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

Y

^{2}= X^{3}+ aX + b
Where x, y, a and b are real numbers and with
the condition

4a

^{3 }+ 27b^{2}≠ 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.