Hardware Realization of Fast Multi-Scalar Elliptic Curve Point Multiplication by Reducing the Hamming Weights Over GF(p)

Full Text (PDF, 486KB), PP.57-63

Views: 0 Downloads: 0


Nagaraja Shylashree 1,* Venugopalachar Sridhar 1

1. Department of E & C Engineering, Mandya, 571401, India

* Corresponding author.

DOI: https://doi.org/10.5815/ijcnis.2014.10.07

Received: 7 Jan. 2014 / Revised: 5 Apr. 2014 / Accepted: 16 May 2014 / Published: 8 Sep. 2014

Index Terms

Sum of products expansion, Hamming Weight, multi-scalar point multiplication, triple-scalar point multiplication, elliptic curve point multiplication, elliptic curve point addition


We present a new hardware realization of fast elliptic curve Multi-Scalar Point Multiplication (MSPM) using the sum of products expansion of the scalars. In Elliptic curve point Multiplication latency depends on the number of one’s (Hamming Weight) in the binary representation of the scalar multiplier. By reducing the effective number of one’s in the multiplier, the multiplication speed is automatically increased. Therefore we describe a new method of effectively reducing the Hamming weight of the scalar multipliers thereby reduces the number of Point Adders when multi scalar multiplication is needed. The increase in speed achieved outweighs the hardware cost and complexity.

Cite This Paper

Nagaraja Shylashree, Venugopalachar Sridhar, "Hardware Realization of Fast Multi-Scalar Elliptic Curve Point Multiplication by Reducing the Hamming Weights Over GF(p)", International Journal of Computer Network and Information Security(IJCNIS), vol.6, no.10, pp.57-63, 2014. DOI:10.5815/ijcnis.2014.10.07


[1]N. Koblitz, “Elliptic curve cryptosystems”, Mathematics of Computation, Vol.48, pp 203-209, 1987.
[2]V. Miller, “Uses of Elliptic Curve in Cryptography”, Advances in Cryptology Crypto’85, LNCS, Vol. 218, pp. 417–426, 1986.
[3]IEEE 1363 standard specifications for public-key cryptography, 1363, Jan 2000.
[4]NIST, Recommended Elliptic Curves for Federal Government Use, May 1999 (http://csrc.nist.gov/encryption).
[5]K.Muthumayil, Dr.V.Rajamani, Dr.S.Manikandan and M.Buvana, “A Group Key Agreement Protocol based on stability and power using Elliptic curve cryptography” IEEE International Conference on Emerging Trends in Electrical and Computer Technology, pp.1051 - 1056,2011.
[6]Raveen R. Goundar, Ken-ichi Shiota, and Masahiko Toyonaga, “A Novel Method for Elliptic Curve Multi-Scalar Multiplication”, World Academy of Science, Engineering and Technology, Vol 33, pp 832-836, 2009.
[7]Darrel Hankerson, Alfred Menezes and Scott Vanstone, “Guide to Elliptic Curve Cryptography” Springer, 2004.
[8]Idrissi, Y.E.H.E., N. Zahid and M. Jedra, “Security analysis of 3GPP (LTE)-WLAN interworking and a new local authentication method based on EAP-AKA”. IEEE International Conference on Future Generation Communication Technology, pp. 137-142, 2012.
[9]Shivkumar,S. and G.Umamaheswari, “Certificate authority schemes using elliptic curve cryptography, RSA and their variants-simulation using ns2”. American Journal of Applied Sciences, vol 11, pp. 171-179, 2014.
[10]Jie. L.K. and H. Kamarulhaili, “Polynomial interpolation
in the elliptic curve Cryptosystem”, Journal of Math. Stat.,
vol 7, pp.326-331, 2011.
[11]Ismail, E.S. and M.S. Hijazi, 2012. Development of a new elliptic curve cryptosystem with factoring problem. American Journal of Applied Sciences, vol 9, pp.1443-1447, 2012.
[12]Ghosh S., M. Alam, I.S. Gupta and D.R. Chowdhury, “A Robust GF(p) parallel arithmetic unit for public key cryptography”, IEEE 10th International conference on Euromicro Digital System Design Architectures, Methods and Tools, pp.109-115, 2007.
[13]G. Orlando, C. Paar, “A Scalable GF(p), Elliptic CurveProcessor Architecture for Programmable Hardware”, in: Cryptographic Hardware and Embedded Systems (CHES), LNCS 2162, pp.348-363, 2001.
[14]De Dormole, G.M. and J.J. Quisquater, “High-speed hardware implementations of elliptic curve cryptography A Survey”, Journal of System Architecture, vol 53, pp. 72-84, 2007.
[15]Sandeep S.V, HameemShanavas.I, Nallusamy.V, Brindha.M, “Hardware Implementation of Elliptic Curve Cryptography over Binary Field”, IJCNIS Vol. 4, PP.1-7, 2012.