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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. [1]
The Diffie–Hellman problem is stated informally as follows: Given an element and the values of and , what is the value of ?. Formally, is a generator of some group (typically the multiplicative group of a finite field or an elliptic curve group) and and are randomly chosen integers.
Elliptic-curve Diffie–Hellman (ECDH) is a key agreement protocol that allows two parties, each having an elliptic-curve public–private key pair, to establish a shared secret over an insecure channel. [1] [2] [3] This shared secret may be directly used as a key, or to derive another key.
The security of the scheme is based on the computational Diffie–Hellman problem. Two variants of IES are specified: Discrete Logarithm Integrated Encryption Scheme (DLIES) and Elliptic Curve Integrated Encryption Scheme (ECIES), which is also known as the Elliptic Curve Augmented Encryption Scheme or simply the Elliptic Curve Encryption Scheme.
An elliptic curve random number generator avoids escrow keys by choosing a point Q on the elliptic curve as verifiably random. Intentional use of escrow keys can provide for back up functionality. The relationship between P and Q is used as an escrow key and stored by for a security domain. The administrator logs the output of the generator to ...
The difficulty of this problem is the basis for the security of several cryptographic systems, including Diffie–Hellman key agreement, ElGamal encryption, the ElGamal signature scheme, the Digital Signature Algorithm, and the elliptic curve cryptography analogues of these.
The curve Hamburg used is an untwisted Edwards curve E d: y 2 + x 2 = 1 − 39081x 2 y 2. The constant d = −39081 was chosen as the smallest absolute value that had the required mathematical properties, thus a nothing-up-my-sleeve number. Curve448 is constructed such that it avoids many potential implementation pitfalls. [7]
Hewlett-Packard holds U.S. patent 6,252,960 on compression and decompression of data points on elliptic curves. It expired in 2018. According to the NSA, Certicom holds over 130 patents relating to elliptic curves and public key cryptography in general. [5] It is difficult to create a complete list of patents that are related to ECC.