Int. J. Advanced Networking and Applications 1329 Volume: 03 Issue: 05 Pages: 1329-1337 (2012) Elliptic Curve Point Multiplication Using MBNR and Point Halving G.N.Purohit Department of Mathematics, Banasthali University Jaipur, Rajasthan,304022, India Email: gn purohitjaipur@yahoo.co.in Asmita Singh Rawat Department of Computer Science, Banasthali University Jaipur, Rajasthan,304022, India Email: singh.asmita27@gmail.com Manoj Kumar Department of Computer Science, KNIT, UPTU Sultanpur, 228188 Uttar Pradesh, India. Email: rajrajmanoj@gmail.com -----------------------------------------------------------------------ABSTRACT---------------------------------------------------------- The fast implementation of elliptic curve cryptosystems relies on the efficient computation of scalar multiplication. As generalization of double base number system of a number k to multi-base number system (MBNR) provides a faster method for the scalar multiplication is most important and costly operation (in terms of time) in ECC, there is always a need of developing a faster method with lower cost. In this paper we optimize the cost of scalar multiplication using halving and add method instead doubling and tripling methods. The cost is reduced from 40% to 50% with respect to the other fastest techniques Keywords: Double base number system, Elliptic curve cryptography, multi-base number system, point halving, W SN. -------------------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: November 12, 2011 Date of Acceptance: January 18, 2012 -------------------------------------------------------------------------------------------------------------------------------------------------- 1. INTRODUCTION Public-key cryptography has been widely studied and used since 1975 when Rivest, Shamir, and Adleman invented RSA public key cryptography. This system heavily depends on integer factorization problem (IFP) using big key bits such as 1024 bits and 2048 bits. Later on Deffie- Hellman in [8] developed the public key exchange algorithm using the discrete logarithm problem (DLP). El Gamal also used DLP in encryption and digital signature scheme. Koblitz and Miller [13,14], independently used EC in cryptography using elliptic curves discrete logarithm problem (ECDLP) . ECC provide a high level of security with much smaller keys in comparison to other popular cryptosystems based on integer factorization. Improving the efficiency of scalar multiplication in EC is one of the main interests of many researchers in the field of cryptography. For this reason, ECC offers a security level equivalent to RSA and DSA while using a much smaller key size. In any implementation of ECC primitives, scalar multiplication is the computationally dominant operation. Several methods have been proposed in the literature to speed-up point multiplication, which use various representations of the base point (affine coordinates, projective coordinates), various representations of the scalar (binary, ternary, NAF, w- NAF), and various curve operations (additions, doublings, halving, tripling). The computational cost (timing) of these curve operations depends on the cost of the arithmetic operations that have to be performed in the underlying field. Many researchers have given more attention to develop the proposed ECC algorithms and improve their efficiency. Improving the efficiency of scalar multiplication in EC is one of the main interests of many researchers in the field of cryptography. Computationally the most expensive operation in ECC is Scalar Multiplication namely given an integer k, and a point P on an elliptic curve curve, the computation of kP = P + · · · + P is called scalar multiplication of point P by scalar k. A key factor for its fast implementation is how to compute the scalar multiplication kP efficiently. Generally the integer k is represented in binary form and the double and add method is applied to calculate kP. It is computed by series of doubling (ECDBL) and addition (ECADD) operation of the point P. A point multiplication is the first sequence of additions, several multiplications, squaring and inversion on a finite field. A strategy that has gained lots of attention in recent years is the use of representations of number k based on double-base and multi-base chains. The use of the so-