Reversible Logic Synthesis Benchmarks Page

Galois field multiplication function gfp^mmult has 2m]log p[ inputs, (a₀, a₁,... a_m-1)=a=a₀+a₁x+...+a_m-1x^m-1 and (b₀, b₁,... b_m-1)=b=b₀+b₁x+...+b_m-1x^m-1, and m]log p[ outputs, (c₀, c₁,... c_m-1)=c=ab=c₀+c₁x+...+c_m-1x^m-1, where each a_i, b_i, and c_i is a mod p number. It computes the field product of two GF(p^m) elements, a and b. Since to say GF(p^m) does not suffice to specify a unique field (up to isomorphism, there is exactly one field GF(p^m), however, depending on the primitive polynomial chosen product of two polynomials may look different), each benchmark specification lists the prime p, and the primitive polynomial used.

GF(p^m) multiplier is required to launch a quantum polynomial time attack on elliptic curve cryptography methods, equivalently, it is used in a quantum polynomial time algorithm that computes discrete logarithm over an elliptic curve group.