A Course in Mathematical Cryptography

10 Non-Commutative Group Based Cryptography

10.1 Group Based Methods

The public key cryptosystems and public key exchange protocols that we have discussed, such as the RSA algorithm, Diffie-Hellman, ElGamal and elliptic curve methods, are number theory based, and hence depend on the structure of abelian groups. As computing machinery has gotten stronger, and computational techniques have become more sophisticated and improved, there have been successful attacks on both RSA and Diffie-Hellman for smaller and specialized parameters (RSA and Diffie- Hellman moduli). Further there are quantum algorithms that specifically break both RSA and Diffie-Hellman. As a consequence, when and if a workable quantum computer is developed, these cryptographic ...