Gaussian Pseudorandom Number Generator Using Linear Feedback Shift Registers in Extended Fields

Guillermo Cotrina,Alberto Peinado,Andrés Ortiz

doi:10.3390/math9050556

Abstract

A new proposal to generate pseudorandom numbers with Gaussian distribution is presented. The generator is a generalization to the extended field GF(2n) of the one using cyclic rotations of linear feedback shift registers (LFSRs) originally defined in GF(2). The rotations applied to LFSRs in the binary case are no longer needed in the extended field due to the implicit rotations found in the binary equivalent model of LFSRs in GF(2n). The new proposal is aligned with the current trend in cryptography of using extended fields as a way to speed up the bitrate of the pseudorandom generators. This proposal allows the use of LFSRs in cryptography to be taken further, from the generation of the classical uniformly distributed sequences to other areas, such as quantum key distribution schemes, in which sequences with Gaussian distribution are needed. The paper contains the statistical analysis of the numbers produced and a comparison with other Gaussian generators.

Highlights

LTE [4], employ pseudorandom numbers; radio frequency identification [5] standards define and recommend the utilization of true random numbers [6].A large part of the pseudo-random number generators (PRNGs) used in cryptography are based on linear feedback shift registers (LFSRs), mainly due to their simplicity, low cost of implementation, good statistical behavior and the possibility of using a mathematical model that allows the generator to be designed for an optimal performance [7]
The present paper describes a Gaussian PRNG based on an LFSR operated and defined in an extension field GF (2n ) instead of using the binary field
It is based on a unique LFSR, using the same approach than the previous proposals [8,9,10], in order to generate a certain number of sequences of uniformly distributed numbers, needed to apply the central limit theorem (CLT)

Summary

Introduction

Another advantage of using LFSRs in cryptography is that the sequences generated have a uniform statistical distribution For all these reasons, there is a lot of published works related to the LFSR, but only a few regarding its utilization to produce numbers with Gaussian distribution. The proposed generator is a way to keep using the LFSR as a basic element to generate pseudorandom numbers in cryptographic areas where other than uniform distribution is required An example of this is quantum key distribution (QKD). QKD (CV-QKD) [18,19,20,21], currently deployed in several countries e.g., China, Japan, Spain and Italy [15] present a lower implementation cost due to the utilization of standard communications components They use coherent detection techniques usually employed in classical optical communications.

LFSR Fundamentals

The Proposed Generator

Distribution Fit Test

Chi Square Test

Results

Conclusions