Elliptic Curve Cryptology
What and Why of ECC?
Elliptic curve cryptography (ECC) is a public key cryptography technique by making use of elliptic curve properties and their algebraic structure of over finite fields. It is one of the efficient ways of providing encryption of cryptographic keys.
Elliptic curves as algebraic/geometric entities have been studied extensively for the past 150 years, and from these studies has emerged a rich and deep theory. Elliptic curve systems as applied to cryptography were first proposed in 1985 independently by Neal Koblitz from the University of Washington, and Victor Miller, who was then at IBM, Yorktown Heights.[1]
These curves have allowed establishment of a new generation of asymmetric cryptographic algorithms. The big win with ECC, as compared to other public-key algorithms, is key size. A fairly typical key size for RSA is 1024 bits--this would take approximately 10^11 MIPs-years to break. A mere 160-bit ECC key offers the same level of security. This advantage only increases with security level--something that will be important as computer power continually grows. A 2048-bit RSA key and a 210-bit ECC key are equivalent.
ECC also has less computational overhead than RSA, primarily because it does not have to analyze prime numbers, a fairly expensive operation.[1]
ECC can be used with SSL scheme, certificates, Diffie-Hellman key agreement, El-Gamal and protocols such ECDSA (Elliptic Curve Digital Signature Algorithm).
This could lead ECC to be a major tool/element of tomorrow’s cryptology. While ECC has not been as extensively researched as RSA, to date all research has confirmed ECC to be secure.[1]
Elliptic curve operations:
The How part
Discrete log cryptosystems are typically descr...
... middle of paper ...
...pute u1 = h(m)w mod n and u2 = r w mod n.
5. Compute u1 P + u2 Q = (x0 , y0 ) and v = x0 mod n.
6. Accept the signature if and only if v = r .[2]
References
[1] www.certicom.com
[2] Neal Koblitz, Alfred Menezes, Scott Vanstone “The State of Elliptic Curve Cryptography”
[3] Nick Sullivan “http://arstechnica.com/security/2013/10/a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptography/”
[4] P.Kocher,”Timing Attacks on Implementations of Diffe-Hellman, RSA, DSS, and Other Systems,”
Advances in Cryptology-CRYPTO’96 Proceedings, Springer-Verlag, 1996, pp. 104-113
[5] Darrel Hankerson, Julio Lopez Hernandez and Alfred Menezes, “Software Implementation of Elliptic Curve Cryptography Over Binary Fields”, Cryptographic Hardware and Embedded Systems, 2000.
[6] Don Johnson and Alfred Menezes, “The Elliptic Curve Digital Signature Algorithm (ECDSA)”, 1999.
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Asymmetric Key Encryption methods are DSA, Diffie Hellman, RSA, Elliptic Curve and DSA. Asymmetric Encryption
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