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22
Matrix Transformation is Complete for the Average Case
 SIAM Journal on Computing
, 1995
"... In the theory of worst case complexity, NP completeness is used to establish that, for all practical purposes, the given NP problem is not decidable in polynomial time. In the theory of average case complexity, average case completeness is supposed to play the role of NP completeness. However, the a ..."
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Cited by 20 (1 self)
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In the theory of worst case complexity, NP completeness is used to establish that, for all practical purposes, the given NP problem is not decidable in polynomial time. In the theory of average case complexity, average case completeness is supposed to play the role of NP completeness. However, the average case reduction theory is still at an early stage, and only a few average case complete problems are known. We present the first algebraic problem complete for the average case under a natural probability distribution. The problem is this: Given a unimodular matrix X of integers, a set S of linear transformations of such unimodular matrices and a natural number n, decide if there is a product of n (not necessarily different) members of S that takes X to the identity matrix. 1 Introduction The theory of NP completeness is very useful. It allows one to establish that certain NP problems are NP complete and therefore, for all practical purposes, not decidable in polynomial time (PTime)....
Dynamical Analysis of a Class of Euclidean Algorithms
"... We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properti ..."
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Cited by 17 (4 self)
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We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise averagecase analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcdlike algorithms together with new results regarding the probable behaviour of their cost functions. 1
Average BitComplexity of Euclidean Algorithms
 Proceedings ICALP’00, Lecture Notes Comp. Science 1853, 373–387
, 2000
"... We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set ..."
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Cited by 15 (6 self)
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We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of an entire class of gcdlike algorithms. Keywords: Averagecase Analysis of algorithms, BitComplexity, Euclidean Algorithms, Dynamical Systems, Ruelle operators, Generating Functions, Dirichlet Series, Tauberian Theorems. 1 Introduction Motivations. Euclid's algorithm was analysed first in the worst case in 1733 by de Lagny, then in the averagecase around 1969 independently by Heilbronn [12] and Dixon [6], and finally in distribution by Hensley [13] who proved in 1994 that the Eu...
Digits and Continuants in Euclidean Algorithms. Ergodic versus Tauberian Theorems
, 2000
"... We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviou ..."
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Cited by 14 (5 self)
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We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters digits and continuants that intervene in an entire class of gcdlike algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.
Towards Practical Deterministic WriteAll Algorithms
 IN PROC., 13TH ACM SYMP. ON PARALLEL ALGORITHMS AND ARCHITECTURES, 2001
, 2001
"... The problem of performing t tasks on n asynchronous or undependable processors is a basic problem in parallel and distributed computing. We consider an abstraction of this problem called the WriteAl l problemusing n processors write 1's into all locations of an array of size t. The most e#cient ..."
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Cited by 9 (4 self)
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The problem of performing t tasks on n asynchronous or undependable processors is a basic problem in parallel and distributed computing. We consider an abstraction of this problem called the WriteAl l problemusing n processors write 1's into all locations of an array of size t. The most e#cient known deterministic asynchronous algorithms for this problem are due to Anderson and Woll. The first class of algorithms has work complexity of O(t ), for n t and any #>0, and they are the best known for the full range of processors (n = t). To schedule the work of the processors, the algorithms use lists of q permutations on [q](q n) that have certain combinatorial properties. Instantiating such an algorithm for a specific # either requires substantial preprocessing (exponential in 1/# )to find the requisite permutations, or imposes a prohibitive constant (exponential in 1/# ) hidden by the asymptotic analysis. The second class deals with the specific case of t = n 2, and these algorithms have work complexity of O(t log t). They also use lists of permutations with the same combinatorial properties. However instantiating these algorithms requires exponential in n preprocessing to find the permutations. To alleviate this costly instantiation Kanellakis and Shvartsman proposed a simple way of computing the permutations. They conjectured that their construction has the desired properties but they provided no analysis. In this paper
Speeding up Subgroup Cryptosystems
, 2003
"... proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. R.A. van Santen, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op woensdag 4 juni 2003 om 16.00 uur door ..."
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Cited by 8 (1 self)
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proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. R.A. van Santen, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op woensdag 4 juni 2003 om 16.00 uur door
Fast and Secure Elliptic Curve Scalar Multiplication over Prime Fields Using Special Addition Chains
, 2006
"... In this paper, we propose a new fast and secure point multiplication algorithm. It is based on a particular kind of addition chains involving only additions (no doubling), providing a natural protection against side channel attacks. Moreover, we propose new addition formulae that take into account t ..."
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Cited by 7 (1 self)
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In this paper, we propose a new fast and secure point multiplication algorithm. It is based on a particular kind of addition chains involving only additions (no doubling), providing a natural protection against side channel attacks. Moreover, we propose new addition formulae that take into account the specific structure of those chains making point multiplication very e#cient.
Continued fractions from Euclid to the present day
, 2000
"... this paper to indicate how continued fractions are relevant to ..."
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Cited by 6 (0 self)
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this paper to indicate how continued fractions are relevant to
Comparison of Simple Power Analysis Attack Resistant Algorithms for an Elliptic Curve
"... Abstract — Side channel attacks such as Simple Power Analysis(SPA) attacks provide a new challenge for securing algorithms from an attacker. Algorithms for elliptic curve point scalar multiplication such as the double and add method are prone to these attacks. The protected double and add algorithm ..."
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Cited by 5 (2 self)
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Abstract — Side channel attacks such as Simple Power Analysis(SPA) attacks provide a new challenge for securing algorithms from an attacker. Algorithms for elliptic curve point scalar multiplication such as the double and add method are prone to these attacks. The protected double and add algorithm provides a simple solution to this problem but is costly in terms of performance. Another class of algorithm for point scalar multiplication that makes use of special addition chains can be used to protect against SPA attacks. A reconfigurable architecture for a cryptographic processor is presented and a number of algorithms for point multiplication are implemented and compared. These algorithms have a degree of parallism within their operations where a number of multiplications can be executed in parallel. Sophisticated scheduling techniques can exploit this parallelism in order to optimize the performance of the calculation. Post place and route results for the processor are given. Index Terms — Cryptography, ellitipic curves, side channel attacks, scheduling techniques I.
A Unifying Framework for the Analysis of a Class of Euclidean Algorithms
 the proceedings of LATIN'2000, LNCS
, 2000
"... . We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on p ..."
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Cited by 4 (2 self)
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. We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise averagecase analyses of four algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods provide a unifying framework for the analysis of an entire class of gcdlike algorithms together with new results regarding the probable behaviour of their cost functions. 1 Introduction Euclid's algorithm, discovered as early as 300BC, was analysed first in the worst case in 1733 by de Lagny, then in the averagecase around 1969 independently by Heilbronn [8] and Dixon [5], and finally in distribut...