The conjgacy problem
To solve an equation of the kind X+A-X=B in a non-commutative group is referred to as the conjugacy problem. Finding a solution to an instance of the conjugacy problem can be quite difficult, if we deal with a large group. The conjugacy problem represents the revearsal of a one-way function, similar to computing the descrete logarithm or factorizing a prime number product.
It is possible to define a key exchange algorithm similar to Diffie-Hellman based on the conjugacy problem. Contrary to Diffie-Hellman, this algorithm is considered secure against quantum computer attacks.
It is out of scope to explain the non-commutative key exchange algorithm in this article. However, I will certainly cover it in my 44CON talk.
Further reading: How to Use a Rubik’s Cube for Encryption