Efficient finite field multiplication in normal basis

摘要

The invention provides improved techniques for multiplication of signals represented in a normal basis of a finite field. An illustrative embodiment includes a first rotator which receives a first input signal representative of a first normal basis field element (a₀ a₁ . . . a_m−1), and a second rotator which receives a second input signal representative of a second normal basis field element (b₀ b₁ . . . b_m−1). A word multiplier receives output signals from the first and second rotators, corresponding to rotated representations of the first and second elements, respectively, and processes the rotated representations w bits at a time to generate an output signal representative of a product of the first and second elements, where w is a word length associated with the word multiplier. The rotated representation of the first element may be given by A[i]=(a_i a_i+1 . . . a_{i+w−1}), the rotated representation of the second element may be given by B[i]=(b_i b_i+1 . . . b_{i+w−1}), and the product may be given by c=(C[0], C[w], C[2w], . . . , C[m−w]), where C[i]=(c_i C_i+1 . . . c_{i +w−1}), m is the degree of the finite field, w is the word length, and i=0, 1, . . . m−1. The invention is particularly well suited for implementation in software, and can provide performance advantages for both general normal basis and optimal normal basis.