Efficient NIZKs for Algebraic Sets

Geoffroy Couteau (1) , Helger Lipmaa , Roberto Parisella , Arne Tobias Ødegaard

1 DI-ENS - Département d'informatique - ENS Paris

Significantly extending the framework of (Couteau and Hartmann, Crypto 2020), we propose a general methodology to construct NIZKs for showing that an encrypted vector χ belongs to an algebraic set, i.e., is in the zero locus of an ideal I of a polynomial ring. In the case where I is principal, i.e., generated by a single polynomial F , we first construct a matrix that is a "quasideterminantal representation" of F and then a NIZK argument to show that F (χ) = 0. This leads to compact NIZKs for general computational structures, such as polynomial-size algebraic branching programs. We extend the framework to the case where I is non-principal, obtaining efficient NIZKs for R1CS, arithmetic constraint satisfaction systems, and thus for NP. As an independent result, we explicitly describe the corresponding language of ciphertexts as an algebraic language, with smaller parameters than in previous constructions that were based on the disjunction of algebraic languages. This results in an efficient GL-SPHF for algebraic branching programs.

Mots clés: Algebraic branching programs; algebraic languages; algebraic sets; NIZK; pairingbased cryptography; SPHF; zero knowledge

Domaines: Cryptographie et sécurité [cs.CR]

Geoffroy Couteau, Helger Lipmaa, Roberto Parisella, Arne Tobias Ødegaard. Efficient NIZKs for Algebraic Sets. ASIACRYPT 2021 - International Conference on the Theory and Application of Cryptology and Information Security, Dec 2021, Online, Singapore. ⟨hal-03374536⟩