Paper 2026/1722 Original: eprint.iacr.org/2026/1722 ↗

Elementary-Vector Modeling for Shorter VOLE-Based Signatures from SD and PKP

Abstract

Zero-knowledge proofs based on VOLE have recently become a promising approach for designing post-quantum signature schemes. Such schemes are constructed by having a signer prove knowledge of a secret input for a prescribed one-way function. In this work, we optimize VOLE-based signature schemes from syndrome decoding (SD) and the permuted kernel problem (PKP). Despite relying on different one-way functions, these schemes share a common modeling method. Elementary vectors are compressed into shorter witnesses and then reconstructed through low-degree polynomials. We propose a unified framework for elementary-vector encoding and reconstruction. By choosing different parameters, this framework captures previous modeling approaches for SD and PKP as special cases. Based on this framework, we propose an optimized encoding and reconstruction approach that reduces the extended witness size and hence the final signature size. At NIST Level I security, our SD- and PKP-based schemes achieve “signature + public key” sizes of 3,496 and 3,341 bytes, respectively, improving over the corresponding 3,775 bytes of SDitH v2.0 and 3,577 bytes of PERK v2.2. Furthermore, we construct a 3,673-byte post-quantum ring signature for rings of 2^20 members, which is 27% and 70% smaller than those of Chiang et al. (CCS ’25) and BLOOM (ASIACRYPT ’22), respectively.