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Asymptotic Behaviour of the Degree of Regularity of SemiRegular Polynomial Systems
 In MEGA’05, 2005. Eighth International Symposium on Effective Methods in Algebraic Geometry
"... We compute the asymptotic expansion of the degree of regularity for overdetermined semiregular sequences of algebraic equations. This degree implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] ..."
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Cited by 42 (24 self)
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We compute the asymptotic expansion of the degree of regularity for overdetermined semiregular sequences of algebraic equations. This degree implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] family of algorithms used by the cryptographic community. 1 Motivations and Results The worstcase complexity of Gröbner bases has been the object of extensive studies. In the most general case, it is well known after work by Mayr and Meyer that the complexity is doubly exponential in the number of variables. For subclasses of polynomial systems, the complexity may be much smaller. Of particular importance is the class of regular sequences of polynomials. There, it is known that after a generic linear change of variables the complexity of the computation for the degreereverselexicographic order is simply exponential in the number of variables. Moreover, in characteristic 0, these systems are generic. Our goal is to give similar complexity bounds for overdetermined systems, for a class of systems that we
Profiles of random trees: correlation and width of random recursive trees and binary search trees
 ADVANCES IN APPLIED PROBABILITY
, 2004
"... We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees, which undergo sharp signchanges when one level is fixed and the other one is varying. An asymptotic estimate for the expected width is also derived. ..."
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Cited by 17 (6 self)
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We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees, which undergo sharp signchanges when one level is fixed and the other one is varying. An asymptotic estimate for the expected width is also derived.
J.M.: All in the XL Family: Theory and Practice
 ICISC 2004. LNCS
, 2005
"... Abstract. The XL (eXtended Linearization) equationsolving algorithm belongs to the same extended family as the advanced Gröbner Bases methods F4/F5. XLanditsrelativesmaybeusedasdirectattacks against multivariate PublicKey Cryptosystems and as final stages for many “algebraic cryptanalysis ” used t ..."
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Cited by 13 (8 self)
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Abstract. The XL (eXtended Linearization) equationsolving algorithm belongs to the same extended family as the advanced Gröbner Bases methods F4/F5. XLanditsrelativesmaybeusedasdirectattacks against multivariate PublicKey Cryptosystems and as final stages for many “algebraic cryptanalysis ” used today. We analyze the applicability and performance of XL and its relatives, particularly for generic systems of equations over mediumsized finite fields. In examining the extended family of Gröbner Bases and XL from theoretical, empirical and practical viewpoints, we add to the general understanding of equationsolving. Moreover, we give rigorous conditions for the successful termination of XL, Gröbner Bases methods and relatives. Thus we have a better grasp of how such algebraic attacks should be applied. We also compute revised security estimates for multivariate cryptosystems. For example, the schemes SFLASH v2 and HFE Challenge 2 are shown to be unbroken by XL variants.
Asymptotics of Poisson approximation to random discrete distributions: an analytic approach
 Advances in Applied Probability
, 1998
"... this paper, we shall describe the asymptotic behaviors of several distances of Poisson approximation to a wide class of discrete distributions covering many examples from number theory, combinatorics and arithmetic semigroups. Our aim is to show that whenever (analytic) generating functions of the r ..."
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Cited by 13 (9 self)
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this paper, we shall describe the asymptotic behaviors of several distances of Poisson approximation to a wide class of discrete distributions covering many examples from number theory, combinatorics and arithmetic semigroups. Our aim is to show that whenever (analytic) generating functions of the random variables in question are available, complexanalytic methods can be used to derive precise asymptotic results for the five distances above. Actually, we shall consider the following generalized distances: let ff ? 0 be a fixed positive number, (X; Y ) = FM (X; Y ) = (X; Y ) = sup K (X; Y ) = sup M (X; Y ) = jP(X = j) \Gamma P(Y = j) Note that d TV = d M . Besides the case ff = 1 (and ff = 1=2 for d M ), only the case d TV was previously studied by Franken [39] for Poisson approximation to the sum of independent but not identically distributed Bernoulli random variables. We take these quantities as our measures of degree of nearness of Poisson approximation, some of which may be interpreted as certain norms in suitable space as many authors did (cf. [12, 22, 23, 74, 96]). For a large class of discrete distributions, we shall derive an asymptotic main term together with an error estimate for each of these distances. Our results are thus "approximation theorems" rather than "limit theorems". The common form of the underlying structure of these distributions suggests the study of an analytic scheme as we did previously for normal approximation and large deviations (cf. [53, 54]). Many concrete examples from probabilistic number theory and combinatorial structures will justify the study of this scheme. Our treatment being completely general, many extensions can be further pursued with essentially the same line of methods. We shall di...
Analysis in Distribution of Two Randomized Algorithms for Finding the Maximum in a Broadcast Communication Model
, 2002
"... The limit laws of three cost measures are derived of two algorithms for finding the maximum in a singlechannel broadcast communication model. Both algorithms use coin flips and comparisons. Besides the ubiquitous normal limit law, the Dickman distribution also appears in a natural way. The method o ..."
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Cited by 5 (4 self)
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The limit laws of three cost measures are derived of two algorithms for finding the maximum in a singlechannel broadcast communication model. Both algorithms use coin flips and comparisons. Besides the ubiquitous normal limit law, the Dickman distribution also appears in a natural way. The method of proof proceeds along the line via the method of moments and the "asymptotic transfers", which roughly bridges the asymptotics of the "conquering cost of the subproblems" and that of the total cost. Such a general approach has proved very fruitful for a number of problems in the analysis of recursive algorithms. 1
LoadBalancing performance of consistent hashing: asymptotic analysis of random node join
 IN IEEE/ACM TRANS. ON NETWORKING
, 2007
"... Balancing of structured peertopeer graphs, including their zone sizes, has recently become an important topic of distributed hash table (DHT) research. To bring analytical understanding into the various peerjoin mechanisms based on consistent hashing, we study how zonebalancing decisions made d ..."
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Cited by 4 (1 self)
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Balancing of structured peertopeer graphs, including their zone sizes, has recently become an important topic of distributed hash table (DHT) research. To bring analytical understanding into the various peerjoin mechanisms based on consistent hashing, we study how zonebalancing decisions made during the initial sampling of the peer space affect the resulting zone sizes and derive several asymptotic bounds for the maximum and minimum zone sizes that hold with high probability. Several of our results contradict those of prior work and shed new light on the theoretical performance limitations of consistent hashing. We use simulations to verify our models and compare the performance of the various methods using the example of recently proposed de
Phase Changes in Random Recursive Structures and Algorithms
 In Proceedings of the Workshop on Probability with Applications to Finance and Insurance (Hong Kong
, 2002
"... A brief survey, based mainly on my recent work with coauthors, is given of the different types of phase changes (or transitions) appearing in random discrete structures and in analysis of algorithms with a recursive character. ..."
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Cited by 4 (2 self)
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A brief survey, based mainly on my recent work with coauthors, is given of the different types of phase changes (or transitions) appearing in random discrete structures and in analysis of algorithms with a recursive character.
Uniform asymptotics of Poisson approximation to the Poissonbinomial distribution. Manuscript submitted for publication
, 2008
"... New uniform asymptotic approximations with error bounds are derived for a generalized total variation distance of Poisson approximations to the Poissonbinomial distribution. The method of proof is also applicable to other Poisson approximation problems. MSC 2000 Subject Classifications: Primary 62E ..."
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Cited by 1 (1 self)
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New uniform asymptotic approximations with error bounds are derived for a generalized total variation distance of Poisson approximations to the Poissonbinomial distribution. The method of proof is also applicable to other Poisson approximation problems. MSC 2000 Subject Classifications: Primary 62E17; secondary 60C05. 1