Efficient Bit-Parallel GF(2^m) Multiplier for a Large Class of Irreducible Pentanomials

This work studies efficient bit-parallel multiplication in GF(2^m) for irreducible pentanomials, based on the so-called shifted polynomial bases (SPBs). We derive a closed expression of the reduced SPB product for a class of polynomials x^m + x^{k s} + x^{k s-1}+ hellip + x^k-1 + 1, with k_s - k₁ les m+1/ 2. Then, we apply the above formulation to the case of pentanomials. The resulting multiplier outperforms, or is as efficient as the best proposals in the technical literature, but it is suitable for a much larger class of pentanomials than those studied so far. Unlike previous works, this property enables the choice of pentanomials optimizing different field operations (for example, inversion), yet preserving an optimal implementation of field multiplication, as discussed and quantitatively proved in the last part of the paper.