AFSRs synthesis with the extended Euclidean rational approximation algorithm

Weihua Liu; Andrew Klapper

doi:10.3934/amc.2017008

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2017, Volume 11, Issue 1: 139-150. Doi: 10.3934/amc.2017008

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AFSRs synthesis with the extended Euclidean rational approximation algorithm

Weihua Liu^1, and
Andrew Klapper^2,

1.
Department of Computer Science, William Paterson University of New Jersey, Wayne, NJ 07470 USA
2.
Department of Computer Science, University of Kentucky, Lexington, KY 40506, USA

Received: June 30, 2015

Published: May 2017

This material is based upon work supported by the National Science Foundation under grants No. CCF-0514660 and CNS-1420227. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation

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Abstract

Pseudo-random sequence generators are widely used in many areas, such as stream ciphers, radar systems, Monte-Carlo simulations and multiple access systems. Generalization of linear feedback shift registers (LFSRs) and feedback with carry shift registers (FCSRs), algebraic feedback shift registers (AFSRs) [7] can generate pseudo-random sequences over an arbitrary finite field. In this paper, we present an algorithm derived from the Extended Euclidean Algorithm that can efficiently find a smallest AFSR over a quadratic field for a given sequence. It is an analog of the Extended Euclidean Rational Approximation Algorithm [1] used in solving the FCSR synthesis problem. For a given sequence $\mathbf{a}$, $2\Lambda(\alpha)+1$ terms of sequence $\mathbf{a}$ are needed to find the smallest AFSR, where $\Lambda(\alpha)$ is a complexity measure that reflects the size of the smallest AFSR that outputs $\mathbf{a}$.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. An algebraic feedback shift register of Length m

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Figure 2. The Extended Euclidean Rational Approximation Algorithm

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