Domingo Gomez

Multiplicative character sums of Fermat quotients and pseudorandom sequences

We prove a bound on sums of products of multiplicative characters of shifted Fermat quotients modulo p. From this bound we derive results on the pseudorandomness of sequences of modular discrete logarithms of Fermat quotients modulo p: bounds on the well-distribution measure, the correlation measure of order ℓ, and the linear complexity.

Attacking the Pollard Generator

Let p be a prime and let c be an integer modulo p. The Pollard generator is a sequence (u_n) of pseudorandom numbers defined by the relation u_n+1equivu_n ²+c mod p. It is shown that if c and 9/14 of the most significant bits of two consecutive values u_n,u_n+1 of the Pollard generator are given, one can recover in polynomial time the initial value u₀ with a probabilistic algorithm. This result is an improvement of a theorem in a recent paper which requires that 2/3 of the most significant bits be known

COMMON FACTORS OF RESULTANTS MODULO p

We show that the multiplicity of a prime p as a factor of the resultant of two polynomials with integer coefficients is at least the degree of their greatest common divisor modulo p. This answers an open question by Konyagin and Shparlinski.

On the linear complexity of the Naor-Reingold sequence with elliptic curves

The Naor-Reingold sequences with elliptic curves are used in cryptography due to their nice construction and good theoretical properties. Here we provide a new bound on the linear complexity of these sequences. Our result improves the previous one obtained by I.E. Shparlinski and J.H. Silverman and holds in more cases.

Cryptanalysis of the quadratic generator

Let p be a prime and let a and c be integers modulo p. The quadratic congruential generator (QCG) is a sequence (vn) of pseudorandom numbers defined by the relation

On finding a shortest path in circulant graphs with two jumps

v_{n+1}\equiv av^{2}_{n}+c mod p

On the linear complexity of the Naor–Reingold sequence

. We show that if sufficiently many of the most significant bits of several consecutive values vn of the QCG are given, one can recover in polynomial time the initial value v0 (even in the case where the coefficient c is unknown), provided that the initial value v0 does not lie in a certain small subset of exceptional values.

On the Distribution of Nonlinear Congruential Pseudorandom Numbers of Higher Orders in Residue Rings

In this paper we present algorithms for finding a shortest path between two vertices of any weighted undirected and directed circulant graph with two jumps. Our shortest path algorithm only requires O(log N) arithmetic steps and the total bit complexity is O(log3N), where N is the number of the graph’s vertices. This method has been derived from some Closest Vector Problems (CVP) of lattices in dimension two and with l1-norm.

Multiplicative Character Sums of Recurring Sequences with Rédei Functions

Abstract We obtain a lower bound on the linear complexity of the Naor–Reingold sequence. This result solves an open problem proposed by Igor Shparlinski and improves known results in some cases.

Cayley Digraphs of Finite Abelian Groups and Monomial Ideals

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present new discrepancy bounds for sequences of s -tuples of successive nonlinear congruential pseudorandom numbers of higher orders modulo a composite integer M .