Elliptic Curve Scalar Multiplication Operation: a Survey Study


  • Ayaat Waleed Department of Mathematics, Faculty of Education for Girls, University of Kufa
  • Najlae Falah Hameed Al Saffar Department of Mathematics, Faculty of Computer Science and Mathematics, University of Kufa https://orcid.org/0000-0001-5599-2885




Elliptic curve scalar multiplication, Signed binary method, NAF, w-NAF, MOF, DRM, CRT


Scalar multiplication is the fundamental operation in the elliptic curve cryptosystem. It involves calculating the integer multiple of a specific elliptic curve point. It involves three levels: field, point, and scalar arithmetic. Scalar multiplication will be significantly more efficient overall if the final level is improved. By reducing the hamming weight or the number of operations in the scalar representation, one can raise the level of scalar arithmetic. This paper reviews some of the algorithms and techniques that improve the elliptic curve scalar multiplication in terms of the third level.



Download data is not yet available.

Author Biography

Najlae Falah Hameed Al Saffar, Department of Mathematics, Faculty of Computer Science and Mathematics, University of Kufa




V. S. Miller, Use of elliptic curves in cryptography, Advances in Cryptology, Proceedings of CRYPTO'85, LNCS, 218 (1986), 417-426.

N. Koblitz, Elliptic curve cryptosystem, Mathematics of Computation, 48 (1987) 203-209.

Booth, A. D. A Signed Binary Multiplication Technique. The Quarterly Journal of Mechanics and Applied Mathematics(1951). 4 (2): 236–24.

A.D. Booth, A signed binary multiplication technique, Journal of Applied Mathematics, 4(2) (1951), 236-240.

G. W. Reitwiesner, Binary Arithmetic, Advances in computers, 1 (1960), 231-308.

F. Morain, J. Olivos, Speeding up the computations on an elliptic curve using addition subtraction chains, RAIRO Theoretical Informatics and Applications, 24 (1990), 531-543.

K. Okeya, Signed binary representations revisited, Proceedings of CRYPTO'04 (2004), 123-139.

M. Joye, S. Yen, Optimal left to right binary signed digit recoding, IEEE Transactions on Computers, 49 (2000), 740-748.

P. Balasubramaniam, E. Karthikeyan, Elliptic curve scalar multiplication algorithm using complementary ISSN

Najlae. F.H: Development of Some Fast And Efficient Methods For Elliptic Curve Scalar Multiplication Over prime fields, Ph.D. Philosophy, Malaysia, Universiti Putra Malaysia, (2015), 39-57

Kavin, B. P., & Ganapathy, S. (2021). A new digital signature algorithm for ensuring data integrity in the cloud using elliptic curves. Int. Arab J. Inf. Technol., 18(2), 180-190.‏

Bos, J. W., Halderman, J. A., Heninger, N., Moore, J., Naehrig, M., & Wustrow, E. (2014). Elliptic curve cryptography in practice. In Financial Cryptography and Data Security: 18th International Conference, FC 2014, Christ Church, Barbados, March 3-7, 2014, Revised Selected Papers 18 (pp. 157-175). Springer Berlin Heidelberg.‏

Revathi, S. Cloud Data security based on Fuzzy Intrusion Detection system with Elliptic Curve Cryptography (ECC) using Non-Adjacent Form (NAF) Algorithm.‏

Hossain, M. R., & Hossain, M. S. (2019, February). Efficient FPGA implementation of modular arithmetic for elliptic curve cryptography. In 2019 International conferencConferencericaElectricalr and communication engineering (ECCE) (pp. 1-6). IEEE.‏

Paliakou, A. Y., Kulikau, V. P., & Stsiapanau, А. А. (2019, November). Resistance projection welding of sheet metal without the formation of a mutual melting zone in the form of a cast nugget. In International Conference on Aviamechanical Engineering and Transport (AviaENT 2019) (pp. 264-270). Atlantis Press.‏

Feldman, N., & Heffetz, O. A grant to every citizen: Survey evidence of the impact of a direct government payment in Israel. National Tax Journal, (2022). 75(2), 229-263.‏

Padhy, S., Shankar, T. N., & Dash, . A Comparison among Fast Point Multiplication Algorithms in Elliptic Curve Cryptosystem(2022).‏

Okeya, K., Schmidt-Samoa, K., Spahn, C., & Takagi, T. (2004, August). Signed binary representations revisited. In CRYPTO (Vol. 2004, pp. 23-139).‏

Maimuţ, D., & Matei, A. C. 0(2022). Speeding-Up Elliptic Curve Cryptography Algorithms. Mathematics, 10(19), 3676.‏

Wu, J., Yu, L., & Khan, Z. (2023). How Do Mutual Dependence and Power Imbalance Condition the Effects of Technological Similarity on Post‐acquisition Innovation Performance Over Time? British Journal of Management, 34(1), 195-219.‏

F. Morain and J. Olivos. Speeding up the computations on an elliptic curve addition-subtraction chains.Theoretical Informatics and Applications,24:531–543, 1990

George W. Reitwiesner. Binary arithmetic.Advances in Computers,1:231–308, 1960

Saffar, N. F. H. A., & Said, M. R. M. (2015). Speeding up the elliptic curve scalar multiplication using a non-adjacent form. Journal of Discrete Mathematical Sciences and Cryptography, 18(6), 801-821.‏

Masson, S., Sanso, A., & Zhang, Z. (2021). Bandersnatch: a fast elliptic curve built over the bls12-381 scalar field. Cryptology ePrint Archive.‏

Bendimerad, M. Y., double Berrich, L., Masoud, M., Jaradat, Y., Jannoud, I., Jaglan, N., ... & Toulali, I. (2015). Communications Antenna and Propagation.‏

Vahdati, Z., Yasin, S., Ghasempour, A., & Salehi, M. (2019). Comparison of ECC and RSA algorithms in IoT devices. Journal of Theoretical and Applied Information Technology, 97(16).‏

Poole, B., Ozair, S., Van Den Oord, A., Alemi, A., & Tucker, G. (2019, May). On variational bounds of mutual information. In International Conference on Machine Learning (pp. 5171-5180). PMLR.‏

Anagreh, M., Vainikko, E., & Laud, P. (2020). Speeding Up the Computation of Elliptic Curve Scalar Multiplication Based on CRT and DRM. In ICISSP (pp. 176-184).‏




How to Cite

Waleed, A., & Al Saffar, N. F. H. (2023). Elliptic Curve Scalar Multiplication Operation: a Survey Study. Journal of Kufa for Mathematics and Computer, 10(2), 173–178. https://doi.org/10.31642/JoKMC/2018/100226

Similar Articles

<< < 1 2 3 4 5 6 7 > >> 

You may also start an advanced similarity search for this article.