Andreas Enge

Building Curves with Arbitrary Small MOV Degree over Finite Prime Fields

Abstract We present a fast algorithm for building ordinary elliptic curves over finite prime fields having arbitrary small MOV degree. The elliptic curves are obtained using complex multiplication by any desired discriminant.

Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time

We provide a subexponential algorithm for solving the discrete logarithm problem in Jacobians of high-genus hyperelliptic curves over finite fields. Its expected running time for instances with genus g and underlying finite field Fq satisfying g ≥ ϑ log q for a positive constant ϑ is given by O(e(f(√1+3/2ϑ + √3/2ϑ) + o(1)) √(g log q) log (g log q)) The algorithm works over any finite field, and its running time does not rely on any unproven assumptions.

The complexity of class polynomial computation via floating point approximations

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. It runs in time

Computing Hilbert class polynomials

O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + \epsilon}) = O ( h^{2 + \epsilon})

Computing modular polynomials in quasi-linear time

for any

Comparing Invariants for Class Fields of Imaginary Quadratic Fields

\epsilon > 0

The arithmetic of Jacobian groups of superelliptic cubics

, where

Fast decomposition of polynomials with known Galois group

D

A general framework for subexponential discrete logarithm algorithms

is the CM discriminant and

Practical Non-Interactive Key Distribution Based on Pairings.

h