A Generalized Method for Constructing Sub-quadratic Complexity

We introduce a generalized method for constructing sub-quadratic complexity multipliers for even characteristic field extensions. The construction is obtained by recursively extending short convolution algorithms and nesting them. To obtain the short convolution algorithms the Winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication. We present a recursive construction technique that extends any d point multiplier into an n = d point multiplier with area that is sub-quadratic and delay that is logarithmic in the bit-length n. We present a thorough analysis that establishes the exact space and time complexities of these multipliers. Using the recursive construction method we obtain six new constructions, among which one turns out to be identical to the Karatsuba multiplier. All six algorithms have sub-quadratic space complexities and two of the algorithms have significantly better time complexities than the Karatsuba algorithm.