We associate via duality a dynamical system to each pair (R S , #), where R S is the ring of S--integers in an A--field k, and # is an element of R S \{0}.

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

Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an r-th power residue for all small factors of p − 1. The corresponding Diffie-Hellman (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

We present an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite. 1

Integer division has been known to lie in P-uniform TC 0 since the mid-1980'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 DLOGTIME-uniform 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 well-known number-theoretic 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].

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 widely-believed number-theoretic 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 non-linear codes, but the linear case is more subtle and has remained open.

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.

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.

constant j-invariant

We study the average multiplicative order of elements modulo n and show that its behaviour is very close to the behaviour of the largest possible multiplicative order of elements modulo n given by the Carmichael function #(n). 2000 Mathematics Subject Classification: Primary 11N37, 11N64; Secondary 20K01 1