: The cryptographic algorithms are studied in to two main directions of private key (symmetric) and public key (asymmetric) . Until 1976 , all known cryptosystems were symmetrical . Diffe and Hellman introduced the idea of using asymmetric cryptosystems in 1976 . The idea of public key cryptography resolvs the problem of key exchange and key distribution in symmetric methods . The use of elliptic curves in cryptography was proposed independently by Koblitz and Miller in 1985 . Elliptic curves are applied in many computational number theory and cryptography applications such as primality proving , integer factorization , Diffe-Hellman key exchange protocol , the digital signature algorithms and pairing based cryptography . Cryptosystems which are based on elliptic curve discrete logarithm problem (ECDLP) have some advantages toward those which are based on discrete logarithm (DLP) over finite fields . The most important advantage is creating equal security with shorter key . The security of various elliptic curve based cryptosystems relies on the intractability of solving the elliptic curve discrete logarithm problem . The generic algorithms for solving the discrete logarithm problems in finite groups are applied to solve the discrete logarithm problems in group of points on elliptic curves over finite fields . Moreover , the index calculus methods that are used for solving the discrete logarithm problems in the multiplicative groups of finite fields can be extended to solve discrete logarithm problems in elliptic curves over finite fields . This approach was first introduced by Kraitchik and latter optimized by Adleman . In general , the index calculus algorithms are composed of the three steps of factor basis definition , sieving step and the linear algebra computations. Over the last 30 years many scientists have proposed and analyzed several techniques to improve the complexity of the algorithms and evaluated the computational security of elliptic curve cryptography . In this thesis , we study the index calculus method for solving the discrete logarithm problems in elliptic curves over binary finite fields . As the requirements , the notion of Weil restriction and summation polynomials are explained to apply in the described algorithms . We show the use of Semaev's summation polynomials and Grobner bases techniques for further speed up in the index calculus algorithms .