A Compact Post-quantum Strong Designated Verifier Signature Scheme from Isogenies

Abstract: Digital signatures are essential cryptographic tools that provide authentication and integrity in digital communications. However, privacy-sensitive applications, such as e-voting and digital cash, require more restrictive verification models to ensure confidentiality and control. Strong Designated Verifier Signature (SDVS) schemes address this need by enabling the signer to designate a specific verifier, ensuring that only this party can validate the signature. Existing SDVS constructions are primarily based on number-theoretic assumptions and are therefore vulnerable to quantum attacks. Although post-quantum alternatives, particularly those based on lattices, have been proposed, they often entail large key and signature sizes. In this work, we introduce $\mathsf{CSI\text{-}SDVS}$, a novel isogeny-based SDVS scheme that offers a compact, quantum-resistant alternative. Our construction builds on the ideal class group action framework of CSIDH and the signature techniques of CSI-FiSh, and relies on the hardness of the Multi-Target Group Action Inverse Problem (MT-GAIP). $\mathsf{CSI\text{-}SDVS}$ achieves strong security guarantees; namely, Strong Unforgeability under Chosen-Message Attacks (SUF-CMA), Non-Transferability (NT), and Privacy of Signer's Identity (PSI), in the random oracle model. Remarkably, both the keys and signatures in $\mathsf{CSI\text{-}SDVS}$ are of size $\mathcal{O}(\lambda)$, representing a significant improvement over the typical $\mathcal{O}(\lambda^2)$ bounds in existing post-quantum SDVS schemes, thereby making it among the most compact PQC-based SDVS schemes and the only post-quantum secure construction based on isogenies.