Since their introduction in constructive cryptographic applications, pairings over (hyper)elliptic curves are at the heart of an ever increasing number of protocols. As they rely critically on efficient implementations of pairing primitives, the study of hardware accelerators has become an active research area. In this paper, we propose two coprocessors for the reduced ηT pairing introduced by ...

For any positive integers n ≥ 3, r ≥ 1 we present formulae for the number of irreducible polynomials of degree n over the finite field F2r where the coefficients of xn−1, xn−2 and xn−3 are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the F2 base field cases, reverse-en...

In this paper, we introduce a new approach to the generation of binary sequences by applying trace functions to elliptic curves over GF (2). We call these sequences elliptic curve pseudorandom sequences (EC-sequence). We determine their periods, distribution of zeros and ones, and linear spans for a class of EC-sequences generated from supersingular curves. We exhibit a class of EC-sequences wh...

The supersingular K3 surface X in characteristic 2 with Artin invariant 1 admits several genus 1 fibrations (elliptic and quasi-elliptic). We use a bijection between fibrations and definite even lattices of rank 20 and discriminant 4 to classify the fibrations, and exhibit isomorphisms between the resulting models of X. We also study a configuration of (−2)-curves on X related to the incidence ...

This paper describes the design of a fast multi-core library for the cryptographic Tate pairing over supersingular elliptic curves. For the computation of the reduced modified Tate pairing over F3509 , we report calculation times of just 2.94 ms and 1.87 ms on the Intel Core2 and Intel Core i7 architectures, respectively. We also try to answer one important design question that surges: how many...

In this paper we investigate the efficiency of the function field sieve to compute discrete logarithms in the finite fields F3n . Motivated by attacks on identity based encryption systems using supersingular elliptic curves, we pay special attention to the case where n is composite. This allows us to represent the function field over different base fields. Practical experiments appear to show t...

Constructing non-supersingular elliptic curves for pairing-based cryptosystems have attracted much attention in recent years. The best previous technique builds curves with ρ = lg(q) / lg(r) ≈ 1 (k = 12) and ρ = lg(q) / lg(r) ≈ 1.25 (k = 24). When k > 12, most of the previous works address the question by representing r(x) as a cyclotomic polynomial. In this paper, we propose a method to find m...

For positive integers 1 ≤ i ≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews-Gordon q-series ∏ 1≤n6≡0,±i (mod 2k+1) 1 1− qn . This study is motivated by their appearance in conformal field theory, where these series are essentially the irreducible characters of (2, 2k + 1) Virasoro minimal models. We determine the vanishing of such W...

Let E be an elliptic curve over Q. We assume that E has good supersingular reduction at a prime p, and for simplicity, assume p is odd and ap = p+ 1− #E(Fp) is zero. Then, as the second author showed, the p-adic L-function Lp,α(E) of E corresponding to α = ±√−p (by Amice-Vélu and Vishik) can be written as Lp,α(E) = f log+p +g logp α by using two Iwasawa functions f and g ∈ Zp[[Gal(Q∞/Q)]] ([20]...

Explicit congruences (mod p) are proved for the class equations or the products of class equations corresponding to discriminants D = −8p,−3p, −12p in the theory of complex multiplication, where p is an odd prime. These congruences are used to give a new proof of a theorem of Ogg, which states that there are exactly 15 primes p for which all j-invariants of supersingular elliptic curves in char...