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A Modified Schnorr Sigma Protocol and Its Application to Isogeny-Based Identification

Authors:
Mahdi Mahdavi
Zaira Pindado
Amineh Sakhaie
Helena Rifà-Pous

Keywords: Schnorr Protocol; Zero-knowledge proofs; Discrete Logarithm Problem; Isogeny-based cryptography; Post-Quantum Cryptography.

Abstract:
Quantum computing threatens classical cryptographic protocols like the Schnorr identification scheme, which relies on the Discrete Logarithm Problem (DLP), vulnerable to quantum attacks. In this paper, we propose a modification to the classical Schnorr protocol by redefining the prover response as $r= cupm x$ mod $q$ instead of $r= u+ cx$ mod $q$. While this adjustment preserves the arithmetic simplicity of the original protocol, it introduces subtle but significant changes to the protocol's security and verifiability. We analyze its soundness, zero-knowledge properties, extractor functionality, and practical viability, and explore its adaptation into a secure digital signature system under standard cryptographic assumptions. To underscore the practical significance of our approach, we implement the modified protocol within an isogeny-based framework, demonstrating its capacity to enhance an existing identification scheme with respect to both security and efficiency. Our findings illustrate that revisiting classical protocols through judicious modifications can yield more robust, quantum-resistant solutions for applications like blockchain.

Pages: 125 to 131

Copyright: Copyright (c) IARIA, 2025

Publication date: October 26, 2025

Published in: conference

ISSN: 2162-2116

ISBN: 978-1-68558-306-4

Location: Barcelona, Spain

Dates: from October 26, 2025 to October 30, 2025