mathematics Article Gaussian Pseudorandom Number Generator Using Linear Feedback Shift Registers in Extended Fields Guillermo Cotrina * , Alberto Peinado and Andrés Ortiz   Citation: Cotrina, G.; Peinado, A.; Ortiz, A Gaussian Pseudorandom Number Generator Using Linear Feedback Shift Registers in Extended Fields. Mathematics 2021, 9, 556. https://dx.doi.org/10.3390/ math9050556 Academic Editor: Luis Hernández Encinas Received: 19 January 2021 Accepted: 25 February 2021 Published: 6 March 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department Ingeniería de Comunicaciones, E.T.S. Ingeniería de Telecomunicación, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain; apeinado@ic.uma.es (A.P.); aortiz@ic.uma.es (A.O.) * Correspondence: gcotrinacuenca@uma.es Abstract: A new proposal to generate pseudorandom numbers with Gaussian distribution is presented. The generator is a generalization to the extended field GF(2 n ) 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(2 n ). 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. Keywords: LFSR; Gaussian distribution; extended fields; central limit theorem 1. Introduction Random number generators are of vital importance in many areas and, particularly, in cryptography. Most cryptographic algorithms and protocols make use of random or pseudorandom numbers. Encryption and authentication schemes in wireless and mobile communications, such as Bluetooth [1], IEEE 802.15.4, IEEE 802.11 WLAN [2], GSM [3] or 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]. In fact, the maximal sequence length generated by an LFSR of m cells is 2 m − 1. However, those sequences suffer from a high predictability in such a way that the whole sequence can be reproduced if an eavesdropper gains access to 2m bits. Despite that, the LFSR is still an important part of the cryptographic generators because those sequences are used to derive more robust ones but keeping the original statistical properties. Two main methods are applied to fix that weakness: nonlinear combination and nonlinear filtering. The former is based on several LFSR, usually with different number of cells [3], and the latter on a unique LFSR whose sequence is processed (filtered) by a nonlinear function [4]. 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. More precisely, in 2010, Kang [8] proposed a Gaussian PRNG, using a LFSR of length N = 4 M bits, to generate pseudorandom numbers of ( M + 4) bits. The numbers were Mathematics 2021, 9, 556. https://doi.org/10.3390/math9050556 https://www.mdpi.com/journal/mathematics