Modular forms and their role in cryptographic constructions
*Tanveer Ahmad WaniCorresponding authortanveer.ahmad@niu.edu.inDepartment of PhysicsNoida International UniversityGreater Noida, Uttar Pradesh, 203201, IndiaView full profile → , A. K. Bhuvaneswaribhuvaneswari@avit.ac.inDepartment of Humanities and Sciences (Mathematics Division)Aarupadai Veedu Institute of TechnologyVinayaka Mission’s Research Foundation (Deemed to be University)Paiyanoor, Chennai, Tamil Nadu, 603104, IndiaView full profile → , Swati Shivkumar Shriyalswati.shriyal@vupune.ac.inDepartment of Artificial IntelligenceVishwakarma UniversityPune, Maharashtra, 411048, IndiaView full profile → , Tushatusha@quantumeducation.inQuantum University Research CenterQuantum UniversityRoorkee, Uttarakhand, 247667, IndiaView full profile → , Yagna B. Adhyaruybadhyaru.ce.hcet@gokuluniversity.ac.inFaculty of EngineeringGokul Global UniversitySidhpur, Gujarat, 384151, IndiaView full profile → , Abrayev Baxrombabrayev@mail.ruDepartment of Information Technology and Exact SciencesTermez University of Economics and ServiceTermez, 190111, UzbekistanView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Apr 2026
- Published Online:
- 31 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-3043
- Pages:
- 4189–4199
Abstract
The paper is concerned with the necessity of new cryptographic systems on classical assumptions and the issue of the secure, quantum resistant communication systems. It suggests a cryptography scheme based on modular forms and exploitation of the transformation invariance, Fourier structure and L-functions to generate strong keys and encrypt. It has been experimentally demonstrated that the performance is best with shorter encryption time (6.7 ms), decryption time (7.4 ms) and protection strength (192 bits) that is better than RSA and ECC. The novelty is that it is a combination of deep number-theoretic features and real world cryptography. It has a contribution of one mathematical-cryptographic framework. The plan proposes efficient and post quantum secure applications that are scalable in the existing digital communications systems.
Keywords
Subject Classifications
References
[1] J.-M. Bohli, M. I. González Vasco, and R. Steinwandt, “Building group key establishment on group theory: A modular approach,” Symmetry, vol. 12, no. 2, Art. no. 197 (2020), doi: 10.3390/sym12020197.
[2] M. Bufalo, D. Bufalo, and G. Orlando, “A note on the computation of the modular inverse for cryptography,” Axioms, vol. 10, no. 2, Art. no. 116 (2021), doi: 10.3390/axioms10020116.
[3] M. Zohourian, A. Pamidimukkala, S. Kermanshachi, and D. Almaskati, “Modular construction: A comprehensive review,” Buildings, vol. 15, no. 12, Art. no. 2020 (2025), doi: 10.3390/buildings15122020.
[4] T.-H. Nguyen, C.-K. Pham, and T.-T. Hoang, “A high-efficiency modular multiplication digital signal processing for lattice-based post-quantum cryptography,” Cryptography, vol. 7, no. 4, Art. no. 46 (2023), doi: 10.3390/cryptography7040046.
[5] R. C. Poonia, R. Almakki, A. K. J. Saudagar, A. Altameem, and M. Albathan, “An optimization approach for real-time object detection in IoT devices through edge computing and deep learning,” Journal of Information and Optimization Sciences, vol. 45, no. 5, pp. 1465–1475 (2024), doi: 10.47974/JIOS-1768.
[6] M. Bufalo, D. Bufalo, and G. Orlando, “Some properties of the computation of the modular inverse with applications in cryptography,” Computation, vol. 11, no. 4, Art. no. 70 (2023), doi: 10.3390/computation11040070.
[7] I. Touil, W. Chikouche, and Dj. Benterki, “Interior-point method for CQP problems employing a novel trigonometric kernel function,” Journal of Information and Optimization Sciences, vol. 47, no. 2, pp. 437–453 (2026), doi: 10.47974/JIOS-1464.
[8] R. Muhenga, F. Sapundzhi, M. Popstoilov, S. Georgiev, and V. Todorov, “Comprehensive analysis of cryptographic algorithms: Implementation and security insights,” Engineering Proceedings, vol. 104, no. 1, Art. no. 43 (2025), doi: 10.3390/engproc2025104043.
[9] A. Ibrahim and F. Gebali, “Enhancing cryptographic solutions for resource-constrained RFID assistive devices: Implementing a resource-efficient field Montgomery multiplier,” Computers, vol. 14, no. 4, Art. no. 135 (2025), doi: 10.3390/computers14040135.
[10] E. Sakalauskas, “Enhanced matrix power function for cryptographic primitive construction,” Symmetry, vol. 10, no. 2, Art. no. 43 (2018), doi: 10.3390/sym10020043.
[11] M. Paunović, A. Büyükkaya, and M. Öztürk, “Modular suprametric spaces and fixed-point principles with applications in fractional burn-healing dynamics,” Mathematics, vol. 14, no. 7, Art. no. 1208 (2026), doi: 10.3390/math14071208.
[12] K. A. Sattar, T. Haider, U. Hayat, and M. D. Bustamante, “An efficient and secure cryptographic algorithm using elliptic curves and max-plus algebra-based wavelet transform,” Applied Sciences, vol. 13, no. 14, Art. no. 8385 (2023), doi: 10.3390/app13148385.
[13] J. I. Escribano Pablos, M. I. González Vasco, M. E. Marriaga, and Á. L. Pérez del Pozo, “Compiled constructions towards post-quantum group key exchange: A design from Kyber,” Mathematics, vol. 8, no. 10, Art. no. 1853 (2020), doi: 10.3390/math8101853.
[14] P.-P. Duong and C.-K. Pham, “Constructing 8 × 8 S-boxes with optimal Boolean function nonlinearity,” Cryptography, vol. 9, no. 4, Art. no. 67 (2025), doi: 10.3390/cryptography9040067.
