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Definition
This section introduces elliptic curves and associated group operations, along with basic structural properties of particular interest in cryptography.
Background
Although classical, elliptic curves have been used in integer factoring algorithms and in primality proving algorithms, and also for designing public-key cryptographic schemes. In cryptography the attractiveness is due, in part, to the apparent resistance (at a given key size) to discrete logarithm attacks compared to methods where security is based on the intractability of integer factorization or finding logarithms in finite fields, especially as security requirements increase.
Theory
An elliptic curveE over a field F is defined by a Weierstrass equation
with a 1, a 2, a 3, a 4, a 6 ∈ F and Δ≠0, where Δ is the discriminant of E and is defined as follows:
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Koblitz N (1994) A course in number theory and cryptography, 2nd edn. Springer, New York
Silverman J (1986) The arithmetic of elliptic curves. Springer, New York
Silverman J (1994) Advanced topics in the arithmetic of elliptic curves. Springer, New York
Washington L (2008) Elliptic curves: number theory and cryptography, 2nd edn. CRC Press, Boca Raton, FL
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Hankerson, D., Menezes, A. (2011). Elliptic Curves. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5906-5_244
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DOI: https://doi.org/10.1007/978-1-4419-5906-5_244
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