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Simple Constructions of Almost kwise Independent Random Variables
, 1992
"... We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the dist ..."
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Cited by 271 (41 self)
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We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ɛ < 1/(k log n)). An additional advantage of our constructions is their simplicity.
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 167 (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.
The discrete logarithm modulo a composite hides O(n) bits
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1993
"... In this paper we consider the oneway function fg�N(X) =g X (modN), where N is a Blum integer. We prove that under the commonly assumed intractability of factoring Blum integers, all its bits are individually hard, and the lower as well as upper halves of them are simultaneously hard. As a result, f ..."
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Cited by 29 (1 self)
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In this paper we consider the oneway function fg�N(X) =g X (modN), where N is a Blum integer. We prove that under the commonly assumed intractability of factoring Blum integers, all its bits are individually hard, and the lower as well as upper halves of them are simultaneously hard. As a result, fg�N can be used in efficient pseudorandom bit generators and multibit commitment schemes, where messages can be drawn according to arbitrary probability distributions.
Fast Generation of Prime Numbers and Secure PublicKey Cryptographic Parameters
, 1995
"... A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudoprime of the same size that passes the MillerRabin test for only one base. The ..."
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Cited by 21 (0 self)
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A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudoprime of the same size that passes the MillerRabin test for only one base. Therefore our algorithm is even faster than presentlyused algorithms for generating only pseudoprimes because several MillerRabin tests with independent bases must be applied for achieving a sufficient confidence level. Heuristic arguments suggest that the generated primes are close to uniformly distributed over the set of primes in the specified interval. Security constraints on the prime parameters of certain cryptographic systems are discussed, and in particular a detailed analysis of the iterated encryption attack on the RSA publickey cryptosystem is presented. The prime generation algorithm can easily be modified to generate nearly random primes or RSAmoduli that satisfy t...
Fast BitLevel, WordLevel and Parallel Arithmetic in Finite Fields for Elliptic Curve Cryptosystems
, 1998
"... ..."
Uses of Randomness in Computation
, 1994
"... Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and perfect hashing. Complexity theory usually considers the worst ..."
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Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and perfect hashing. Complexity theory usually considers the worstcase behaviour of deterministic algorithms, but it can also consider averagecase behaviour if it is assumed that the input data is drawn randomly from a given distribution. Rabin popularised the idea of &quot;probabilistic &quot; algorithms, where randomness is incorporated into the algorithm instead of being assumed in the input data. Yao showed that there is a close connection between the complexity of probabilistic algorithms and the averagecase complexity of deterministic algorithms. We give examples of the uses of randomness in computation, discuss the contributions of Rabin, Yao and others, and mention some open questions.
Easy numbers for the Elliptic Curve Primality Proving Algorithm
, 1992
"... We present some new classes of numbers that are easier to test for primality with the Elliptic Curve Primality Proving algorithm than average numbers. It is shown that this is the case for about half the numbers of the Cunningham project. Computational examples are given. ..."
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We present some new classes of numbers that are easier to test for primality with the Elliptic Curve Primality Proving algorithm than average numbers. It is shown that this is the case for about half the numbers of the Cunningham project. Computational examples are given.