Cryptographic schemes using elliptic curves over finite fields require the computation of the cardinality of the curves. Dramatic progress have been achieved recently in that field. The aim of this article is to highlight part of these improvements and to describe an efficient implementation of them in the particular case of the field GF (2n), for n ≤ 500.

j=1 hk,j |8(k, j) 2 [p]⇥[Nk], hk,j 2Hk , and the union of all such families GF,n = S |N|=n GF,N. Fix ⇢ > 0. For a fixed N, the Rademacher complexity of GF,N can be bounded as follows for any m 1: Rm(GF,N)  1 n Pp k=1 Nk Rm(Hk). Thus, the following standard margin-based Rademacher complexity bound holds (Koltchinskii & Panchenko, 2002). For any > 0, with probability at least 1 , for all g 2 GF,...

For an integer n ≥ 3, a rank-n matroid is called an n-spike if it consists of n three-point lines through a common point such that, for all k ∈ {1, 2, . . . , n− 1}, the union of every set of k of these lines has rank k + 1. Spikes are very special and important in matroid theory. Wu [13] found the exact numbers of n-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for ...

This paper examines several modular inverse algorithms in GF (p) that have been proposed in literature. A survey of these algorithms attempts to study the evolution of modular inversion methods and trace key areas of improvement for hardware implementation efficiency.

A construction is given for associating to any symmetric (v, k, X) design a self-dual code of length v+ 1 over GF (p), where p is any divisor of the square-free part of k — k. These codes are then used to obtain a substantial strengthening of a result of Hughes on automorphisms of designs. Specifically, suppose a symmetric {v, k, X) design possesses an automorphism a of odd prime order q. If an...

The present study characterizes the neural correlates of noun and verb imageability and addresses the question of whether components of the neural network supporting word recognition can be separately modified by variations in grammatical class and imageability. We examined the effect of imageability on BOLD signal during single-word comprehension of nouns and verbs. Subjects made semantic simi...

Abstract This article investigates the evolutionary and spatial dynamics of typological characters in 117 Indo-European languages. We partition types change (i.e., gain or loss) for each variant according to whether they bring about a simplification morphosyntactic patterns that must be learned, are neutral neither simplifying nor introducing complexity) introduce more complex pattern. find cha...

This squib aims to further our understanding of the relation between invariants of grammar and grammatical categories. Keenan and Stabler (2003) propose to formalize of the notion of ‘structural/ grammatical’ in terms of automorphism invariance, based on a very general notion of formal grammars (Bare Grammars). It is natural to think that category labels (more precisely: sets of language expres...