Paper 2026/1635 Original: eprint.iacr.org/2026/1635 ↗

Limber: Low Overhead SNARKs for Integers from Any PCS

Abstract

In real-world applications of SNARKs, non-native arithmetic, the emulation of computation outside of the specified SNARK field, is a key bottleneck. It introduces large overheads, and proof system designers often resort to non-standard SNARK-friendly hash-functions or other means like elliptic curve cycles to mitigate its costs. Besides performance concerns, non-native circuit arithmetization is also a major cause of implementation errors. In a collection of 27 critical bugs in real world ZK systems (0xPARC/zkbugtracker), 9 were related to non-native arithmetization. We tackle these challenges by constructing a minimal overhead SNARK for integer computation that generically handles non-native arithmetic. We follow the recipe of Zaratan (PKC 26), which proves an integer relation such as $a\cdot b = c + u\cdot m$ by fingerprinting---reducing it to the same relation but over a randomly sampled prime field. Realizing this recipe requires an integer mod-PCS that commits to integer polynomials and opens their evaluations modulo a random prime, which is crucially chosen after the underlying PCS's setup and commitment phases. Our central contribution is Limber, the first practical integer mod-PCS construction that asymptotically has $o(1)$ multiplicative commitment overhead and can be instantiated with any standard field polynomial commitment scheme, including ones over small fields. Combining Limber with a PIOP for integer R1CS over the random prime yields our SNARK. We demonstrate its practicality by implementing our scheme and showing that we can prove an RSA accumulator benchmark $15$-$30\times$ faster than circuit-based approaches, and non-native Poseidon2 hashing $24\times$ faster.