Anshel–Anshel–Goldfeld protocol, also known as a commutator key exchange, is a key-exchange protocol using nonabelian groups.
Unlike other group-based protocols, it does not employ any commuting or commutative subgroups of a given platform group and can use any nonabelian group with efficiently computable normal forms.
It is often discussed specifically in application of braid groups, which notably are infinite (and the group elements can take variable quantities of space to represent).
The computed shared secret is an element of the group, so in practice this scheme must be accompanied with a sufficiently secure compressive hash function to normalize the group element to a usable bitstring.
Alice's public/private information: Bob's public/private information: Transitions: Shared key: The key shared by Alice and Bob is the group element
, and the conjugated public keys
A direct attack then consists of trying to find a suitable
directly, which would require some additional special structure of the group.)
For this reason the public keys
must be chosen to generate a large subgroup of
— ideally, they form a full set of generators, so that
cannot be constrained just by knowing that is generated from
given the conjugation relations is called the conjugation problem, and substantial research has been done on attacks to the conjugacy problem on braid groups, although no full efficient solution has achieved.