In late 2012 and early 2013 the discrete logarithm problem (DLP) in finite fields of small characteristic underwent a dramatic series of breakthroughs, culminating in a heuristic quasipolynomial time algorithm, due to Barbulescu, Gaudry, Joux and Thomé. Using these developments, Adj, Menezes, Oliveira and Rodŕıguez-Henŕıquez analysed the concrete security of the DLP, as it arises from pairings ...

The Elliptic Curve Discrete Logarithm Problem(ECDLP) is known to be an exponential time problem except the cases of smooth curves, supersingular curves and anomalous curves. Recently, several new methods to solve ECDLP on a prime eld were proposed. All of them try to solve ECDLP on a prime eld by lifting a given elliptic curve to low rank elliptic curves de ned over the rationals. In this exten...

Let C be a supersingular genus-2 curve over an algebraically closed field of characteristic 3. We show that if C is not isomorphic to the curve y = x + 1 then up to isomorphism there are exactly 20 degree-3 maps φ from C to the elliptic curve E with j-invariant 0. We study the coarse moduli space of triples (C, E, φ), paying particular attention to questions of rationality. The results we obtai...

Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D < 0 such that the reduction map modulo a prime above p from elliptic curves with CM by OD to supersingular elliptic curves in characteristic p. In the algorithm we first determine an explicit constant Dp so that |D| > Dp implies that the map is necessarily surjective and then we compu...

This paper proposes explicit formulae for the addition step and doubling step in Miller’s algorithm to compute Tate pairing on Jacobi quartic curves. We present a geometric interpretation of the group law on Jacobi quartic curves, which leads to formulae for Miller’s algorithm. The doubling step formula is competitive with that for Weierstrass curves and Edwards curves. Moreover, by carefully c...

The most significant pairing-based cryptographic protocol to be proposed so far is undoubtedly the Identity-Based Encryption (IBE) protocol of Boneh and Franklin. In their paper [6] they give details of how their scheme might be implemented in practise on certain supersingular elliptic curves of prime characteristic. They also point out that the scheme could as easily be implemented on certain ...