Paper 2023/396

Monomial Isomorphism for Tensors and Applications to Code Equivalence Problems

Abstract

Starting from the problem of $d$-Tensor Isomorphism ($d$-TI), we study the relation between various Code Equivalence problems in different metrics. In particular, we show a reduction from the sum-rank metric (CE${}_{sr}$) to the rank metric (CE${}_{rk}$). To obtain this result, we investigate reductions between tensor problems. We define the Monomial Isomorphism problem for $d$-tensors ($d$-TI${}^*$), where, given two $d$-tensors, we ask if there are $d-1$ invertible matrices and a monomial matrix sending one tensor into the other. We link this problem to the well-studied $d$-TI and the TI-completeness of $d$-TI${}^*$ is shown. Due to this result, we obtain a reduction from CE${}_{sr}$ to CE${}_{rk}$. In the literature, a similar result was known, but it needs an additional assumption on the automorphisms of matrix codes. Since many constructions based on the hardness of Code Equivalence problems are emerging in cryptography, we analyze how such reductions can be taken into account in the design of cryptosystems based on CE${}_{sr}$.