Mathematical foundations of code-based cryptography for quantum-resistant systems
Khursheed Alamkhursheed.alam@sharda.ac.inDepartment of Mathematics & Data ScienceThe A. H. Siddiqi Centre for Advanced Research in Applied Mathematics & Physics (CARAMP)Sharda UniversityGreater Noida, Uttar Pradesh, 201310, IndiaView full profile → , Supriya Maheshwarisupriya.maheshwari@niet.co.inDepartment of MathematicsNoida Institute of Engineering and TechnologyGreater Noida, Uttar Pradesh, 201310, IndiaView full profile → , A. K. Singhaksinghrbs@yahoo.comDepartment of MathematicsRBS Engineering Technical CampusKhandari, Agra, Uttar Pradesh, 282002, IndiaView full profile → , *Masood AlamCorresponding authormalam@squ.edu.omDepartment of Mathematics & Information TechnologyCentre for Preparatory StudiesSultan Qaboos UniversityMuscat, 123, Sultanate of OmanView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Jan 2026
- Published Online:
- 30 Sep 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-2856
- Pages:
- 3379–3389
Abstract
This paper looks at the theoretical bases of code-based cryptographic techniques, specifically looking to determine if they can be used as a method to develop new post-quantum resistant systems. To do this we look at linear block codes, Hamming distance, and Goppa codes as the base in which to create secure communications. These are then used to build the McEliece and Niederreiter cryptosystems, where we illustrate the dependence on the computational difficulty of solving the Syndrome Decoding Problem (SDP), and the General Decoding Problem (GDP). Using both analytical methods, and by providing examples using numbers, we demonstrate how an intentional introduction of errors, and their subsequent correction, form the basis of encrypting information, and decrypting information. This is done to ensure that confidentiality, and robustness are provided. We also perform a comparison of these systems with RSA and ECC, and demonstrate that while the classical systems are compromised by Shor’s Algorithm, the code-based systems have exponential hardness. Figures and Tables help to visually understand the complexity analysis of these systems, the flow of encryption, and the process of error correction. Our results support our conclusion that McEliece and Niederreiter are viable candidate post-quantum cryptographic systems, as they are able to provide efficient, reliable, and provably secure alternatives to current systems.
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References
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