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39
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}. ..."
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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}.
On the statistical properties of Diffie–Hellman distributions
 MR 2001k:11258 Zbl 0997.11066
"... Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an rth power residue for all small factors of p − 1. The corresponding DiffieHellman (DH) distribution is (ϑ x, ϑ y, ϑ xy) where x, y are randomly chosen from Z ∗ p. A recently formulated assumption is that giv ..."
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Cited by 29 (10 self)
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Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an rth power residue for all small factors of p − 1. The corresponding DiffieHellman (DH) distribution is (ϑ x, ϑ y, ϑ xy) where x, y are randomly chosen from Z ∗ p. A recently formulated assumption is that given p, ϑ of the above form it is infeasible to distinguish in reasonable time between DH distribution and triples of numbers chosen
On certain exponential sums and the distribution of DiffieHellman triples
 J. London Math. Soc
, 1999
"... Let g be a primitive root modulo a prime p. It is proved that the triples (gx,gy,gxy), x,y�1,…,p�1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε�0 be fixed. Then p−� x,y=� exp0 2πiagx�bgy�cgxy ..."
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Cited by 26 (14 self)
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Let g be a primitive root modulo a prime p. It is proved that the triples (gx,gy,gxy), x,y�1,…,p�1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε�0 be fixed. Then p−� x,y=� exp0 2πiagx�bgy�cgxy
PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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Cited by 26 (2 self)
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
Uniform Circuits for Division: Consequences and Problems
 Electronic Colloquium on Computational Complexity 7:065
, 2000
"... Integer division has been known to lie in Puniform TC 0 since the mid1980's, and recently this was improved to L uniform TC 0 . At the time that the results in this paper were proved and submitted for conference presentation, it was unknown whether division lay in DLOGTIMEuniform TC 0 (also ..."
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Cited by 13 (6 self)
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Integer division has been known to lie in Puniform TC 0 since the mid1980's, and recently this was improved to L uniform TC 0 . At the time that the results in this paper were proved and submitted for conference presentation, it was unknown whether division lay in DLOGTIMEuniform TC 0 (also known as FOM). We obtain tight bounds on the uniformity required for division, by showing that division is complete for the complexity class FOM + POW obtained by augmenting FOM with a predicate for powering modulo small primes. We also show that, under a wellknown numbertheoretic conjecture (that there are many "smooth" primes), POW (and hence division) lies in FOM. Building on this work, Hesse has shown recently that division is in FOM [17].
Limits to List Decodability of Linear Codes
 In Proc. 34th ACM Symp. on Theory of Computing
, 2002
"... We consider the problem of the best possible relation between the list decodability of a binary linear code and its minimum distance. We prove, under a widelybelieved numbertheoretic conjecture, that the classical "Johnson bound" gives, in general, the best possible relation between the list decod ..."
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Cited by 10 (5 self)
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We consider the problem of the best possible relation between the list decodability of a binary linear code and its minimum distance. We prove, under a widelybelieved numbertheoretic conjecture, that the classical "Johnson bound" gives, in general, the best possible relation between the list decoding radius of a code and its minimum distance. The analogous result is known to hold by a folklore random coding argument for the case of nonlinear codes, but the linear case is more subtle and has remained open.
On the density of primes in arithmetic progression having a prescribed primitive root
, 1999
"... ..."
Composition factors from the group ring and Artin's theorem on orders of simple groups
 Proc. London Math. Soc
, 1990
"... The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. Th ..."
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Cited by 8 (2 self)
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The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups. 1.
Ordinary elliptic curves of high rank over ¯ Fp(x) with constant jinvariant
"... constant jinvariant ..."
On Hooley's Theorem With Weights
, 1995
"... We adapt Hooley's proof that the Generalized Riemann Hypothesis implies the Artin Conjecture for primitive roots to various other problems. We consider the sum p#x f (i p ) where i p is the index of 2 modulo p and f is a given function. In various cases we establish asymptotic formulas for such a ..."
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Cited by 7 (2 self)
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We adapt Hooley's proof that the Generalized Riemann Hypothesis implies the Artin Conjecture for primitive roots to various other problems. We consider the sum p#x f (i p ) where i p is the index of 2 modulo p and f is a given function. In various cases we establish asymptotic formulas for such a sum and analyse the constants. While we claim no originality, we outline the method to approach this problem in a fairly general case.