International Association
for Cryptologic Research

An extractable one-way function (EOWF), introduced by Canetti and Dakdouk (ICALP 2008) and generalized by Bitansky et al. (SIAM Journal on Computing vol. 45), is an OWF that allows for efficient extraction of a preimage for the function. We study (generalized) EOWFs that have a public image verification algorithm. We call such OWFs verifiably-extractable and show that several previously known constructions satisfy this notion. We study how such OWFs relate to subversion zero-knowledge (Sub-ZK) NIZKs by using them to generically construct a Sub-ZK NIZK from a NIZK satisfying certain additional properties, and conversely show how to obtain them from any Sub-ZK NIZK. Prior to our work, the Sub-ZK property of NIZKs was achieved using concrete knowledge assumptions.

Significantly extending the framework of (Couteau and Hartmann, Crypto 2020), we propose a general methodology to construct NIZKs for showing that an encrypted vector $\vec{\chi}$ belongs to an algebraic set, i.e., is in the zero locus of an ideal $\mathscr{I}$ of a polynomial ring. In the case where $\mathscr{I}$ 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 $F (\vec{\chi}) = 0$. This leads to compact NIZKs for general computational structures, such as polynomial-size algebraic branching programs. We extend the framework to the case where $\IDEAL$ is non-principal, obtaining efficient NIZKs for R1CS, arithmetic constraint satisfaction systems, and thus for $\mathsf{NP}$. 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.