1970-01-01

(2026-05) Zinc+; SNARKs for Polynomial Rings

[[Alexander Abdugafarov]]
[[Albert Garreta]]
[[Amit Kumar]]
[[Michał Osadnik]]
[[Psi Vesely]]
[[Ilia Vlasov]]
[[Kai Zhe Zheng]]

2026-05-01

Abstract

Nearly all succinct proof systems express computations as algebraic constraints over a finite field. Operations not native to this field, such as bitwise manipulation, modular arithmetic, and lattice-ring operations, require an arithmetization step that can inflate the witness size by one or more orders of magnitude.

We introduce Universal Constraint Systems (UCS) and Zinc $+$ . The first is a relation that can express the above constraints with minimal overhead. The second is a framework for building SNARKs for UCS. Concretely, UCS consists of algebraic constraints and ideal membership predicates over multiple polynomial rings simultaneously, such as $\mathbb{F}_q[X], \mathbb{Q}[X], \mathbb{Z}[X]$ , etc.

Zinc $+$ SNARKs are built from 1) a PIOP for UCS, and 2) a hash-based IOPP for multilinear polynomials over $R=\mathbb{Q}[X]$ or $R=\mathbb{F}_q[X]$ . For 1), we provide a general compiler that takes standard finite-field PIOPs and turns them into a PIOP for UCS. The IOPP in 2) depends on $R$ : for $R=\mathbb{F}_q[X]$ , we construct it via a black-box lift of any existing IOPP for $\mathbb{F}_q$ , and for $R=\mathbb{Q}[X]$ , we present a novel tensor IOPP design, instantiated with the new code family below.

We introduce Integer Pseudo-Reed Solomon (IPRS) codes, a new family of MDS codes over $\mathbb{Q}$ and $\mathbb{Q}[X]$ . While not Reed-Solomon codes, these codes have optimal MDS relative minimal distance, support efficient FFT-based encoding, and have bounded norm growth when encoding (unlike a naïve lift of Reed-Solomon codes to the integers).

Our unoptimized, open-source, implementation proves 7 SHA-256 compressions followed by the multi-scalar multiplication (MSM) part of an ECDSA verification (the bulk of the work), with the following performance, benchmarked on a MacBook Air M4, without zero-knowledge:

Prover time: 40.6 ms, Verifier time: 7.0 ms, Proof size: 198 KB.

Zinc $+$ can be instantiated end-to-end or as a lightweight extension to any existing hash-based SNARK over~ $\mathbb{F}_q$ .