GF(2n) Bit-Parallel Squarer Using Generalized Polynomial Basis For a New Class of Irreducible Pentanomials

Introduction: Squarer is an important circuit building block in squareand-multiply-based exponentiation and inversion circuits. When GF (2n) 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 (2n) elements. For practical applications where values of n are often in the range of [1, 10000], GF (2n) can be defined by either an irreducible trinomial or an irreducible pentanomial. Paar et al. and Wu presented explicit squaring formulae of polynomial basis squarers for an arbitrary irreducible trinomial respectively [1], [2] and [3]. Using Montgomery’s presentation with the factor xk, Wu also proposed an optimized Montgomery squarer [4]. On the other hand, Hariri and Reyhani-Masoleh presented a Montgomery squarer for a special class of irreducible pentanomials xn + xk+1 + xk + xk−1 + 1 (3< k < (n− 3)/2) [5]. For an arbitrary irreducible pentanomial, Park derived explicit formulae and complexities of squarers based on weakly dual basis [6]. 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 work, we consider bit-parallel squarers based on a new GF (2n) representation – generalized polynomial basis (GPB), which is defined by Cilardo and is a generalisation of the shifted polynomial basis [7].