Five, six, and seven-term Karatsuba-like formulae

The Karatsuba-Ofman algorithm starts with a way to multiply two 2-term (i.e., linear) polynomials using three scalar multiplications. There is also a way to multiply two 3-term (i.e., quadratic) polynomials using six scalar multiplications. These are used within recursive constructions to multiply two higher-degree polynomials in subquadratic time. We present division-free formulae, which multiply two 5-term polynomials with 13 scalar multiplications, two 6-term polynomials with 17 scalar multiplications, and two 7-term polynomials with 22 scalar multiplications. These formulae may be mixed with the 2-term and 3-term formulae within recursive constructions, leading to improved bounds for many other degrees. Using only the 6-term formula leads to better asymptotic performance than standard Karatsuba. The new formulae work in any characteristic, but simplify in characteristic 2. We describe their application to elliptic curve arithmetic over binary fields. We include some timing data.