Paper 2024/1344 Original: eprint.iacr.org/2024/1344 ↗

Quantum Security of a Compact Multi-Signature

Abstract

With the rapid advance in quantum computing, quantum security is now an indispensable property for any cryptographic system. In this paper, we study how to prove the security of a complex cryptographic system in the quantum random oracle model. We first give a variant of Zhandry's compressed quantum random oracle (${\bf CStO}$), called compressed quantum random oracle with adaptive special points ({\bf CStO}$_s$). Then, we extend the on-line extraction technique of Don et al (EUROCRYPT'22) from {\bf CStO} to ${\bf CStO}_s$. We also extend the random experiment technique of Liu and Zhandry (CRYPTO'19) for extracting the ${\bf CStO}$ query that witnesses the future adversarial output. With these preparations, a systematic security proof in the quantum random oracle model can start with a random {\bf CStO} experiment (that extracts the witness for the future adversarial output) and then convert this game to one involving ${\bf CStO}_s$. Next, the on-line extraction technique for ${\bf CStO}_s$ can be applied to extract the witness for any on-line commitment. With this strategy, we give a security proof of our recent compact multi-signature framework that is converted from any weakly secure linear ID scheme. We also prove the quantum security of our recent lattice realization of this linear ID scheme, by iteratively applying the weakly collapsing protocol technique of Liu and Zhandry (CRYPTO 2019). Combining these two results, we obtain the first quantum security proof for a compact multi-signature.