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10
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].
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.
Counting points modulo p for some finitely generated subgroups of algebraic group
 Bull. London Math. Soc
, 1982
"... We begin by explaining the basic idea of this paper in a simple case. We write np for the order of 2 modulo the prime p, so that np is the number of powers of 2 which are distinct mod p. We have the elementary bounds logp < £ np ^ p1. ..."
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Cited by 7 (0 self)
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We begin by explaining the basic idea of this paper in a simple case. We write np for the order of 2 modulo the prime p, so that np is the number of powers of 2 which are distinct mod p. We have the elementary bounds logp < £ np ^ p1.
Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j
, 2003
"... Using an elementary approach based on careful handlings of Cauchy integrals, we give precise effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. Moreover, we adapt an old result of Rademacher to get a convergent series expansion of these Fourier coecie ..."
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Cited by 5 (0 self)
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Using an elementary approach based on careful handlings of Cauchy integrals, we give precise effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. Moreover, we adapt an old result of Rademacher to get a convergent series expansion of these Fourier coecients and we show that this expansion allows to nd again these estimates. Our results improve previous ones by K. Mahler and O. Herrmann. In particular, we show that the Fourier coefficients of j are smaller than their asymptotically equivalent given by Petersson and Rademacher.
Smooth Orders and Cryptographic Applications
 Lecture Notes in Comptuer Science
, 2002
"... We obtain rigorous upper bounds on the number of primes x for which p1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker re ..."
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Cited by 5 (1 self)
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We obtain rigorous upper bounds on the number of primes x for which p1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker result for almost all odd numbers n. We also discuss some cryptographic applications.
On the Average Value of Divisor Sums in Arithmetic Progressions
, 2005
"... We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that “on average” these sums are close to the expected value. We also give applications of our result to sums of the divisor function twisted with characters (both additive and multiplica ..."
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Cited by 4 (2 self)
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We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that “on average” these sums are close to the expected value. We also give applications of our result to sums of the divisor function twisted with characters (both additive and multiplicative) taken on the values of various functions, such as rational and exponential functions; in particular, we obtain upper bounds for such twisted sums. 1
On the Order of Finitely Generated Subgroups of Q*(mod ρ) and Divisors of ρ1
, 1996
"... Introduction Let r be a positive integer. We say that r nonzero integers a 1 , ..., a r are multiplicatively independent if whenever there exist m 1 , ..., m r # Z such that r =1, it follows that m 1 =}}}=m r =0. We assume that none of a 1 , ..., a r is a perfect square or \1; let 1 denote t ..."
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Introduction Let r be a positive integer. We say that r nonzero integers a 1 , ..., a r are multiplicatively independent if whenever there exist m 1 , ..., m r # Z such that r =1, it follows that m 1 =}}}=m r =0. We assume that none of a 1 , ..., a r is a perfect square or \1; let 1 denote the subgroup of Q* generated by a 1 , ..., a r and let 1 p  denote the order of such a group 1 (mod p). In the case r=1, 1=(a), let ord p (a) denote the order of a (mod p). The famous Artin Conjecture for primitive roots (see [1]) states that ord p (a)=p&1 for infinitely many primes p. Artin's Conjecture has been proved under the assumption of the Generalized Riemann Hypothesis by C. Hooley (See [13]). In his paper it is implicitly shown (unconditionally) that ord p (a)>p#log p (1.1) for all but O(x#log x) primes p#x. article no. 0044 207 0022314X#96 #18.00 Copyright # 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. * Supported in part by C.I.C.M.A.
Artin’s conjecture for primitive root §1. Artin’s original conjecture
, 2004
"... possible generalization of ..."
The binary Goldbach problem
, 903
"... with arithmetic weights attached to one of the variables ..."
The ternary Goldbach problem
, 904
"... with arithmetic weights attached to two of the variables ..."