Algebraic properties of cryptosystem PGM |
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Authors: | Spyros S. Magliveras Nasir D. Memon |
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Affiliation: | (1) Department of Computer Science and Engineering, University of Nebraska-Lincoln, 68588-0115 Lincoln, NE, USA |
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Abstract: | In the late 1970s Magliveras invented a private-key cryptographic system calledPermutation Group Mappings (PGM). PGM is based on the prolific existence of certain kinds of factorization sets, calledlogarithmic signatures, for finite permutation groups. PGM is an endomorphic system with message space ℤ|G| for a given finite permutation groupG. In this paper we prove several algebraic properties of PGM. We show that the set of PGM transformations ℐ G is not closed under functional composition and hence not a group. This set is 2-transitive on ℤ|G| if the underlying groupG is not hamiltonian and not abelian. Moreover, if the order ofG is not a power of 2, then the set of transformations contains an odd permutation. An important consequence of these results is that the group generated by the set of transformations is nearly always the symmetric group ℒ|G|. Thus, allowing multiple encryption, any permutation of the message space is attainable. This property is one of the strongest security conditions that can be offered by a private-key encryption system. S. S. Magliveras was supported in part by NSF/NSA Grant Number MDA904-82-H0001, by U.S. West Communications, and by the Center for Communication and Information Science of the University of Nebraska. |
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Keywords: | Cryptography Cryptology Finite permutation group Permutation group mappings (PGM) Multiple encryption Logarithmic signatures |
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