An improved transformation between Fibonacci FSRs and Galois FSRs

TL;DR

This paper presents an improved method for transforming Fibonacci FSRs into Galois FSRs, allowing for fewer stages and more efficient implementations in stream cipher design.

Contribution

It introduces a transformation that does not require equal stages, calculates the total number of equivalent Galois FSRs, and develops an algorithm for minimal-stage Galois FSRs.

Findings

Equivalent Galois FSRs can have fewer stages than Fibonacci FSRs.

An algorithm for minimal-stage Galois FSRs is proposed.

Numerical examples validate the transformation strategies.

Abstract

Feedback shift registers (FSRs), which have two configurations: Fibonacci and Galois, are a primitive building block in stream ciphers. In this paper, an improved transformation is proposed between Fibonacci FSRs and Galois FSRs. In the previous results, the number of stages is identical when constructing the equivalent FSRs. In this paper, there is no requirement to keep the number of stages equal for two equivalent FSRs here. More precisely, it is verified that an equivalent Galois FSR with fewer stages cannot be found for a Fibonacci FSR, but the converse is not true. Furthermore, the total number of equivalent Galois FSRs for a given Fibonacci FSR with n stages is calculated. In order to reduce the propagation time and memory, an effective algorithm is developed to find equivalent Galois FSR and is proved to own minimal operators and stages. Finally, the feasibility of our proposed strategies, to mutually transform Fibonacci FSRs and Galois FSRs, is demonstrated by numerical examples.