For short data sequences, Winograd's convolution algorithms attaining the minimum number of multiplications also attain a low number of additions, making them very efficient. However, for longer lengths they require a larger number of additions. Winograd's approach is usually extended to longer lengths by using a nesting approach such as the Agarwal-Cooley or Split-Nesting algorithms. Although these nesting algorithms are organizationally quite simple, they do not make the greatest use of the factorability of the data sequence length. The algorithm proposed it this paper adheres to Winograd's original approach more closely than do the nesting algorithms. By evaluating polynomials over simple matrices we retain, in algorithms for longer lengths, the basic structure and strategy of Winograd's approach.