The Berlekamp–Massey algorithm over finite rings, modules, and bimodules

Abstract

We give an algorithm for finding a monic polynomial of the least degree that generates a given sequence w(0,/ -1) of length / with complexity O(l 2 ) operations as / -» <*>. We consider the sequences w(0, / -1 ) over a finite ring R with identity, over a finite module χΜ , or over finite bimodule , where A and Β are finite rings with identities.

An algorithm for construction of the annihilator of a polylinear recurring sequence over a finite commutative ring

“…It follows by Proposition 1 that if u sn OEF sn D 0, then f sn .x/ 2 Ann u. sn .x/. They must satisfy conditions similar to (5). The last substep j gives F sn , u sn OEF sn , and f sn .x/.…”

“…Beforehand we represent the ring R as the union (3) of classes of associated elements, choose some monomial ordering , and order the elements of F C according to (4).…”

“…In [9], this algorithm is developed for sequences over residue rings Z m , and in [4,5,6], for sequences over an arbitrary finite ring R with identity. The BerlekampMassey algorithm is generalised to polylinear (multidimensional) sequences over a field in [10]- [16], [18], and to polylinear sequences over residue rings Z m in [17].…”

“…The BerlekampMassey algorithm is generalised to polylinear (multidimensional) sequences over a field in [10]- [16], [18], and to polylinear sequences over residue rings Z m in [17]. A survey on the Berlekamp-Massey algorithm can be found in [5,3].…”

“…As a base of our algorithm we take the algorithm from [4,5] for sequences over R and generalise it to the case of k-sequences. The algorithm in essence is a special form of Gaussian exclusion applied to the shifts of u.…”