To make sure that your cryptosystem is secure, then you have to do cryptanalysis.
The known attacks for ECC are:
* The Pohlig-Hellman algorithm (which reduces the problem to subgroups of prime order)
* Shanks’ baby-step-giant-step method
* Pollard’s methods, the rho method and the kangaroo method, both of which have parallel versions due to van Oorschot and Wiener
* The Menezes-Okamoto-Vanstone (MOV) attack using the Weil pairing
* The Frey-Rueck attack using the Tate pairing
* The attacks on anomalous elliptic curves (i.e., elliptic curves over F_p which have p points) due to Semaev, Satoh-Araki and Smart
* Weil descent (for some special finite fields)
* Algebro-geometric attack
Next step will be to choose which attacks to be used for measuring the level of security of ECC.
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