Fully homomorphic encryption must be fat or ugly?

In 1978, Rivest, Adleman and Dertouzos asked for algebraic systems for which useful privacy homomorphisms exist. To date, the only acknownledged result is noise based encryption combined with bootstrapping. Before that, there were several failed attempts. We prove that fully homomorphic schemes are impossible for several algebraic structures. Then we develop a characterisation of all fully homomorphic schemes and use it to analyse three examples. Finally, we propose a conjecture stating that secure FHE schemes must either have a significant ciphertext expansion or use unusual algebraic structures.