Paper 2026/2141

BitZ: proofs and commitments in arbitrary rings through binary fields

Abstract

We introduce BitZ, a hash-based Polynomial Commitment Scheme (PCS) for committing to multilinear polynomials $\mathbf{f}$ with coefficients in an arbitrary finitely generated ring $S$, e.g.\ a finite field $\mathbb{F}$, the integers $\mathbb{Z}$, a cyclotomic ring, etc. Moreover, given another arbitrary ring $R$ and a ring homomorphism $\psi:S\to R$, BitZ then proves evaluation claims over $R$ for the polynomial $\psi(\mathbf{f})$. BitZ's costs depend almost exclusively on the number of bits in the coefficients of $\mathbf{f}$, and not on $S$, $R$ or $\psi$. Moreover, BitZ provides range checks (or more generally, bit-size checks) essentially for free. BitZ can thus be used as a PCS in essentially any proof system. We do so to build a SNARK, called BitZ-SNARK, for integer polynomial constraints, following the fingerprinting technique of Campanelli and Hall-Andersen, where one commits over $\mathbb{Z}$ and proves the constraints over a random prime field $\mathbb{F}_q$, i.e. BitZ is deployed with $S=\mathbb{Z}$, $R=\mathbb{F}_q$, and $\psi$ reduction modulo $q$. BitZ applies equally to other ring-based proof systems, or field-based ones. To commit to $\mathbf{f}$, BitZ first decomposes $\mathbf{f}$ into a string of bits, and then commits to it over a binary field $\mathbb{F}_{2^{\nu}}$, in packed form. The scheme then proves the linear claim on $\psi(\mathbf{f})$ over the arbitrary ring $R$, even though it committed to the bits forming $\mathbf{f}$ over a binary field. We implement BitZ-SNARK and use it to prove, among others, SHA-256 hashing followed by ECDSA signature verification; RSA modular exponentiation and Poseidon hashing; integer multiplication; and SHA-256 hashing followed by multiplication modulo $2^{32}$, consistently obtaining better performance than prior approaches on most tasks. As an example, we achieve a throughput of $5$ million proved 32-bit integer multiplications per second on a MacBook Air M5 24 GB (10 threads, CPU-only) with proof sizes under $150$ kB. We prove a SHA-256 hash of a $2$ kB ($2^5$ compressions) message followed by a P-256 ECDSA signature verification with $50$ ms and $3.3$ ms prover and verifier time, respectively, and with a proof of $76$ kB, single-threaded. With $10$ threads the times are $20$ ms and $3.4$ ms.

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