Cryptography provides a method for securing and authenticating the transmission of information over the insecure channels. Elliptic Curve [EC] Cryptography is a public key cryptography .It replaces RSA because of its increased security with lesser number of key bits .Elliptic Curve scalar multiplication module will be available in  majority of secure communication systems. The most important operation in Elliptic Curve Cryptosystem is the computation of scalar multiplication using   karatsuba multiplier. In scalar multiplication of kP for given integer k and point P on elliptic curve. This work aims to design and implement elliptic curve scalar multiplier on a single field programmable gate array (FPGA).The hardware complexity is reduced using polynomial  basis representation of finite field and projective co-ordinate representation of elliptic curves.