### Abstract

In the late 1970s Magliveras invented a private-key cryptographic system called Permutation Group Mappings (PGM). PGM is based on the prolific existence of certain kinds of factorization sets, called logarithmic signatures, for finite permutation groups. PGM is an endomorphic system with message space ℤ_{|G|} for a given finite permutation group G. 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 group G is not hamiltonian and not abelian. Moreover, if the order of G 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.

Original language | English (US) |
---|---|

Pages (from-to) | 167-183 |

Number of pages | 17 |

Journal | Journal of Cryptology |

Volume | 5 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1992 |

### Fingerprint

### Keywords

- Cryptography
- Cryptology
- Finite permutation group
- Logarithmic signatures
- Multiple encryption
- Permutation group mappings (PGM)

### ASJC Scopus subject areas

- Theoretical Computer Science
- Applied Mathematics
- Electrical and Electronic Engineering
- Computational Theory and Mathematics

### Cite this

*Journal of Cryptology*,

*5*(3), 167-183. https://doi.org/10.1007/BF02451113

**Algebraic properties of cryptosystem PGM.** / Magliveras, Spyros S.; Memon, Nasir D.

Research output: Contribution to journal › Article

*Journal of Cryptology*, vol. 5, no. 3, pp. 167-183. https://doi.org/10.1007/BF02451113

}

TY - JOUR

T1 - Algebraic properties of cryptosystem PGM

AU - Magliveras, Spyros S.

AU - Memon, Nasir D.

PY - 1992/10

Y1 - 1992/10

N2 - In the late 1970s Magliveras invented a private-key cryptographic system called Permutation Group Mappings (PGM). PGM is based on the prolific existence of certain kinds of factorization sets, called logarithmic signatures, for finite permutation groups. PGM is an endomorphic system with message space ℤ|G| for a given finite permutation group G. 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 group G is not hamiltonian and not abelian. Moreover, if the order of G 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.

AB - In the late 1970s Magliveras invented a private-key cryptographic system called Permutation Group Mappings (PGM). PGM is based on the prolific existence of certain kinds of factorization sets, called logarithmic signatures, for finite permutation groups. PGM is an endomorphic system with message space ℤ|G| for a given finite permutation group G. 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 group G is not hamiltonian and not abelian. Moreover, if the order of G 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.

KW - Cryptography

KW - Cryptology

KW - Finite permutation group

KW - Logarithmic signatures

KW - Multiple encryption

KW - Permutation group mappings (PGM)

UR - http://www.scopus.com/inward/record.url?scp=0026987219&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026987219&partnerID=8YFLogxK

U2 - 10.1007/BF02451113

DO - 10.1007/BF02451113

M3 - Article

VL - 5

SP - 167

EP - 183

JO - Journal of Cryptology

JF - Journal of Cryptology

SN - 0933-2790

IS - 3

ER -