Non-commutative cryptography is the area of cryptology where the cryptographic primitives, methods and systems are based on algebraic structures like semigroups, groups and rings which are non-commutative. One of the earliest applications of a non-commutative algebraic structure for cryptographic purposes was the use of braid groups to develop cryptographic protocols. Later several other non-commutative structures like Thompson groups, polycyclic groups, Grigorchuk groups, and matrix groups have been identified as potential candidates for cryptographic applications. In contrast to non-commutative cryptography, the currently widely used public-key cryptosystems like RSA cryptosystem, Diffie–Hellman key exchange and elliptic curve cryptography are based on number theory and hence depend on commutative algebraic structures.

Non-commutative cryptographic protocols have been developed for solving various cryptographic problems like key exchange, encryption-decryption, and authentication. These protocols are very similar to the corresponding protocols in the commutative case.

In these protocols it would be assumed that G is a non-abelian group. If w and a are elements of G the notation w^a would indicate the element a⁻¹wa.

The following protocol due to Ko, Lee, et al., establishes a common secret key K for Alice and Bob.

This a key exchange protocol using a non-abelian group G. It is significant because it does not require two commuting subgroups A and B of G as in the case of the protocol due to Ko, Lee, et al.

In the original formulation of this protocol the group used was the group of invertible matrices over a finite field.

This protocol describes how to encrypt a secret message and then decrypt using a non-commutative group. Let Alice want to send a secret message m to Bob.

Let Bob want to check whether the sender of a message is really Alice.