Engineering >> Computer Science & EngineeringElliptic Curve DiffieHellman Cryptography on the Real Number Plane (R)by Shawn Joseph
Submitted : Spring 2015 Elliptic Curve Cryptography is a form of publickey cryptography that uses the elliptic curve graph’s design for encryption, digital signatures and pseudorandom generators. In 1977, the RSA algorithm and the DiffieHellman key exchange algorithm were introduced. The algorithms were the first to present viable cryptographic system based on number theory  before that cryptography relied on secret codes kept in books that had to be securely transported everywhere to ensure complete security. This paper seeks to examine the elliptic curve’s general function and an actual elliptic curve on a real number (R) plane, and find the solutions of both in four of the following cases:
1. If P and Q are two distinct points on the curve with different x values 2. If P and Q are two distinct points on the curve with the same x values 3. If P = Q and the slope of the line tangent to the curve is a real number 4. If P = Q and the slope of the tangent line to the curve is infinity ,
to eventually compute R = P+Q. The paper will also seek to show the process and mathematical steps of the DiffieHellman key exchange, and explain why computing the private key from the public key is nearly impossible.
