Impact of Intelʼs new instruction sets on software implementation of GF(2)[x]GF(2)[x] multiplication

PCLMULQDQ, a new instruction that supports GF(2)[x]GF(2)[x] multiplication, was introduced by Intel in 2010. This instruction brings dramatic change to software implementation of multiplication in GF(2m)GF(2m) fields. In this paper, we present improved Karatsuba formulae for multiplying two small binary polynomials, compare different strategies for PCLMULQDQ-based multiplication in the five GF(2m)GF(2m) fields recommended by NIST and conclude the best design approaches to software implementation of GF(2)[x]GF(2)[x] multiplication.

► We present improved Karatsuba formulae for multiplying small binary polynomials.
► We compare different strategies for PCLMULQDQ-based multiplication in the NIST GF(2m)GF(2m) fields.
► Optimized codes based on multiple-term Karatsuba formulae have best performance.
► Current AVX instructions bring slight speedup to GF(2)[x]GF(2)[x] multiplication.