The addition operation in ECC is the counterpart of modular multiplication in RSA, and multiple addition is the counterpart of modular exponentiation.

**ELLIPTIC CURVE CRYPTOGRAPHY**

The
addition operation in ECC is the counterpart of modular multiplication in RSA,
and multiple addition is the counterpart of modular exponentiation. To form a
cryptographic system using elliptic curves, we need to find a "hard
problem" corresponding to factoring the product of two primes or taking
the discrete logarithm.

Consider
the equation Q = kP where Q, P in E_{p}(a, b) and k < p. It is
relatively easy to calculate Q given k and P, but it is relatively hard to
determine k given Q and P. This is called the discrete logarithm problem for
elliptic curves.

Because
9P = (4, 5) = Q, the discrete logarithm Q = (4, 5) to the base P = (16, 5) is k
= 9. In a real application, k would be so large as to make the brute-force
approach infeasible.

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Cryptography and Network Security : Elliptic Curve Cryptography |

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