GF(2) bit-parallel squarer using generalised polynomial basis for new class of irreducible pentanomials

Introduction: The squarer is an important circuit building block in square-and-multiply-based exponentiation and inversion circuits. When GF(2) elements are represented in a normal basis, squaring is simply a circular shift operation. Therefore, most previous works on squarers focused on other representations of GF(2) elements. For practical applications where values of n are often in the range of [1, 10 000], GF(2) can be defined by either an irreducible trinomial or an irreducible pentanomial. Paar et al. [1] and Wu [2, 3] presented explicit squaring formulae of polynomial basis squarers for an arbitrary irreducible trinomial. Using Montgomery’s presentation with the factor x, Wu also proposed an optimised Montgomery squarer [4]. On the other hand, Hariri and Reyhani-Masoleh [5] presented a Montgomery squarer for a special class of irreducible pentanomials x + x + 1 + x + x + 1(3 < k < (n − 3)/2). For an arbitrary irreducible pentanomial, Park [6] derived explicit formulae and complexities of squarers based on a weakly dual basis. The numbers of XOR gates used in these pentanomial-based squarers are about 1.5n and the gate delays of these squarers are 2TX, where TX is the delay of one 2-input XOR gate. In this Letter, we consider bit-parallel squarers based on a new GF(2) representation – the generalised polynomial basis (GPB), which is defined by Cilardo [7] and is a generalisation of the shifted polynomial basis.