Non-interactive Zero-Knowledge from LPN and MQ

We give the first construction of non-interactive zero-knowledge (NIZK) arguments from post-quantum assumptions other than Learning with Errors. In particular, we achieve NIZK under the polynomial hardness of the Learning Parity with Noise (LPN) assumption, and the exponential hardness of solving random under-determined multivariate quadratic equations (MQ). We also construct NIZK satisfying statistical zero-knowledge assuming a new variant of LPN, Dense-Sparse LPN, introduced by Dao and Jain (CRYPTO 2024), together with exponentially-hard MQ.

The main technical ingredient of our construction is an extremely natural (but only in hindsight!) construction of correlation-intractable (CI) hash functions from MQ, for a NIZK-friendly sub-class of constant-degree polynomials that we call concatenated constant-degree polynomials. Under exponential security, this hash function also satisfies the stronger notion of approximate CI for concatenated constant-degree polynomials. The NIZK construction then follows from a prior blueprint of Brakerski-Koppula-Mour (CRYPTO 2020). In addition, we show how to construct (approximate) CI hashing for degree-$d$ functions from the (exponential) hardness of solving random degree-$d$ equations, a natural generalization of MQ. To realize NIZK with statistical zero-knowledge, we design a lossy public-key encryption scheme with approximate linear decryption and inverse-polynomial decryption error from Dense-Sparse LPN. These constructions may be of independent interest.

Our work therefore gives a new way to leverage MQ with uniformly random equations, which has found little cryptographic applications to date. Indeed, most applications in the context of encryption and signature schemes make use of structured variants of MQ, where the polynomials are not truly random but posses a hidden planted structure. We believe that the MQ assumption may plausibly find future use in the designing other advanced proof systems.