Point Counting

Synonyms

Elliptic curve trace computation; Zeta function computation

Related Concepts

Elliptic Curve Cryptography

Definition

Point counting is the task of computing the cardinality of the set formed by the points of an elliptic curve.

Background

Let E be an elliptic curve defined over a finite field ${\mathbb{F}}_{q}$. In order to build a cryptosystem that uses E, the corresponding discrete logarithm problem must be difficult. The first threat comes from the Pohlig–Hellman algorithm whose complexity depends on the factorization of the number of points of E. The point-counting problem is to determine this cardinality, so as to assess the difficulty of the discrete logarithm problem. This is a one-time computation: an organization (governmental or standardizing) that wants to publish a recommended elliptic curve for cryptographic use must do this point-counting task and publish the result so that users can verify the claimed security level.

Theory and Applications

An approach to get...