Elliptic Curve Crypto-system Using Baby-step Giant-step Algorithm

Although the Elgamal Elliptic Curve Cryptosystem (EECC) has proven to be the most outstanding cryptographic system, the increasing complexity of cryptographic attacks necessitates the development and analysis of more secure cryptographic systems. One way to achieve that is through the exploration of other approaches to EECC security system. Hence, the topic- Baby-Step Giant-Step Algorithm using group theoretic approach. As most used cryptographic systems assume that the discrete logarithm is extremely difficult to compute, the more difficult it is, the more security it provides a data transfer. The algorithm we presented in this paper is based on a space-time trade-off. It is proven to be a simple modification of trial multiplication, the naive method of finding discrete logarithms. The results show that the best way to speed up the baby-step giant-step algorithm is to use an efficient table lookup scheme. Most especially, the hashing is done on the second component in the pair (j, αj). Hence, the baby-step giant-step algorithm could be used by an eavesdropper to derive the private key generated in DiffieHellman key exchange, when the modulus is a prime number p that is not too large.