Evaluating a polynomial and its reverse

Abstract

The reverse of an n th degree polynomial p(x) is rev(p) (x) = x n p(x -1 ). We show that one can evaluate p and rev(p) in only n+0(log n) multiplications modulo {x -1 }. The method uses an algorithm to evaluate reciprocal polynomials of degree n in only [n/2] + 1 + l ([n/2]) multiplications modulo {x -1 }, where l (k) denotes the length of a shortest addition chain for k.