Minghua Qu and S.A.Vanstone [2] have proposed a public key cryptosystem (FGM) which is based on factorisations of a binary vector space (i.e. transversal logarithmic signatures of an elementary abelian 2-group). In this paper, a generalised (basis-independent) decryption algorithm is given, which shows that there are many equivalent private keys, and a method of efficiently obtaining such an equivalent private key is given. The FGM cryptosystem is thus rendered insecure. Although the FGM cryptosystem is defined in terms of linear algebra, the attack given here is essentially group--theoretic in nature. Thus this attack throws doubt on any cryptosystem which relies on the security of transversal logarithmic signatures. Key Words. Public Key Cryptosystems, Finite Group Mappings, Permutation Group Mappings, Logarithmic Signatures. This author was supported by S.E.R.C. research grant GR/H23719 1 The paper is organised as follows. Section 1 gives a description of the Finite Group Mapp...