose variance is particularly small for scores that are close to M, i.e., for approximate occurrences of the pattern in the text. No assumptions are made about the probabilistic characteristics of the input, or about the number of different symbols appearing in the text and in the pattern (i.e., the alphabet size need not be much smaller than M). The solution extends to string matching with classes, class complements, "never match" and "always match" symbols, to the weighted case and to higher dimensions. We also perform an experimental comparison to a naive string matching algorithm and to a classical algorithm by Baeza-Yates and Gannet, with the conclusion that our algorithm is faster for patterns of typical length a few thousands or more.
We establish a fundamental isomorphism between discrete-time balanced urn processes and certain o... more We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differen- tial systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over
Fast Software Encryption, 2004
We study the differential probability adp ⊕ of exclusive-or when differences are expressed using ... more We study the differential probability adp ⊕ of exclusive-or when differences are expressed using addition modulo 2 N. This function is important when analysing symmetric primitives that mix exclusive-or and addition-especially when addition is used to add in the round keys. (Such primitives include idea, Mars, rc6 and Twofish.) We show that adp ⊕ can be viewed as a formal rational series with a linear representation in base 8. This gives a linear-time algorithm for computing adp ⊕ , and enables us to compute several interesting properties like the fraction of impossible differentials, and the maximal differential probability for any given output difference. Finally, we compare our results with the dual results of Lipmaa and Moriai on the differential probability of addition modulo 2 N when differences are expressed using exclusive-or.
Algorithmica, 2001
We give a randomized algorithm in deterministic time O(N log M) for estimating the score vector o... more We give a randomized algorithm in deterministic time O(N log M) for estimating the score vector of matches between a text string of length N and a pattern string of length M , i.e., the vector obtained when the pattern is slid along the text, and the number of matches is counted for each position. A direct application is approximate string matching. The randomized algorithm bases on convolution to find an estimator of the scores and can be viewed as a randomization of an algorithm by Fischer and Paterson. The variance of our estimator is particularly small for scores that are close to M , i.e., for approximate occurrences of the pattern in the text. No assumption is made about the probabilistic characteristics of the input, or about the size of the alphabet. The solution extends to string matching with classes, class complements, "never match" and "always match" symbols, to the weighted case and to higher dimensions.
Discrete Mathematics & Theoretical Computer Science, Oct 31, 2007
We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which origi... more We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which originates from differential cryptanalysis, using tools from probability theory, and from analytic number theory. We derive a Fourier representation of a first-order summation function obtained by interpreting this rational series as a non-classical rational sequence via the octal numeration system. The method is applicable to a wide class of sequences rational with respect to a numeration system essentially under the condition ...
Gazette Des Mathematiciens, 2011
Dmtcs, 2007
We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which origi... more We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which originates from differential cryptanalysis, using elementary tools from analysis and linear algebra, and more sophisticated tools from analytic number theory. We show that a probability distribution function describes the asymptotic behaviour of the rational series according to the length of words. As a result, the non-classical rational sequence, obtained by interpreting this rational series via the octal numeration system, admits an oscillating asymptotic behaviour for its first-order summation function. The distribution function and the periodic function are differentiable almost everywhere and not differentiable on an everywhere dense set. We compute the moments of the distribution function using a functional equation, which brings to light a self-similarity phenomenon, and we derive a Fourier representation of the periodic function using a Dirichlet series with vector coefficients. The method is applicable to a wide class of sequences rational with respect to a numeration system essentially under the condition that they admit a linear representation with nonnegative coefficients.
We give a randomized algorithm for estimating the score vector of matches between a text string o... more We give a randomized algorithm for estimating the score vector of matches between a text string of length N and a pattern string of length M; this is the vector obtained when the pattern is slid along the text, and the number of matches is counted for each position. The randomized algorithm takes deterministic time O«NjM)CODV(M)) where Conv(M) is the time for performing a convolution of two vectors of size M each. In particular, using the fast Fourier transform for convolut.ions and thus assuming that all arit.hmetic operations take constant time yields a time complexity of O(N log M). The algorithm finds an unbiased estimator of the scores, whose variance is particularly small for scores that are close to M, i.e., for approximate occurrences of the pattern in the text. No assumptions are made about the probabilistic characteristics of the input, or about the number of different symbols appearing in the text and in the pattern (i.e., the alphabet size need not be much smaller than M). The solution extends to string matching with classes, class complements, "never match" and "always match" symbols, to the weighted case and to higher dimensions. We also perform an experimental comparison to a naive string matching algorithm and to a classical algorithm by Baeza-Yates and Gannet, with the conclusion that our algorithm is faster for patterns of typical length a few thousands or more.
We establish a fundamental isomorphism between discrete-time balanced urn processes and certain o... more We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differen- tial systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over
Fast Software Encryption, 2004
We study the differential probability adp ⊕ of exclusive-or when differences are expressed using ... more We study the differential probability adp ⊕ of exclusive-or when differences are expressed using addition modulo 2 N. This function is important when analysing symmetric primitives that mix exclusive-or and addition-especially when addition is used to add in the round keys. (Such primitives include idea, Mars, rc6 and Twofish.) We show that adp ⊕ can be viewed as a formal rational series with a linear representation in base 8. This gives a linear-time algorithm for computing adp ⊕ , and enables us to compute several interesting properties like the fraction of impossible differentials, and the maximal differential probability for any given output difference. Finally, we compare our results with the dual results of Lipmaa and Moriai on the differential probability of addition modulo 2 N when differences are expressed using exclusive-or.
Algorithmica, 2001
We give a randomized algorithm in deterministic time O(N log M) for estimating the score vector o... more We give a randomized algorithm in deterministic time O(N log M) for estimating the score vector of matches between a text string of length N and a pattern string of length M , i.e., the vector obtained when the pattern is slid along the text, and the number of matches is counted for each position. A direct application is approximate string matching. The randomized algorithm bases on convolution to find an estimator of the scores and can be viewed as a randomization of an algorithm by Fischer and Paterson. The variance of our estimator is particularly small for scores that are close to M , i.e., for approximate occurrences of the pattern in the text. No assumption is made about the probabilistic characteristics of the input, or about the size of the alphabet. The solution extends to string matching with classes, class complements, "never match" and "always match" symbols, to the weighted case and to higher dimensions.
Discrete Mathematics & Theoretical Computer Science, Oct 31, 2007
We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which origi... more We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which originates from differential cryptanalysis, using tools from probability theory, and from analytic number theory. We derive a Fourier representation of a first-order summation function obtained by interpreting this rational series as a non-classical rational sequence via the octal numeration system. The method is applicable to a wide class of sequences rational with respect to a numeration system essentially under the condition ...