Efficient scalar multiplication techniques on elliptic curves
Bhawani Sankar Panigrahibspanigrahi.cse@gmail.comDepartment of Computer Science & EngineeringGITAM School of Computer Science and EngineeringGITAM Deemed to be UniversityVishakhapatnam, Andhra Pradesh, 530045, IndiaView full profile → , *Rahul Kumar SinghCorresponding authorrahul.singh1@niu.edu.inDepartment of Computer Science & EngineeringNoida International UniversityGreater Noida, Uttar Pradesh, 203201, IndiaView full profile → , Praney Madanpraney.madan.orp@chitkara.edu.inCentre for Research Impact and OutcomeChitkara UniversityRajpura, Punjab, 140417, IndiaView full profile → , Nyalam Ramunyalamramu.nr@proton.meCentre for Multidisciplinary ResearchAnurag UniversityHyderabad, Telangana, 500088, IndiaView full profile → , Anisha Chaudharyanisha.chaudhary@quantumeducation.inQuantum University Research CenterQuantum UniversityRoorkee, Uttarakhand, 247667, IndiaView full profile → , Manoj I. Patelmanoj.patel@gokuluniversity.ac.inFaculty of Computer Science and ApplicationGokul Global UniversitySiddhpur, Gujarat, 384151, India0009-0009-0731-9528View 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-3054
- Pages:
- 4321–4331
Abstract
The elliptic curve cryptography is based upon the efficient scalar multiplication, which directly affects the performance and security of a calculation. The article is an in-depth analysis of the advanced techniques of multiplication of scalars to reduce time complexity and resource consumption. It analyzes algorithmic optimization strategies, including, but not limited to, precomputation techniques, windowing techniques and memorycomputation trade-offs. The parallel processing and pipelining are addressed to enhance the throughput features of hardware and software programs. The use of the GLV and GLS methods of endomorphisms are examined with the aim of hastening the breakdown of scalars. The binary method and its computational complexity is also taken into consideration along with the optimized point addition and doubling algorithms in the context of the group law. According to the experimental findings, the time spent on execution, the amount of energy used, and the scalability have improved positively, which demonstrates the applicability of the proposed optimizations to the recent cryptographic systems.
Keywords
Subject Classifications
References
[1] F.-J. Kan, Y.-H. Chen, J.-J. Wang, and C.-D. Lee, “Efficient scalar multiplication of ECC using lookup table and fast repeating point doubling,” Mathematics, vol. 13, no. 6, Art. no. 924 (2025), doi: 10.3390/math13060924.
[2] N. H. Sabbry and A. B. Levina, “An optimized point multiplication strategy in elliptic curve cryptography for resource-constrained devices,” Mathematics, vol. 12, no. 6, Art. no. 881 (2024), doi: 10.3390/math12060881.
[3] C. A. Lara-Nino, A. Diaz-Perez, and M. Morales-Sandoval, “Energy/area-efficient scalar multiplication with binary Edwards curves for the IoT,” Sensors, vol. 19, no. 3, Art. no. 720 (2019), doi: 10.3390/s19030720.
[4] Y. Hao, S. Zhong, M. Ma, R. Jiang, S. Huang, J. Zhang, and W. Wang, “Lightweight architecture for elliptic curve scalar multiplication over prime field,” Electronics, vol. 11, no. 14, Art. no. 2234 (2022), doi: 10.3390/electronics11142234.
[5] Y.-D. Ning, Y.-H. Chen, C.-S. Shih, and S.-I. Chu, “Lookup table-based design of scalar multiplication for elliptic curve cryptography,” Cryptography, vol. 8, no. 1, Art. no. 11 (2024), doi: 10.3390/cryptography8010011.
[6] Z. Zhang, W. Wang, J. Zhang, X. He, M. Ma, S. Ren, and H. Dang, “Area-time-efficient secure comb scalar multiplication architecture based on recoding,” Micromachines, vol. 15, no. 10, Art. no. 1238 (2024), doi: 10.3390/mi15101238.
[7] J. Kwon, S. C. Seo, and S. Hong, “Efficient implementations of four-dimensional GLV-GLS scalar multiplication on 8-bit, 16-bit, and 32-bit microcontrollers,” Applied Sciences, vol. 8, no. 6, Art. no. 900 (2018), doi: 10.3390/app8060900.
[8] V. Gaikwad Shrimant, R. Halam, G. Tiwari, S. Chilukuri, R. Ramesh, and A. Singla, “Developing secure communication protocols using elliptic curve cryptography for IoT devices,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 29, no. 2-A, pp. 533–541 (2026), doi: 10.47974/JDMSC-2493.
[9] A. Sajid, O. S. Sonbul, M. Rashid, and M. Y. I. Zia, “A hybrid approach for efficient and secure point multiplication on binary Edwards curves,” Applied Sciences, vol. 13, no. 9, Art. no. 5799 (2023), doi: 10.3390/app13095799.
[10] W. Eid, T. F. Al-Somani, and M. C. Silaghi, “Efficient elliptic curve operators for Jacobian coordinates,” Electronics, vol. 11, no. 19, Art. no. 3123 (2022), doi: 10.3390/electronics11193123.
[11] N. Dehgardi, M. Azari, and Y. Shang, “Relationship between the inverse sum indeg index of graph operations and their factors,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 29, no. 3, pp. 1181–1199 (2026), doi: 10.47974/JDMSC-2193.
[12] V. Tanksale, “Efficient elliptic curve Diffie–Hellman key exchange for resource-constrained IoT devices,” Electronics, vol. 13, no. 18, Art. no. 3631 (2024), doi: 10.3390/electronics13183631.
[13] M. Haritha, M. Nithin Venkat Sai, M. Sree Rishik, M. Venkata Sriram, M. S. Siva Srinivas, and S. V. V. S. Kumar, “Intrusion detection and prevention using Snort and Python for detecting DDOS attack and DNS flood,” Scholaria Publications, vol. 7, no. 2 (2026), doi: 10.64264/ijisea/0730.
