The elliptic curve discrete logarithm problem (ECDLP) forms the basis of numerous cryptosystems today. The security of these cryptosystems usually depends on the choice of curves. In this thesis, we give a summary of recent attacks on the ECDLP, find the criteria of weak curves, and suggest the conditions that a secure curve should have. We will also discuss some attacks which works on the DLP but may fail to the ECDLP.
