Fast parallel computation of the polynomial shift

Abstract

Given an n-degree polynomial f(x) over an arbitrary ring, the shift of f(x) by c is the operation which computes the coefficients of the polynomial f(x+c). In this paper, we consider the case when the shift by the given constant c has to be performed several times (repeatedly). We propose a parallel algorithm that is suited to an SIMD architecture to perform the shift in O(1) time if we have O(n/sup 2/) processor elements available. The proposed algorithm is easy to generalize to multivariate polynomial shifts. The possibility of applying this algorithm to polynomials with coefficients from non-commutative rings is discussed, as well as the bit-wise complexity of the algorithm.