Classification of all the minimal bilinear algorithms for computing the coefficients of the product of two polynomials modulo a polynomial. Part II: The algebra G[u]/〈uⁿ〉

In this paper we classify all the minimal bilinear algorithms for computing the coefficients of (Σ^n-1_i=0 x_iuⁱ) (Σ^n-1_i=0 y_iuⁱ) mod Q(u)^l where deg Q(u)=j, jl=n and Q(u) is irreducible (over G) is studied. The case where l = 1 was studied in [8]. For l > 1 the main results are that we have to distinguish between two cases: j > 1 and j = 1. The case where j > 1 was studied in [1]. For j = 1 it is shown that up to equivalence, every minimal (2n - 1 multiplications) bilinear algorithm for computing the coefficients of (Σ^n-1_i=0 x_iuⁱ) (Σ^n-1_i=0 y_iuⁱ) mod uⁿ is done either by first computing the coefficients of (Σ^n-1_i=0 x_iuⁱ) (Σ^n-1_i=0 y_iuⁱ) and then reducing them modulo uⁿ or by first computing the coefficients (Σ^n-2_i=0 x_iuⁱ) (Σ^n-1_i=0 y_iuⁱ) and then reducing them modulo uⁿ and adding x_n-1y₀u^n-1 or by first computing the coefficients (Σ^n-2_i=0 x_iuⁱ) (Σ^n-2_i=0 y_iuⁱ) and then reducing them modulo uⁿ and adding (x_n-1y₀ + x₀y_n-1)uⁿ-1.