Vectorial Conception of FCSR

Abstract

In this paper, we investigate the structure of FCSR made by Goresky and Klapper. Using a vectorial construction of the objects and of the register, we extend the analysis of FCSRs. We call these registers vectorial FCSRs or VFCSRs. We obtain similar results to those of analysis of FCSRs and of d-FCSRs generating binary sequences or p-ary sequences. In fact, the AFSRs built over finite fields $\mathbb{F}_{p^{n}}$ with n ≥ 2 suffer from an very difficult and formal analysis. But if you analyze these registers with a vectorial structure, you can decompose the output sequence into a vector of binary sequences or p-ary sequences. This method allows us to obtain very easily the period, the behavior of memory with interval optimized , the maximal period, the existence of l-sequences and the calculations become explicit and easily implementable. At the end of this paper, we implement the quadratic case ($\mathbb{F}_{2^{2}}$ case) and present the conclusions about pseudorandom properties of quadratic l-sequences which are tested by NIST STS package. In conclusion, VFCSRs are easy to implement in software and hardware and have excellent pseudorandomn property.