Logarithmic-Size (Linkable) Ring Signatures from Lattice Isomorphism Problems

The Lattice Isomorphism Problem (LIP) asks whether two given lattices are isomorphic via an orthogonal linear transformation. At Eurocrypt 2022, Ducas and van Woerden provide a solid foundation for LIP as a promising candidate for post-quantum cryptography. They then propose a digital signature HAWK from LIP in the hash-then-sign framework, whose module version was recently investigated by Ducas et al. at Asiacrypt 2022. HAWK is one of the brightest prospects at round one of the NIST for additional digital signatures. In this paper, we build the first (linkable) ring signature schemes based on the hardness of LIP. The proposed signatures have the logarithmic size in the number of ring users. Our signature size is significantly smaller than several ring signatures based on other underlying problems when the number of users in the ring is large. To this end, we leverage group action properties of LIP and follow the Merkle tree-based construction of Beullens, Katsumata and Pintore at Asiacrypt 2020 in the context of isogeny-based cryptography, with suitable adaptions to lattice isomorphism group actions.