Discrete logarithm

A discrete logarithm is an exponent x satisfying g^x = h in a group. Finding x can be computationally difficult even when computing g^x is efficient.

Cyclic group: A group generated by repeated applications of one element. Powers of a generator enumerate the group, making discrete logarithms meaningful.
Diffie–Hellman key exchange: A protocol that lets two parties derive a shared secret over a public channel. Its security in common groups depends on the difficulty of recovering exponents from public powers.
Discrete logarithm problem: The computational problem of finding an exponent from a group element and its known base. This is the search task whose hardness underpins many discrete-logarithm cryptosystems.
John Napier: A Scottish mathematician who introduced logarithms in the early seventeenth century. His logarithms supplied the classical analogy behind naming the discrete problem.
Group order: The number of elements in a finite group. It bounds the exponents that need consideration in a finite group.
ElGamal encryption: A public-key encryption scheme based on exponentiation in a finite group. Recovering its secret key from the public key is a discrete logarithm problem.
Computational Diffie–Hellman assumption: The assumption that computing a shared power from two public powers is computationally infeasible. It is related to, but not known to be equivalent to, solving discrete logarithms.
Évariste Galois: A French mathematician whose work founded group theory and connected groups with algebraic equations. Group theory provides the algebraic setting in which discrete logarithms are defined.
Generator (group theory): An element whose powers generate a cyclic group. The base g is commonly chosen to generate the group in which the logarithm is sought.
Digital Signature Algorithm: A digital signature scheme whose security relies on discrete logarithms in a finite field. Its public key exposes a group power while keeping the signing exponent secret.

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