Let G be a cyclic group with generator g and order jGj with known factorization. We prove that, given an appropriate auxiliary group H p for each large prime factor p of jGj, breaking the Diie-Hellman protocol and computing discrete logarithms are polynomial-time equivalent for G. Appropriate auxiliary groups H p are elliptic curves over GF (p) or extension elds, subgroups of the multiplicative...

In this paper, I will develop the formal foundations of a theory of complexity that underlies theory of grammatical induction. The initial concern will be the learning theoretic foundations of linguistic locality. That is, I will develop a theory that will place bounds on the amount a learner can draw from an input text. These bounds will limit the amount of variation that could potentially be ...

This study aimed at investigating the effects of grammatical error correction on EFL learners’ accuracy. Twenty-two male and female senior students were chosen randomly to respond to a questionnaire investigating their beliefs about immediate grammatical error correction.  Actually, the study was conducted in order to answer this question: what is the effect of grammatical error feedback on stu...

Shannon gave a lower bound in 1959 on the binary rate of spherical codes of given minimum Euclidean distance ρ. Using nonconstructive codes over a finite alphabet, we give a lower bound that is weaker but very close for small values of ρ. The construction is based on the Yaglom map combined with some finite sphere packings obtained from nonconstructive codes for the Euclidean metric. Concatenat...

We consider a quantum polynomial-time algorithm which solves the discrete logarithm problem for points on elliptic curves over GF (2m). We improve over earlier algorithms by constructing an efficient circuit for multiplying elements of binary finite fields and by representing elliptic curve points using a technique based on projective coordinates. The depth of our proposed implementation is O(m...

An efficient algorithm for the multiplication in GF (2) was introduced by Mastrovito. The space complexity of the Mastrovito multiplier for the irreducible trinomial x +x+1 was given as m − 1 XOR and m AND gates. In this paper, we describe an architecture based on a new formulation of the multiplication matrix, and show that the Mastrovito multiplier for the generating trinomial x + x + 1, wher...

We give complexity analysis of the class of short generating functions (GF). Assuming #P 6⊆FP/poly, we show that this class is not closed under taking many intersections, unions or projections of GFs, in the sense that these operations can increase the bit length of coefficients of GFs by a super-polynomial factor. We also prove that truncated theta functions are hard in this class.

We consider nondeterministic D-way branching programs computing functions f : D → {0, 1} in time at most kn. In the boolean case, where D = {0, 1}, no exponential lower bounds are known even for k = 2. Ajtai has proved such lower bounds for explicit functions over domains D of size at least n, and Beame, Saks and Thathachar for explicit functions over domains of size at least 2 k . In this note...

This paper presents a system for the automatic generation of Galois-field (GF) arithmetic circuits, named the GF Arithmetic Module Generator (GF-AMG). The proposed system employs a graph-based circuit description called the GF Arithmetic Circuit Graph (GF-ACG). First, we present an extension of the GF-ACG to handle GF(pm) (p ≥ 3) arithmetic circuits, which can be efficiently implemented by mult...