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Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography to provide equivalent security.
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic algorithms based on mathematical problems termed one-way functions. Security of public-key cryptography depends on keeping the private key secret; the public key can be openly distributed without compromising security.
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In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms. Dedicated to the discrete logarithm in where is a prime, index calculus leads to a family of algorithms adapted to finite fields and to some families of elliptic curves. The algorithm collects relations among the discrete logarithms of small primes, computes them by a linear algebra procedure and finally expresses the desired discrete logarithm with respect to the discrete logarithms of small primes.
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Discrete logarithm records are the best results achieved to date in solving the discrete logarithm problem, which is the problem of finding solutions x to the equation given elements g and h of a finite cyclic group G. 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. Common choices for G used in these algorithms include the multiplicative group of integers modulo p, the multiplicative group of a finite field, and the group of points on an elliptic curve over a finite field.
In cryptography, security level is a measure of the strength that a cryptographic primitive — such as a cipher or hash function — achieves. Security level is usually expressed as a number of "bits of security", where n-bit security means that the attacker would have to perform 2n operations to break it, but other methods have been proposed that more closely model the costs for an attacker. This allows for convenient comparison between algorithms and is useful when combining multiple primitives in a hybrid cryptosystem, so there is no clear weakest link. For example, AES-128 is designed to offer a 128-bit security level, which is considered roughly equivalent to a RSA using 3072-bit key.
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Notes
- 1 2 3 Blake, Ian; Menezes, Alfred J.; Stinson, Doug (2015), "Guest editorial: Special issue in honor of Scott A. Vanstone", Designs, Codes and Cryptography, 77 (2–3): 287–299, doi: 10.1007/s10623-015-0106-2
- ↑ Blake, Ian; Fuji-Hara, R.; Mullin, Ron; Vanstone, Scott A. (1984), "Computing logarithms in finite fields of characteristic two", SIAM J. Algebr. Discrete Methods, 5 (2): 276–285, doi:10.1137/0605029
- 1 2 "Prof. Scott Vanstone, FRSC, FIACR, 1947-2014" . Retrieved 9 April 2018.
- ↑ "Certicom Founder Receives Security Award for Mathematics from RSA" . Retrieved 9 April 2018.
- ↑ "In Memory of Scott Alexander Vanstone". J. Scott Early funeral home web site. Archived from the original on March 4, 2014.
- ↑ Blake, Ian; Menezes, Alfred; Stinson, Doug (2015-12-01). "Guest Editorial: Special Issue in Honor of Scott A. Vanstone". Designs, Codes and Cryptography. 77 (2): 287–299. doi: 10.1007/s10623-015-0106-2 . ISSN 1573-7586.
