Multiplicative complexity of polynomial multiplication over finite fields

Let Mq(n) denote the number of multiplications required to compute the coefficients of the product of two polynomials of degree n over a q-element field by means of bilinear algorithms. It is shown that Mq(n) ≥ 3n - o(n). In particular, if q/2 ≪ n ≤ q + 1, we establish the tight bound Mq(n) = 3n + 1 - ⌊q/2⌋. The technique we use can be applied to analysis of algorithms for multiplication of polynomials modulo a polynomial as well.