The Berlekamp-Massey Algorithm via Minimal Polynomials

We present a recursive minimal polynomial theorem for finite sequences over a commutative integral domain D. This theorem is relative to any element of D. The ingredients are: the arithmetic of Laurent polynomials over D, a recursive ’index function’ and simple mathematical induction. Taking reciprocals gives a ’Berlekamp-Massey theorem’ i.e. a recursive construction of the polynomials arising in the Berlekamp-Massey algorithm, relative to any element of D. The recursive theorem readily yields the iterative minimal polynomial algorithm due to the author and a transparent derivation of the iterative Berlekamp-Massey algorithm. We give an upper bound for the sum of the linear complexities of s which is tight if s has a perfect linear complexity profile. This implies that over a field, both iterative algorithms require at most 2⌊ 2 4 ⌋ multiplications.