Alternative to the karatsuba algorithm for software implementations of GF(2n) multiplications

In [11], a new approach to subquadratic space complexity multiplication for extended finite fields has recently been proposed for hardware implementation. In this article, we develop the corresponding algorithm for software implementation. Compared to the Karatsuba algorithm, the proposed algorithm has a lower theoretical time complexity when the size of the input is greater than a fixed integer. While its recursive implementation is as simple as that of the Karatsuba algorithm, it requires less memory to store the look-up table than the latter, e.g., 512 bytes vs. 128 kilobytes in our implementation. To the best of our knowledge, this is the first better alternative to the Karatsuba algorithm for software implementation dealing with ”intermediate” sized finite fields. May 18, 2006 Index Terms Finite field, subquadratic time complexity multiplication algorithm, coordinate transformation, shifted polynomial basis, Toeplitz matrix.