In this paper, we present a low complexity bit-parallel Montgomery multiplier for GF(2m) generated with a special class of irreducible pentanomials xm + xm−1 + xk + x + 1. Based on a combination of generalized polynomial basis (GPB) squarer and a newly proposed square-based divide and conquer approach, we can partition field multiplications into a composition of sub-polynomial multiplications a...

We consider polynomials over the binary field, F 2. A polynomial f of degree m is called primitive if k = 2 m − 1 is the smallest positive integer such that f divides x k + 1. We consider polynomials over the binary field, F 2. A polynomial f of degree m is called primitive if k = 2 m − 1 is the smallest positive integer such that f divides x k + 1.

Let C f n denote the set of all subintervals of a binary shift-register sequence with length n generated by a primitive polynomial f of degree m, where m< n≤ 2m, together with the zero vector of length n. Munemasa [8] showed that if the polynomial f is a trinomial, then C f n corresponds to an orthogonal array of strength 2. His result is based on a proof that very few trinomials of degree at m...

finite fields, irreducible polynomials A table of low-weight irreducible polynomials over the finite field F2 is presented. For each integer n in the range 2 = n = 10,000, a binary irreducible polynomial f(x) of degree n and minimum posible weight is listed. Among those of minimum weight, the polynomial listed is such that the degree of f(x) – xn is lowest (similarly, subsequent lower degrees a...

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 ...

Arithmetic operations over binary extension fields GF(2 m ) have many important applications in domains such as cryptography, code theory and digital signal processing. These must be fast, so low-delay implementations of arithmetic circuits are required. Among operations, field multiplication is...

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...

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...

Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f . Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polyn...