Editing Elliptic-curve cryptography (section)

=== Application to cryptography ===
[[Public-key cryptography]] is based on the [[Intractability (complexity)#Intractability|intractability]] of certain mathematical [[Computational hardness assumption|problems]]. Early public-key systems, such as [[RSA_(cryptosystem)|RSA]]'s 1983 patent, based their security on the assumption that it is difficult to [[Integer factorization|factor]] a large integer composed of two or more large prime factors which are far apart. 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 [[elliptic curve point multiplication|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.<ref name=":0" /> For example, a 256-bit elliptic curve public key should provide [[Security level|comparable security]] to a 3072-bit RSA public key.