Elliptic curves

The elliptic curves algorithm is based on the discrete logarithm problem discussed earlier but in the context of elliptic curves. An elliptic curve is an algebraic cubic curve over a field, which can be defined by the following equation. The curve is non-singular, which means that it has no cusps or self-intersections. It has two variables a and b, as well as a point of infinity.

Here, a and b are integers whose values are elements of the field on which the elliptic curve is defined. Elliptic curves can be defined over real numbers, rational numbers, complex numbers, or finite fields. For cryptographic purposes, an elliptic curve over prime finite fields is used instead of real numbers. Additionally, the prime should be greater than 3. Different curves can be generated by varying the value of a and/or b.

The most prominently used cryptosystems based on elliptic curves are the Elliptic Curve Digital Signature Algorithm (ECDSA) and the Elliptic Curve Diffie-Hellman (ECDH) key exchange.

To understand public key cryptography, the key concept that needs to be explored is the concept of public and private keys.