AFSRs synthesis with the extended Euclidean rational approximation algorithm

Weihua Liu,Andrew Klapper

doi:10.3934/amc.2017008

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 $\mathbf{a}$.

Highlights

Algebraic feedback shift registers (AFSRs), proposed by Klapper and Xu [7], are pseudo-random sequence generators that can produce sequences over the quotient ring R/(π), where R is an integral domain and π is an element in R
The other method is based on low-dimensional lattice basis reduction [13], and is called the lattice rational ap√proximation algorithm [11]. It works for AFSRs over quadratic integer rings Z[ d] with quadratic time complexity
We apply the extended Euclidean algorithm on a norm-Euclidean imaginary quadratic field to find a smallest AFSR for a given sequence a. It is more efficient than the lattice rational approximation algorithm in that only 2Λ(α) + 1 terms of sequence a are needed

Summary

Introduction

Algebraic feedback shift registers (AFSRs), proposed by Klapper and Xu [7], are pseudo-random sequence generators that can produce sequences over the quotient ring R/(π), where R is an integral domain and π is an element in R. Key words and phrases: AFSR synthesis, rational approximation, the extended Euclidean algorithm, sequences, stream ciphers. The other method is based on low-dimensional lattice basis reduction [13], and is called the lattice rational ap√proximation algorithm [11] It works for AFSRs over quadratic integer rings Z[ d] with quadratic time complexity. We apply the extended Euclidean algorithm on a norm-Euclidean imaginary quadratic field to find a smallest AFSR for a given sequence a It is more efficient than the lattice rational approximation algorithm in that only 2Λ(α) + 1 terms of sequence a are needed. It is different from the measure φπ(a) in lattice rational approximation algorithm but the difference is at most 1

Preliminaries

The extended Euclidean rational approximation algorithm

Conclusions