Cryptography is a constantly evolving field – elliptic curve groups have been a major part of this field since 1987 when Koblitz and Miller separately proposed their use in cryptosystems such as RSA.
- Elliptic Curve: A curve defined by the equation: where A and B are less than a prime p, and where the discriminant is not equal to zero. We are concerned with the curves where A = 0.
- Elliptic Curve Group (ECG): A group whose elements are defined by the elliptic curve and the operation in question. Elliptic curves naturally qualify as groups due simply to their nature.
- Elliptic Pair of Primes: Two primes (p,q) such that an ECG defined by the following equation has an order q, and the ECG defined by a different equation mod q has the order p. These pairs are the main topic of our study, and learning more about them and their effects on RSA security is our primary goal.
- Discrete Log Problem (DLP): Solving for x in the equation:
NOTE: Operations such as addition and multiplication are defined differently when used in terms of an ECG operation. An example can be found below.
Dr. Liljana Babinkostova