A fast point doubling and point addition operations on an elliptic curve over prime field are proposed. This occur when we use a special coordinates system (to represent any point on elliptic curve over prime field. Using this system improved the elliptic curve point arithmetic by reducing the computation cost for point doubling and point addition operation. | International Journal of Computer Networks and Communications Security VOL. 2, NO. 12, DECEMBER 2014, 462–471 Available online at: ISSN 2308-9830 Improved Arithmetic on Elliptic Curves over Prime Field NAJLAE FALAH HAMEED AL SAFFAR1 AND MOHAMAD RUSHDAN MD SAID2 1 2 Institute for Mathematical Research, Universiti Putra Malaysia,Malaysia Department of Mathematics, Faculty of Mathematics and Computer Science,Kufa University,Iraq 2 Institute for Mathematical Research, Universiti Putra Malaysia,Malaysia E-mail: 1najlae_falah@, 2mrushdan@ ABSTRACT A fast point doubling and point addition operations on an elliptic curve over prime field are proposed. This occur when we use a special coordinates system (to represent any point on elliptic curve over prime field. Using this system improved the elliptic curve point arithmetic by reducing the computation cost for point doubling and point addition operation. Keywords: Elliptic curve cryptosystem, point arithmetic of elliptic curve, affine coordinates, projective coordinates and Jacobian coordinates. 1 INTRODUCTION Elliptic curves are used for several kinds of cryptosystems, even it involved in key exchange protocols and digital signature algorithms [11], since it independently presented by Miller [20] and Koblitz [15] in the 1980s. Elliptic curve cryptography ECC has attracted attention in recent years due to it’s dependence on the difficulty of the elliptic curve discrete logarithm problem ( ECDLP ). Since there are no known subexponential time algorithms to solve the ECDLP , ECC supplies the same level of security with a shorter key size comparing with the well known public key cryptosystems based on the discrete logarithm problem ( DLP ) and the integer factoring problem ( IFP ) over finite fields such as RSA [23], DSA [18] and ElGamal [9]. Because of this singularity (requires a shorter key sizes are translated to less power and storage requirements, and reduced computing time .
