# Deterministically broken Periodicity of Linear Congruential Generators using Quasi-Crystals

• Published 1999

#### Abstract

We describe the design of a family of aperiodic pseudorandom number generator (APRNG). These deterministic generators are based on linear congruential generators (LCGs) and, unlike any other deterministic PRNG, lead to nonperiodic pseudorandom sequences. An APRNG consists of several LCGs whose combination, controlled by a quasicrystal, forms an infinite aperiodic sequence of pseudorandom numbers. Keyworks : aperiodic pseudo-random number generators, cryptography, design of algorithm, linear congruential generators, pseudo-random number generators, quasi-crystals. Résumé Nous décrivons la conception d’une famille de générateurs apériodique de nombres pseudo-aléatoires (GANPA). Ces générateurs déterministes utilisent des générateurs congruents linéaires (GCLs) et, contrairement à tout autre GNPA, engendrent des suites pseudo-aléatoires apériodiques. Un GANPA est formé de GCLs dont la combinaison, déterminée à l’aide d’un quasicristal, forme une suite infinie et apériodique de nombres pseudo-aléatoires. The aperiodic pseudo-random number generator (APRNG) is a deterministic PRNG based on linear congruential generators (LCGs) which, unlike any deterministic PRNG, leads to non-periodic pseudo-random sequence. The APRNG is of interest, for example, in computer programming and intensive simulations, and it is a first step in constructing a cryptographic system using quasi-crystals (QC). A LCG is a pseudo-random sequence generator of the form xn = (axn−1 + b) (mod m) for all integers n > 0 , (1) with a given“seed”x0. When the parameters (a, b,m) are chosen properly, the sequence (xn) +∞ n=0 has the maximal period m, and for any i ∈ N and sufficiently suitable N , (xn) n=i shows good statistical behavior with respect to most reasonable empirical tests [4]. LCGs with small prime moduli are very fast and easily implementable, unfortunately they have small period and are cryptographically insecure [3, 5, 6]. The APRNG consists of several LCGs whose combination, controlled by a QC, forms an infinite aperiodic sequence of pseudo-random numbers. The aperiodicity of this sequence follows from the properties of cut-and-project sequences (CPSs) [8, 9, 10, 2]. These sequences are mathematical models of quasi-crystals which are constructed by projection of chosen points from a two dimensional lattice on a straight line. These points are chosen from a strip along the straight line as illustrated in figure 1. The position and width of the strip are free parameters. According to the 3-gap theorem [14], each such projection determines a tiling of the real line where distances between neighboring points have at most three different values. It is possible to choose the slope of the straight line in such a way that this geometric definition can be formulated in a simple algebraic way [11].

### Cite this paper

@inproceedings{Guimond1999DeterministicallyBP, title={Deterministically broken Periodicity of Linear Congruential Generators using Quasi-Crystals}, author={Louis-Sebastien Guimond and Jǐŕı Patera and Zuzana Mas{\'a}kov{\'a} and Edita Pelantov{\'a}}, year={1999} }