Subgroup

Related Concepts

Group; Order

Definition

A subgroup is a subset ${\textrm{ S}}^{{\prime}}$ of the elements of a group G = (S, ×) such that S ^′ is itself a group with respect to the same group operation.

Theory

If G = (S, ×) is a group, then for any g ∈ S, the set of elements

$$g,\ {g}^{2},\ {g}^{3},\ \ldots $$

(together with the multiplication operation) is also a group. This set is one example of a subgroup of G: a subset of the elements of the group that follows the group axioms (closure, associativity, identity, inverse).

The order of any subgroup of a group G divides the order of the group G itself; this is known as Lagrange’s theorem.

Applications

The orderand structure of a subgroup in which cryptographic operations are computed can often have a significant impact on the security level. A small-subgroup attack, for instance, exploits the fact that the set of elements generated by a cryptographic parameter is too small and therefore can be searched exhaustively to determine a...