Implementation of point-counting algorithms on genus 2 hyperelliptic curves based on the birthday paradox

Abstract

Our main contribution is an efficient implementation of the Gaudry - Schost and Galbraith - Ruprai point-counting algorithms on Jacobians of hyperelliptic curves. Both of them are low memory variants of Matsuo - Chao - Tsujii (MCT) Baby-Step Giant-Step-like algorithm. We present an optimal memory restriction (a time-memory tradeoff) that minimizes the runtime of the algorithms. This tradeoff allows us to get closer in practical computations to theoretical bounds of expected runtime at 2.45√N and 2.38a√N for the Gaudry - Schost and Galbraith - Ruprai algorithms, respectively. Here N is the size of the 2-dimensional searching space, which is as large as the Jacobian group order, divided by small modulus m, precomputed by using other techniques. Our implementation profits from the multithreaded regime and we provide some performance statistics of operation on different size inputs. This is the first open-source parallel implementation of 2-dimensional Galbraith - Ruprai algorithm.