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A Zero-Knowledge Proof of Knowledge for Subgroup Distance Problem

Cansu Betin Onur

TL;DR

This paper addresses zero-knowledge identification by basing security on the Subgroup Distance Problem (SDP) in the $Hamming$ metric over the symmetric group $S_n$. It proposes Subgroup Distance Zero Knowledge Proof (SDZKP), a three-round identification protocol that masks secrets with a cryptographically secure pseudorandom number generator and employs a Stern-type framework to enable a witness extraction. The authors prove perfect completeness, $3$-special-soundness, and statistical zero-knowledge under standard cryptographic assumptions, presenting a robust, code-based, post-quantum-friendly alternative for secure authentication. The work expands the ZKP toolkit by leveraging SDP hardness and code-based techniques to deliver a practical, quantum-resilient identification mechanism with strong security guarantees.

Abstract

In this study, we introduce a novel zero-knowledge identification scheme based on the hardness of the subgroup distance problem in the Hamming metric. The proposed protocol, named Subgroup Distance Zero Knowledge Proof (SDZKP), employs a cryptographically secure pseudorandom number generator to mask secrets and utilizes a Stern-type algorithm to ensure robust security properties.

A Zero-Knowledge Proof of Knowledge for Subgroup Distance Problem

TL;DR

This paper addresses zero-knowledge identification by basing security on the Subgroup Distance Problem (SDP) in the metric over the symmetric group . It proposes Subgroup Distance Zero Knowledge Proof (SDZKP), a three-round identification protocol that masks secrets with a cryptographically secure pseudorandom number generator and employs a Stern-type framework to enable a witness extraction. The authors prove perfect completeness, -special-soundness, and statistical zero-knowledge under standard cryptographic assumptions, presenting a robust, code-based, post-quantum-friendly alternative for secure authentication. The work expands the ZKP toolkit by leveraging SDP hardness and code-based techniques to deliver a practical, quantum-resilient identification mechanism with strong security guarantees.

Abstract

In this study, we introduce a novel zero-knowledge identification scheme based on the hardness of the subgroup distance problem in the Hamming metric. The proposed protocol, named Subgroup Distance Zero Knowledge Proof (SDZKP), employs a cryptographically secure pseudorandom number generator to mask secrets and utilizes a Stern-type algorithm to ensure robust security properties.
Paper Structure (9 sections, 1 theorem, 3 equations, 1 figure)

This paper contains 9 sections, 1 theorem, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subgroup Distance Problem (SDP)
  4. Basic Cryptographic Definitions
  5. Subgroup Distance Zero Knowledge Proof (SDZKP)
  6. Setup
  7. Protocol Design
  8. Security Analysis
  9. Conclusion

Key Result

Theorem 4.1

Protocol SDZKP is a black-box statistical zero knowledge proof of knowledge protocol with knowledge soundness error $\frac{2}{3}.$

Figures (1)

  • Figure 1: The message flow diagram depicting Subgroup Distance Zero Knowledge Proof (SDZKP).

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: Zero-knowledge proof of knowledge (ZKPoK)
  • Definition 2.7
  • Definition 2.8
  • Theorem 4.1
  • Proof 4.1