Results 1  10
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22
Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
SInteger Dynamical Systems: Periodic Points
 J. Reine Angew. Math
"... We associate via duality a dynamical system to each pair (R S , #), where R S is the ring of Sintegers in an Afield k, and # is an element of R S \{0}. ..."
Abstract

Cited by 34 (22 self)
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We associate via duality a dynamical system to each pair (R S , #), where R S is the ring of Sintegers in an Afield k, and # is an element of R S \{0}.
A Sender Verifiable MixNet and a New Proof of a Shuffle, Cryptology ePrint Archive, Report 2005/137
, 2005
"... Abstract. We introduce the first El Gamal based mixnet in which each mixserver partially decrypts and permutes its input, i.e., no reencryption is necessary. An interesting property of the construction is that a sender can verify noninteractively that its message is processed correctly. We call t ..."
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Cited by 8 (1 self)
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Abstract. We introduce the first El Gamal based mixnet in which each mixserver partially decrypts and permutes its input, i.e., no reencryption is necessary. An interesting property of the construction is that a sender can verify noninteractively that its message is processed correctly. We call this sender verifiability. The mixnet is provably UCsecure against static adversaries corrupting any minority of the mixservers. The result holds under the decision DiffieHellman assumption, and assuming an ideal bulletin board and an ideal zeroknowledge proof of knowledge of a correct shuffle. Then we construct the first proof of a decryptionpermutation shuffle, and show how this can be transformed into a zeroknowledge proof of knowledge in the UCframework. The protocol is sound under the strong RSAassumption and the discrete logarithm assumption. Our proof of a shuffle is not a variation of existing methods. It is based on a novel idea of independent interest, and we argue that it is at least as efficient as previous constructions. 1
Sur un problème de Gelfond: la somme des chiffres des nombres premiers
, 2010
"... In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of prime numbers written in a basis q> 2 is equidistributed in arithmetic progressions (except for some well known degenerate cases). We prove also that the sequence.˛sq.p/ / where p runs through the p ..."
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Cited by 7 (1 self)
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In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of prime numbers written in a basis q> 2 is equidistributed in arithmetic progressions (except for some well known degenerate cases). We prove also that the sequence.˛sq.p/ / where p runs through the prime numbers is equidistributed modulo 1 if and only if ˛ 2 � n �.
Segmented Information Dispersal
 R. Cipolla on Computer Vision, Cambridge (England
, 1996
"... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii I Segmented Information Dispersal . . . . . . . . . . . . . . . . . . . . . . . 1 A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 B. A Combinatorial Design Formulation . . . . . . . . . . . ..."
Abstract

Cited by 6 (3 self)
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii I Segmented Information Dispersal . . . . . . . . . . . . . . . . . . . . . . . 1 A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 B. A Combinatorial Design Formulation . . . . . . . . . . . . . . . . . . 8 C. CollisionFree Digraphs (CFDs) . . . . . . . . . . . . . . . . . . . . . 9 1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. CFDs and SID Solutions . . . . . . . . . . . . . . . . . . . . . . . 12 3. The Size of CFDs . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 D. The SDS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1. Constructing CFDs . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2. SDSbased SID Solutions . . . . . . . . . . . . . . . . . . . . . . . 22 E. The GR Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1. The Basic Construction . . . . . . . . . . . . . . . . . . ...
Distribution of generalized Fermat prime numbers
 Math. Comp
, 1999
"... Abstract. Numbers of the form Fb,n = b2n +1 are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form 2m −1. The theoretical distributions of GFN primes, for fixed n, are derived and c ..."
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Cited by 5 (3 self)
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Abstract. Numbers of the form Fb,n = b2n +1 are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form 2m −1. The theoretical distributions of GFN primes, for fixed n, are derived and compared to the actual distributions. The predictions are surprisingly accurate and can be used to support Bateman and Horn’s quantitative form of “Hypothesis H” of Schinzel and Sierpiński. A list of the current largest known GFN primes is included. 1.
On the pdivisibility of Fermat quotients
 Math.Comp.66 (1997), 1353–1365. http://users.utu.fi/taumets/fermat/fermat.htm. MR 97i:11003
"... Abstract. The authors carried out a numerical search for Fermat quotients Qa =(a p−1 −1)/p vanishing mod p, for1≤a≤p−1, up to p<10 6. This article reports on the results and surveys the associated theoretical properties of Qa. The approach of fixing the prime p rather than the base a leads to some a ..."
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Cited by 5 (0 self)
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Abstract. The authors carried out a numerical search for Fermat quotients Qa =(a p−1 −1)/p vanishing mod p, for1≤a≤p−1, up to p<10 6. This article reports on the results and surveys the associated theoretical properties of Qa. The approach of fixing the prime p rather than the base a leads to some aspects of the theory apparently not published before. 1.
Large Sophie Germain primes
 Math. Comp
, 1996
"... Abstract. If P is a prime and 2P+1 is also prime, then P is a Sophie Germain prime. In this article several new Sophie Germain primes are reported, which are the largest known at this time. The search method and the expected search times are discussed. 1. ..."
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Cited by 4 (1 self)
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Abstract. If P is a prime and 2P+1 is also prime, then P is a Sophie Germain prime. In this article several new Sophie Germain primes are reported, which are the largest known at this time. The search method and the expected search times are discussed. 1.
Eigenvalue statistics in quantum ideal gases. Emerging applications of number theory
 IMA Vol. Math. Appl
, 1996
"... The eigenvalue statistics of quantum ideal gases with single particle energies en = n α are studied. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density. For integer α> 1 one expects and finds nu ..."
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Cited by 3 (0 self)
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The eigenvalue statistics of quantum ideal gases with single particle energies en = n α are studied. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density. For integer α> 1 one expects and finds number theoretic degeneracies and deviations from the Poissonian spacing distribution. By semiclassical arguments, the length spectrum of the classical system is shown to be related to sums of integers to the power α/(α − 1). In particular, for α = 3/2, the periodic orbits are related to sums of cubes, for which one again expects number theoretic degeneracies, with consequences for the two point correlation function. 1