Cayley digraphs of finite abelian groups and monomial ideals

In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph’s node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider its representations via its minimal system of generators or its irredundant irreducible decomposition. From this last piece of information, we can compute the graph’s diameter and average minimum distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded monomial ordering. This result permits the use of Gr¨obner bases for computing the ideal and finding an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter d, all of them associated to irreducible monomial ideals

A strategy for recovering roots of bivariate polynomials modulo a prime

Let $p$ be a prime and \F_p the finite field with $p$ elements. We show how, when given an irreducible bivariate polynomial f \in \F_p[X,Y] and approximations to (v_0,v_1) \in \F_p^2 such that $f(v_0,v_1)=0$, one can recover $(v_0,v_1)$ efficiently, if the approximations are good enough. This result has been motivated by the predictability problem for non-linear pseudorandom number generators and, other potential applications to cryptography

Cryptanalysis of the Quadratic Generator

Abstract. 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 vn+1 ≡ av 2 n +c mod p. 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 routing on circulant graphs of degree four

In this paper we present the first polynomial time deterministic algorithm to compute the shortest path between two vertices of a circulant graph of degree four. Our spectacular algorithm only requires O(log 3 N) bit operations, where N is the number of the vertices and it is based on shortest vector problems in a special class of lattices for L1-norm. Moreover, the technique can be extended to weighted and directed circulant graphs, the so called double-loop networks. Our main tools are results and methods from the geometry of numbers and computer algebra.