We introduce two new infinite families of APN functions, one on fields of order 22k for k not divisible by 2, and the other on fields of order 23k for k not divisible by 3. The polynomials in the first family have between three and k+ 2 terms, the second family’s polynomials have three terms. © 2007 Elsevier Inc. All rights reserved.

In the previous paper [FLSU], we determined the Galois groups associated with some polynomials constructed through the use of circulant matrices. In the process of determining the Galois groups, the irreducibility of the trinomial x + x +m was established, where p represents an odd prime and m an integer ≥ 2. The approach there was based on a method of Lebesgue [Le]. In this paper, we discuss t...

By embedding a Toeplitz matrix-vector product (MVP) of dimension n into a circulant MVP of dimension N = 2n+ δ− 1, where δ can be any nonnegative integer, we present a GF(2) multiplication algorithm. This algorithm leads to a new redundant representation, and it has two merits: 1. The flexible choices of δ make it possible to select a proper N such that the multiplication operation in ring GF(2...

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.

In this paper, a new bit-parallel Montgomery multiplier for GF (2) is presented, where the field is generated with an irreducible trinomial. We first present a slightly generalized version of a newly proposed divide and conquer approach. Then, by combining this approach and a carefully chosen Montgomery factor, the Montgomery multiplication can be transformed into a composition of small polynom...

It is well known that Stickelberger-Swan theorem is very important for determining reducibility of polynomials over a binary field. Using this theorem it was determined the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on. We discuss this problem for type II pentanomials n...

In this contribution, we derive a novel parallel formulation of the standard Itoh-Tsujii algorithm for multiplicative inverse computation overGF(2m). The main building blocks used by our algorithm are: field multiplication, field squaring and field square root operators. It achieves its best performance when using a special class of irreducible trinomials, namely, P (X) = Xm +Xk + 1, withm and ...

We present a method for computing pth roots using a polynomial basis over finite fields Fq of odd characteristic p, p ≥ 5, by taking advantage of a binomial reduction polynomial. For a finite field extension Fqm of Fq our method requires p− 1 scalar multiplication of elements in Fqm by elements in Fq. In addition, our method requires at most (p − 1)dm/pe additions in the extension field. In cer...

We present algorithms revealing new families of polynomials allowing sub-exponential detection of p-adic rational roots, relative to the sparse encoding. For instance, we show that the case of honest n-variate (n+ 1)-nomials is doable in NP and, for p exceeding the Newton polytope volume and not dividing any coefficient, in constant time. Furthermore, using the theory of linear forms in p-adic ...

Montgomery multiplication in GF(2m) 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(2m), and r(x) is a fixed field element in GF(2m). In this paper, first we present a generalized Montgomery multiplication algorithm in GF(2m). Then by choosing r(X) according to f(x), we show that efficient architectur...