A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant ∆n(a, b) of the trinomial fn,a,b(t) = t n + at+ b, where n ≥ 5 is a fixed integer. In particular, it is shown that, under the abc-conjecture, for every n ≡ 1 (mod 4), the quadratic fields Q (√ ∆n(a, b) ) are pairwise distinct for a positive proportion of such discriminants with i...

Montgomery multiplication in GF(2) is defined by a(x)b(x) r−1(x) mod f(x), where the field is generated by irreducible polynomial f(x), a(x) and b(x) are two field elements in GF(2), and r(x) is a fixed field element in GF(2). In this paper, first we present a generalized Montgomery multiplication algorithm in GF(2). Then by choosing r(x) according to f(x), we show that efficient architecture f...

Suppose q is a prime power and f ∈ Fq[x] is a univariate polynomial with exactly t nonzero terms and degree <q−1. To establish a finite field analogue of Descartes’ Rule, Bi, Cheng and Rojas (2013) proved an upper bound of 2(q − 1) t−2 t−1 on the number of cosets in F∗ q needed to cover the roots of f in F∗ q . Here, we give explicit f with root structure approaching this bound: For q a t power...

This paper presents two efficient implementations of fast and pipelined bit-parallel polynomial basis multipliers over GF (2m) by irreducible pentanomials and trinomials. The architecture of the first multiplier is based on a parallel and independent computation of powers of the polynomial variable. In the second structure only even powers of the polynomial variable are used. The par...

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 ...