On Minimal Factorisations of Sporadic Groups

For many years cryptographers have used large abelian finite groups but some are now turning their attention to non-abelian ones. They feel that these could be a good source of “trap doors” that can be used in public key encryption [Magliveras 02]. One proposed system is MST1 [Magliveras 02]. This uses a certain type of group factorisation to encode messages which can only be decoded by the recipient. In [Vasco et al. 03], González Vasco et al. conjecture that minimal factorisations of this type exist for all finite groups. Proofs are given in [Magliveras 02] that they exist for L2(q) for any prime power q and all alternating groups An. In [Vasco et al. 03] it is proved that they exist for the Mathieu sporadic groups, the group U3(3), and any group with a factorisation into Sylow subgroups, and hence that one exists for any group of order less than |J1|. Later in this section we define minimal factorisations and related concepts. Section 2 lists some sufficient conditions for the existence of minimal factorisations. In Section 3 we prove the existence of minimal factorisations for the sporadic groups J1, J2, HS, McL, He, and Co3. We also show that the existence of minimal factorisations for certain smaller groups implies their existence for Ru and Suz. We draw our conclusions in Section 4.