The discrete logarithm problem in cokernels of $\mathcal{O}_K$-matrices

Abstract

In 2009 and 2010, Blackburn and Shokrieh independently found that the discrete logarithm can be computed efficiently on the sandpile group of a graph, meaning that sandpile groups are not secure for cryptography. We generalize this problem to cokernels of matrices with entries in the ring of integers $\mathcal{O}_K$ of a number field $K$. When $K$ has nontrivial class group, the failure of the Euclidean algorithm in $\mathcal{O}_K$ is an obstacle to generalizing previous methods. For $M$ in $\mathrm{M}_{n\times m}(\mathcal{O}_K)$, we overcome this obstacle to efficiently compute discrete logarithms in $\mathrm{cok}(M) = \mathcal{O}_K^n/M\mathcal{O}_K^m$. In particular, we find an algorithm with time complexity $\tilde{O}((m+n)^{\omega+1})$, where $\omega$ is an exponent of matrix multiplication, to compute discrete logarithms in $\mathrm{cok}(M)$ when $\mathrm{cok}(M)$ is viewed either as an $\mathcal{O}_K$-module or as a group. When $M$ is Hermitian with respect to a Galois involution $\sigma$ and nonsingular, we improve the time complexity to $\tilde{O}(n^\omega)$.