Bit-Parallel GF(2n) Squarer Using Shifted Polynomial Basis

We present explicit formulae and complexities of bit-parallel shifted polynomial basis (SPB) squarers in finite field GF (2)s generated by general irreducible trinomials x+x+1 (0 < k < n) and type-II irreducible pentanomials x + x + x + xk−1 + 1 (3 < k < (n − 3)/2). The complexities of the proposed squarers match or slightly outperform the previous best results. These formulae can also be used to design polynomial basis Montgomery squarers without any change. Furthermore, we show by examples that XOR gate numbers of SPB squarers are different when different shift factors in the SPB definition, i.e., parameter v in {xi−v|0 ≤ i ≤ n−1}, are used. This corrects previous misinterpretation.