text: Application to cryptography Public-key cryptography is based on the intractability of certain mathematical problems. Early public-key systems, such as RSA's 1983 patent, based their security on the assumption that it is difficult to factor a large integer composed of two or more large prime factors. For later elliptic-curve-based protocols, the base assumption is that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible (the computational Diffie–Hellman assumption): this is the "elliptic curve discrete logarithm problem" (ECDLP). The security of elliptic curve cryptography depends on the ability to compute a point multiplication and the inability to compute the multiplicand given the original point and product point. The size of the elliptic curve, measured by the total number of discrete integer pairs satisfying the curve equation, determines the difficulty of the problem.
  The primary benefit promised by elliptic curve cryptography over alternatives such as RSA is a smaller key size, reducing storage and transmission requirements. For example, a 256-bit elliptic curve public key should provide comparable security to a 3072-bit RSA public key.
  Cryptographic schemes
  Several discrete logarithm-based protocols have been adapted to elliptic curves, replacing the group  with an elliptic curve:
  The Elliptic-curve Diffie–Hellman (ECDH) key agreement scheme is based on the Diffie–Hellman scheme,
  The Elliptic Curve Integrated Encryption Scheme (ECIES), also known as Elliptic Curve Augmented Encryption Scheme or simply the Elliptic Curve Encryption Scheme,
  The Elliptic Curve Digital Signature Algorithm (ECDSA) is based on the Digital Signature Algorithm,
  The deformation scheme using Harrison's p-adic Manhattan metric,
  The Edwards-curve Digital Signature Algorithm (EdDSA) is based on Schnorr signature and uses twisted Edwards curves,
  The ECMQV key agreement scheme is based on the MQV key agreement scheme,
  The ECQV implicit certificate scheme.
  Implementation
  Some common implementation considerations include: