In cryptography

Cryptography

Cryptography is the practice and study of techniques for secure communication in the presence of third parties...

, a subgroup confinement attack (or small subgroup confinement attack) on a cryptographic method that operates in a large finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

 is where an attacker attempts to compromise the method by forcing a key to be confined to an unexpectedly small subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

 of the desired group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

.

Several methods have been found to be vulnerable to subgroup confinement attack, including some forms or applications of Diffie-Hellman key exchange

Diffie-Hellman key exchange

Diffie–Hellman key exchange Synonyms of Diffie–Hellman key exchange include:*Diffie–Hellman key agreement*Diffie–Hellman key establishment*Diffie–Hellman key negotiation...

 and DH-EKE

Encrypted key exchange

Encrypted Key Exchange is a family of password-authenticated key agreement methods described by Steven M. Bellovin and Michael Merritt...

.