Abstract: Group-based cryptography is concerned with the role of nonabelian groups in
cryptography. Since its origins in the 1980s, there have been numerous cryptographic
proposals based on nonabelian groups, many of which have been broken.
This thesis adds to the cryptanalytic literature by demonstrating the insecurity of
several group-based cryptosystems. We cryptanalyse several key establishment
protocols based on matrix groups, due to (i) Baumslag, Camps, Fine, Rosenberger
and Xu, (ii) Habeeb, Kahrobaei, and Shpilrain, (iii) Romanczuk and Ustimenko,
and (iv) a modified version of a scheme by Stickel. We also cryptanalyse the MST3
public key cryptosystem and treat the Tillich–Zémor hash function.