We investigate some of the algebraic properties of the SAFER block cipher when the message space is considered as a Z-module. In particular we consider the invariant Z-submodules of the PHT layer and show how these invariant Z-submodules give potential cryptographic weaknesses. Key Words. Block cipher, SAFER, Cryptanalysis, Invariant Z-submodules. The author acknowledges the support of the Nu eld Foundation. 1