Elliptic Curves Cryptography

Page #435547 of Chapter: Crypto-for-the-rest-of-us

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The mathematical attributes used in Diffie-Hellman, and RSA were also found in a strange mathematical construct known as "Elliptic Curves".

This allowed one to replicate the same capabilities using ECC -- Elliptic Curve Cryptography.

An Elliptic curve is a finite set of points satisfying:
  • P + O = O + P = P
  • P+(-P)=O
  • P+Q = Q +P
  • (P+Q)+R=P+(Q+R)

    where P, and Q are two arbitrary points in the set, and "O" is a special "zero" construct.

    Let a point Q in the ECC set be the result of adding a point P n times to itself:

  • Q = P + P + P + ....... + P (n times)

    We can then write: Q = nP, and n = logp(Q)

    P,Q, and n are three values such that any two determine the third. We found an easy way to compute Q, and no easy way to compute n, and hence we can use the ECC math to build the ECC version for Diffie and Hellman, and RSA.

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