We present a space-efficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D | 1/2+ɛ log P) space and has an expected running time of O(|D | 1+ɛ). We describe practical optimizations that allow us to handle larger discriminants than other methods, with |D | as large as 1013 and h(D) up to 106. We apply these results to construct pairing-friendly elliptic curves of prime order, using the CM method.

This report describes and analyzes the MD6 hash function and is part of our submission package for MD6 as an entry in the NIST SHA-3 hash function competition 1. Significant features of MD6 include: • Accepts input messages of any length up to 2 64 − 1 bits, and produces message digests of any desired size from 1 to 512 bits, inclusive, including

We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N 1/12) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over low-degree extension fields, where in genus 2 we find Jacobians over F p 2 and trace zero varieties over F p 3 with near-prime orders up to 372 bits in size. For p =2 61 − 1, the average time to find a group with 244-bit near-prime order is under an hour on a PC.

Abstract. We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig–Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without using a relation matrix. The problem of computing a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian group G is addressed, yielding a Monte Carlo algorithm to compute the structure of G using O(|G | 1/2) group operations. These results also improve generic algorithms for extracting pth roots in G. 1.