Elliptic Curve Cryptography (ECC) is a variant of public-key cryptography, which is based on the algebraic structure of elliptic curves over finite fields. One of the advantages of ECC in comparison to other cryptosystems such as RSA, is its shorter key. In many elliptic curve cryptosystems an efficient hash function from bit strings into set of points on an elliptic curve over a finite field is required. Constructing such a hash function which satisfies the cryptographic properties is a difficult task and the best way to have an efficient hash function into elliptic curves is applying the encoding functions. On the other hand, if a encoding function does not cover all points of the given elliptic curve, the resulting construction of hash function can be distinguished from a random oracle to elliptic curves. Therefore, when somebody sends bit-strings, the adversary can analyze the transmitted data and attack the system. For this reason, In this thesis is studied above candidates for constructing hash functions to elliptic curves which are indifferentiable from a random oracle to elliptic curve, that both construction can be plugged into any cryptosystem which is provably secure in random oracle model, and the resulting cryptosystem remains secure in the random oracle model. کلیدواژه فارسی