For a primitive root g modulo a prime p ≥ 1 we obtain upper bounds on the gaps between the residues modulo p of the N consecutive powers agn , n = 1, . . . , N , which is uniform over all integers a with gcd(a, p)= 1. §

with initial terms u0 ux = ••• = ur_2 = 0, ur_1 1. Then (u) is called a unit sequence with coefficients al5 a25 -.., ar. For a positive integer AT, the primitive period of (u) modulo AT, denoted by K(M), is the least positive integer m such that un + m = un (mod AT) for all nonnegative integers n greater than or equal to some fixed integer n0. It is known that the primitive period modulo M of a...

Let S(f) denote the set of integral ideals / such that / is a permutation polynomial modulo i", where / is a polynomial over the ring of integers of an algebraic number field. We obtain a classification for the sets S which may be written in the form S(f). Introduction. A polynomial f(x) with coefficients in a commutative ring R is said to be a permutation polynomial modulo an ideal I oi R (abb...

At CRYPTO 2003, Rubin and Silverberg introduced the concept of torus-based cryptography over a finite field. We extend their setting to the ring of integers modulo N . We so obtain compact representations for cryptographic systems that base their security on the discrete logarithm problem and the factoring problem. This results in smaller key sizes and substantial savings in memory and bandwidt...

In coding theory, Gray isometries are usually defined as mappings between finite Frobenius rings, which include the ring Zm of integers modulo m and the finite fields. In this paper, we derive an isometric mapping from Z8 to Z4 2 from the composition of the Gray isometries on Z8 and on Z4 . The image under this composition of a Z8-linear block code of length n with homogeneous distance d is a (...

We show that, for sufficiently large integers m and X, for almost all a = 1,. .. , m the ratios a/x and the products ax, where |x| X, are very uniformly distributed in the residue ring modulo m. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over r and s, ranging over relatively short intervals, the distribution of Kloosterman sums K r,s (...

Matching families are one of the major ingredients in the construction of locally decodable codes (LDCs) and the best known constructions of LDCs with a constant number of queries are based on matching families. The determination of the largest size of any matching family in Zm, where Zm is the ring of integers modulo m, is an interesting problem. In this paper, we show an upper bound of O((pq)...

Michael Klemm has recently studied the conditions satisfied by the complete weight enumerator of a self-dual code over Z 4 , the ring of integers modulo 4. In the present paper we deduce analogous theorems for the ‘‘symmetrized’’ weight enumerator (which ignores the difference between + 1 and − 1 coordinates) and the Hamming weight enumerator. We give a number of examples of self-dual codes, in...

We show that, for sufficiently large integers m and X, for almost all a = 1,. .. , m the ratios a/x and the products ax, where |x| X, are very uniformly distributed in the residue ring modulo m. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over r and s, ranging over relatively short intervals, the distribution of Kloosterman sums K r,s (...