Elliptic Curve Cryptography

Elliptic Curve Cryptography was proposed by Niel Koblitz and Victor Miller in 1985. This is based on algebraic of elliptic curves over finite fields and provided public key infrastructure to end user and it is considered as strong public key cryptosystem comparing with other algorithms like RSA. RSA is based on Integer Factorization Problem which means it is impossible or very difficult to find factor of very large prime numbers and that is an assumption. While ECC is based on elliptic curve discrete logarithm problem which means it is infeasible to determine discrete logarithm of point with multiplier in elliptic curve based on finite fields. Because of ECDLP feature elliptic curve cryptography can provide same security with small key size with related to other algorithms which can result speed computation with low memory, lower network bandwidth and lower power.  Most valuable feature in ECC is that it has highest strength per bit with related to other well-known cryptosystems. Therefore it is considered as high secure cryptosystem suitable for mobile devices.

The mathematical formula of ECC over the elliptic curve is

Y² = X³ + aX + b

Where x, y, a and b are real numbers and with the condition

4a³ + 27b² ≠ 0

By changing the values of ‘a’ and ‘b’ different elliptic curves can be generated. All the points which satisfy the above equation lie on the elliptic curve. Private Key is generated by random number generations while public key is obtained by multiplying the private key with a constant base point G (Scalar multiplication) in the curve. Public key is obtained as a point in this curve. ECC biggest advantage is its small key size, a 160 bit key size of ECC is equivalent in security to 1024 bit key size of RSA.

Implementation.