Paper 2026/671

A note on the Unsuitability of LIGA for Linkable Ring Signatures: The perils of non-commutativity

Václav Gregor, University of Bordeaux

Florian L’écu Leal, University of Bordeaux

Abstract

In this work, we study the proposal for a linkable ring signature (LRS) in [KTS+24]. It is instantiated from the group action based framework described in [BKP20], using the Lattice Isomorphism Group Action (LIGA), meaning that the security of the signature rests on the famous Lattice Isomorphism Problem (LIP). We will show that this signature does not in fact fit the requirements to be a linkable ring signature, despite the guarantees of the [BKP20] framework, due to it straying from that framework by using a non-commutative group for the group actions. More specifically, we will show that the signature from [KTS+24] satisfies neither the property of correctness nor linkability, which are required of a LRS. This further damages the signature, as it was already shown in [BCF25] that the linkable anonymity property of [KTS24+] isn't satisfied. The group used in LIGA is the group of invertible integer matrices: $\mathrm{GL}_n(\mathbb{Z})$. As the main obstacle in successfully applying the framework mentioned above to construct a LRS based on LIP is the fact that this group is non-commutative, we try fixing the signature by restricting the secret key space to a commutative subgroup of $\mathrm{GL}_n(\mathbb{Z})$. However, we will see that finding a commutative subgroup of $\mathrm{GL}_n(\mathbb{Z})$ that maintains the hardness of the underlying LIP, and that at the same time is realistic to use is not as easy as it may seem.

@misc{cryptoeprint:2026/671,
      author = {Iñigo Diaz Iribarnegaray and Václav Gregor and Florian L’écu Leal},
      title = {A note on the Unsuitability of {LIGA} for Linkable Ring Signatures: The perils of non-commutativity},
      howpublished = {Cryptology {ePrint} Archive, Paper 2026/671},
      year = {2026},
      url = {https://eprint.iacr.org/2026/671}
}