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We develop a public key cryptosystem based on invariants of diagonalizable groups and investigate properties of such cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on Euclidean algorithm).2010 Mathematics Subject Classification. 94A60(primary), and 11T71(secondary). Key words and phrases. cryptosystem, invariants, diagonalizable group, number field, supergroup. This publication was made possible by a NPRF award NPRP 6 -1059 -1 -208 from the Qatar National Research Fund (a member of The Qatar Foundation). The statements made herein are solely the responsibility of the authors. 1 2.1. Design of cryptosystems based on invariants of groups. To design a cryptosystem, Alice needs to choose a finitely generated subgroup G of GL(V ) for some vector space V = F n and a set {g 1 , . . . , g s } of generators of G. Alice also chooses an n× n matrix a. Alice needs to know an invariant f of this representation of G. Depending on this invariant f , Alice chooses a set M = {v 0 , . . . , v r−1 } of vectors from V such that the set S = aM = {av 0 , . . . , av r−1 } is separated by the invariant f . This means that f (av i ) = f (av j ) whenever i = j. The set M represents messages Alice can receive and elements of the set S are bijectively assigned to elements of M . The sets S is a part of the public key.Alice also chooses a set of randomly generated elements g 1 , . . . , g m of G (say, by multiplying some of the given generators of G), which generates a subgroup of G that will be denoted by G s .