Elliptic Curve CryptologyWhat and Why ECC?Elliptic Curve Cryptography (ECC) is a public key cryptography technique using the properties of elliptic curve and their algebraic structure over finite fields. It is one of the effective ways to encrypt cryptographic keys. Elliptic curves as algebraic/geometric entities have been studied extensively over 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 of the University of Washington and Victor Miller, then at IBM, Yorktown Heights.[1]These curves allowed the establishment of a new generation of asymmetric cryptographic systems. algorithms. The big advantage of ECC, compared to other public key algorithms, is the size of the key. A fairly typical key size for RSA is 1024 bits - this would take around 10^11 MIP-years to break. A simple 160-bit ECC key provides the same level of security. This advantage only increases with the level of security, which will become important as computer power continually increases. A 2048-bit RSA key and a 210-bit ECC key are equivalent. ECC also has less computation time than RSA, mainly because it does not need to parse prime numbers, a fairly expensive operation.[1]ECC can be used with SSL schema, certificates, Diffie key agreement -Hellman, El-Gamal and protocols such as ECDSA (Elliptic Curve Digital Signature Algorithm). This could lead ECC to be a major tool/element of tomorrow's cryptology. Although ECC has not been as extensively researched as RSA, to date all research has confirmed the safety of ECC. ..pute u1 = h(m)w mod n and u2 = rw mod n.5. Calculate 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, “Timed 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 on Binary Fields,” Cryptographic Hardware and Embedded Systems, 2000.[6] Don Johnson and Alfred Menezes, “The Elliptic Curve Digital Signature Algorithm (ECDSA)”, 1999.