Karatsuba multiplication in galois field for the implementation of elliptic curve cryptography

Abstract

From the initial stages of public key cryptography, mainly two types of cryptosystems were used to overthrow the attacks. Therefore these two cryptosystems called as RSA and El Gamal are mainly preferred and commonly used. They can be used for encryption and decryption as well as digital signatures. In 1985, Victor Miller and Neal Koblitz invented the Elliptic Curve Cryptography that received more attention widely and formed a suitable substitution for the traditional public key cryptosystems like RSA in the application level. For realizing the protocols such as Elliptic Curve Digital Signature Algorithm(ECDSA), Diffie-Hellman key Exchange, Elgamal Encryption and Decryption etc the Elliptic Curve Cryptography is preferred. An algorithm based on modular multiplication called as Karatsuba multiplication is used here for developing the elliptic curve cryptosystem in the Galois field. The finite field operations in the elliptic curve is used in implementing the Elliptic Curve Digital Signature Algorithm. The algorithm confirmed the suitability of the VLSI implementation of the Elliptic Curve Cryptography.