International Association
for Cryptologic Research

The Linear Code Equivalence (LCE) Problem has received increased attention in recent years due to its applicability in constructing efficient digital signatures. Notably, the LESS signature scheme based on LCE is under consideration for the NIST post-quantum standardization process, along with the MEDS signature scheme that relies on an extension of LCE to the rank metric, namely Matrix Code Equivalence (MCE) Problem. Building upon these developments, a family of signatures with additional properties, including linkable ring, group, and threshold signatures, has been proposed. These novel constructions introduce relaxed versions of LCE (and MCE), wherein multiple samples share the same secret equivalence. Despite their significance, these variations have often lacked a thorough security analysis, being assumed to be as challenging as their original counterparts. Addressing this gap, our work delves into the sample complexity of LCE and MCE --- precisely, the sufficient number of samples required for efficient recovery of the shared secret equivalence. Our findings reveal, for instance, that one should not use the same secret twice in the LCE setting since this enables a polynomial time (and memory) algorithm to retrieve the secret. Consequently, our results unveil the insecurity of two advanced signatures based on variants of the LCE Problem.

Group actions are becoming a viable option for post-quantum cryptography assumptions. Indeed, in recent years some works have shown how to construct primitives from assumptions based on isogenies of elliptic curves, such as CSIDH, on tensors or on code equivalence problems. This paper presents a bit commitment scheme, built on non-transitive group actions, which is shown to be secure in the standard model, under the decisional Group Action Inversion Problem. In particular, the commitment is computationally hiding and perfectly binding, and is obtained from a novel and general framework that exploits the properties of some orbit-invariant functions, together with group actions. Previous constructions depend on an interaction between the sender and the receiver in the commitment phase, which results in an interactive bit commitment. We instead propose the first non-interactive bit commitment based on group actions. Then we show that, when the sender is honest, the constructed commitment enjoys an additional feature, i.e., it is possible to tell whether two commitments were obtained from the same input, without revealing the input. We define the security properties that such a construction must satisfy, and we call this primitive linkable commitment. Finally, as an example, an instantiation of the scheme using tensors with coefficients in a finite field is provided. In this case, the invariant function is the computation of the rank of a tensor, and the cryptographic assumption is related to the Tensor Isomorphism problem.