Abstract—The rise of quantum computing is putting tremendous pressure on existing cryptographic systems, particularly digital signature schemes based on Rivest-Shamir-Adleman (RSA) and Elliptic Curve Cryptography. Quantum algorithms such as Shor and Grover have demonstrated the ability to severely weaken traditional security assumptions, highlighting the urgent need to develop new quantum-resistant digital signature schemes. Rather than relying on standard approaches such as lattice-based or multivariate-based cryptography, this paper explores an alternative direction: leveraging a new nonlinear exponentiation problem defined over finite fields, whose mathematical structure is designed to render Shor's algorithm inapplicable and to reduce the effectiveness of Grover’s algorithm. Based on this newly proposed hard problem, we introduce multiple digital signature schemes, each with a distinct structure in its signing and verification algorithms. Although these schemes differ in operational mechanisms, they all maintain correctness, remain secure against classical attacks, offer strong quantum resistance, and are fully compatible with existing Public Key Infrastructure system. Through both theoretical analysis and performance evaluation, we demonstrate that diversifying digital signature constructions from a single underlying hard problem is not only feasible but also offers practical advantages: it allows selecting a design best suited for specific application environments while maintaining post-quantum security. This result opens a promising new path toward the development of flexible, efficient, and long-term secure digital signature schemes for the post-quantum era.

Keywords—post-quantum digital signature, new hard problem, shor’s algorithm, grover’s algorithm, Public Key Infrastructure (PKI), non-standard assumption

Cite: Tuan Nguyen Kim, Luu Hong Dung, Hoang Duc Tho, and Ha Nguyen Hoang, “Quantum Resistant Digital Signatures Constructed from a Nonstandard Discrete Logarithm Problem over Finite Fields," Journal of Communications, vol. 21, no. 1, pp. 1-11, 2026.

Copyright © 2026 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).