zkDaily - 2026-09-07

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ZKP Frontier Tracker 🎯

Mon

09.07

2026

Subring VOLE over Galois Rings with Applications to ZK over

\mathbb{Z}_{p^k}

Paper

https://eprint.iacr.org/2026/1864

subring VOLEGalois rings

Cascudo and Liu proposed subring VOLE (srVOLE) over Galois rings, constructed maliciously secure protocols, and built publicly verifiable ZK proofs over rings with reduced communication.

NotesNotes要点

Proposed subring VOLE (srVOLE), generalizing subfield VOLE to Galois rings
Constructed two maliciously secure srVOLE protocols: one PCG-like with low amortized communication, one SoftSpoken-like compatible with VOLEitH
Can be used to build publicly verifiable VOLE-based ZK
Shown that Moz $\mathbb{Z}_{2^k}$ arella's VOLE correlation is a special case of srVOLE
Adapted QuickSilver protocol to any Galois ring
Circuit satisfiability: communication reduced to one subring element per multiplication gate

Collected by @icerdesign

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Q&A Deep Dive 💬

Mon

09.07

2026

What aspects of existing ZK are improved?

It reduces communication and achieves public verifiability, especially for ZK over integer rings.

Why can communication be halved for circuit satisfiability?

Using srVOLE, each multiplication gate requires communicating just one subring element rather than a more expensive field element.

What restriction is overcome for polynomial satisfiability?

Prior schemes were mostly limited to degree-2 relations; this protocol supports arbitrary low-degree polynomials.

Prompted by @icerdesign