International Association
for Cryptologic Research

<p>The Crossbred algorithm is currently the state-of-the-art method for solving overdetermined multivariate polynomial systems over $\mathbb{F}_2$. Since its publication in 2017, several record breaking implementations have been proposed and demonstrate the power of this hybrid approach. Despite these practical results, the complexity of this algorithm and the choice of optimal parameters for it are difficult open questions. In this paper, we prove a bivariate generating series for potentially admissible parameters of the Crossbred algorithm. </p>

Parallel versions of collision search algorithms require a significant amount of memory to store a proportion of the points computed by the pseudo-random walks. Implementations available in the literature use a hash table to store these points and allow fast memory access. We provide theoretical evidence that memory is an important factor in determining the runtime of this method. We propose to replace the traditional hash table by a simple structure, inspired by radix trees, which saves space and provides fast look-up and insertion. In the case of many-collision search algorithms, our variant has a constant-factor improved runtime. We give benchmarks that show the linear parallel performance of the attack on elliptic curves discrete logarithms and improved running times for meet-in-the-middle applications.