## Related Concepts

## Definition

The ECDLP is a special case of the discrete logarithm problem

## Background

Let *E* be an elliptic curve defined over a finite field \({\mathbb{F}}_{q}\), and let \(P \in E({\mathbb{F}}_{q})\) be a point of order *n*. Given \(Q \in \langle P\rangle\), the elliptic curve discrete logarithm problem (ECDLP) is to find the integer *l*, \(0 \leq l \leq n - 1\), such that *Q* = *lP*. The ECDLP is a special case of the discrete logarithm problem in which the cyclic group *G* is represented by the group \(\langle P\rangle\) of points on an elliptic curve. It is of cryptographic interest because its apparent intractability is the basis for the security of elliptic curve cryptography.

## Theory

If the order *n* of the base point *P* is composite and its factorization is known, then the Pohlig–Hellman algorithm [19] (see the discrete logarithm problem entry) can be used to efficiently reduce the ECDLP in \(\langle P\rangle\)to...

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Hankerson, D., Menezes, A. (2011). Elliptic Curve Discrete Logarithm Problem. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5906-5_246

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