Revisiting And Improving Algorithms For The 3xor Problem

The 3SUM problem is a well-known problem in computer science and many geometric problems have been reduced to it. We study the 3XOR variant which is more cryptologically relevant. In this problem, the attacker is given black-box access to three random functions F, G and H and she has to find three inputs x, y and z such that F(x) circle plus G(y) circle plus H(z) = 0. The 3XOR problem is a difficult case of the more-general k-list birthday problem.Wagner's celebrated k-list birthday algorithm, and the ones inspired by it, work by querying the functions more than strictly necessary from an information-theoretic point of view. This gives some leeway to target a solution of a specific form, at the expense of processing a huge amount of data.However, to handle such a huge amount of data can be very difficult in practice. This is why we first restricted our attention to solving the 3XOR problem for which the total number of queries to F, G and H is minimal. If they are n-bit random functions, it is possible to solve the problem with roughly O(2(n/3)) queries. In this setting, the folklore quadratic algorithm finds a solution after O(2(2n/3)) operations. We present a 3XOR algorithm that generalizes an idea of Joux, with complexity O(2(2n/3) /n) in times and O(2(n/3)) in space. This algorithm is practical: it is up to 3x faster than the quadratic algorithm. Furthermore, using Bernstein's "clamping trick", we show that it is possible to adapt this algorithm to any number of queries, so that it will always be at least as good as, if not better than, Wagner's descendants in the same settings.We also revisit a 3SUM algorithm by Baran-Demaine-Patrascu which is asymptotically n(2) / log(2) n times faster than the quadratic algorithm when adapted to the 3XOR problem, but is otherwise completely impractical.To gain a deeper understanding of these problems, we embarked on a project to solve actual 3XOR instances for the SHA256 hash function. We believe that this was very beneficial and we present practical remarks, along with a 96-bit 3XOR for SHA256.