Not Enough LESS: An Improved Algorithm for Solving Code Equivalence Problems over Fq

Abstract

Recently, a new code based signature scheme, called LESS, was proposed with three concrete instantiations, each aiming to provide 128 bits of classical security [3]. Two instantiations (LESS-I and LESS-II) are based on the conjectured hardness of the linear code equivalence problem, while a third instantiation, LESS-III, is based on the conjectured hardness of the permutation code equivalence problem for weakly self-dual codes. We give an improved algorithm for solving both these problems over sufficiently large finite fields. Our implementation breaks LESS-I and LESS-III in approximately 25 s and 2 s respectively on an Intel i5-8400H CPU. Since the field size for LESS-II is relatively small (F7) our algorithm does not improve on existing methods. Nonetheless, we estimate that LESS-II can be broken with approximately 244 row operations.