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.
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.
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.
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.
The basis for the security and strength of the various protocols presented above is the difficulty of the following two problems: If no algorithm is known to solve the conjugacy search problem, then the function x → ux can be considered as a one-way function.
The following is a list of the properties expected of G. Let n be a positive integer.