This paper describes carry-less arithmetic operations modulo an integer 2^M-1 in the thousand-bit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementation on a cluster of PlayStation 3 game consoles a new record was set for the elliptic curve method for integer factorization.

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.

Integer division remains expensive on today's processors as the cost of integer multiplication declines. We present code sequences for division by arbitrary nonzero integer constants and run-time invariants using integer multiplication. The algorithms assume a two's complement architecture. Most also require that the upper half of an integer product be quickly accessible. We treat unsigned division, signed division where the quotient rounds towards zero, signed division where the quotient rounds towards -∞, and division where the result is known a priori to be exact. We give some implementation results using the C compiler GCC.