Elliptic Curves Cryptography (ECC) is one of the most efficient public key cryptosystem . The security of ECC is based on the difficulty of solving discrete logarithm problem in the group of points on an elliptic curve over a finite field . The main applications of elliptic curves cryptography are in Diffie - Hellmann key exchange algorithm (ECDH) and Elliptic Curve Digital Signature Algorithm (ECDSA) , that are widely used in commercial standards , such as IPsec or Traort Layer Security (TLS) protocols . Every elliptic curve can be represented by the Weierstrass equation . The set of Weierstrass elliptic curve points under the chord and tangent process form an Abelian group . The addition law is said to be complete , if the projective addition formulas compute the sum of every two points on the curve . In other words , the addition formulas do not have exceptional pairs of two points where their sum are not well computed . To implement secure ECC , complete addition formulas are required to prevent side-channel attacks based on exceptional cases .
