Algebraic properties of generalized Rijndael-like ciphers

Liljana Babinkostova; Kevin W. Bombardier; Matthew C. Cole; Thomas A. Morrell; Cory B. Scott

doi:10.1515/gcc-2014-0004

Algebraic properties of generalized Rijndael-like ciphers

Liljana Babinkostova, Kevin W. Bombardier, Matthew C. Cole, Thomas A. Morrell, Cory B. Scott

Mathematics Department

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We provide conditions under which the set of Rijndael-like functions considered as permutations of the state space and based on operations of the finite field GF(p^k) (p ≥ 2) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen the Advanced Encryption Standard (AES), a Rijndael cipher with specific block sizes. In [39], R. Sparr and R. Wernsdorf provided conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF(2^k) is equal to the alternating group on the state space. In this paper we provide conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF(p^k) (p ≥ 2) is equal to the symmetric group or the alternating group on the state space.

Original language	English
Pages (from-to)	37-54
Number of pages	18
Journal	Groups, Complexity, Cryptology
Volume	6
Issue number	1
DOIs	https://doi.org/10.1515/gcc-2014-0004
State	Published - May 2014

Keywords

Finite fields
Group operation
Imprimitivity
Rijndael cipher
Symmetric groups

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abstract = "We provide conditions under which the set of Rijndael-like functions considered as permutations of the state space and based on operations of the finite field GF(pk) (p ≥ 2) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen the Advanced Encryption Standard (AES), a Rijndael cipher with specific block sizes. In [39], R. Sparr and R. Wernsdorf provided conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF(2k) is equal to the alternating group on the state space. In this paper we provide conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF(pk) (p ≥ 2) is equal to the symmetric group or the alternating group on the state space.",

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T1 - Algebraic properties of generalized Rijndael-like ciphers

AU - Babinkostova, Liljana

AU - Bombardier, Kevin W.

AU - Cole, Matthew C.

AU - Morrell, Thomas A.

AU - Scott, Cory B.

PY - 2014/5

Y1 - 2014/5

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AB - We provide conditions under which the set of Rijndael-like functions considered as permutations of the state space and based on operations of the finite field GF(pk) (p ≥ 2) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen the Advanced Encryption Standard (AES), a Rijndael cipher with specific block sizes. In [39], R. Sparr and R. Wernsdorf provided conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF(2k) is equal to the alternating group on the state space. In this paper we provide conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF(pk) (p ≥ 2) is equal to the symmetric group or the alternating group on the state space.

KW - Finite fields

KW - Group operation

KW - Imprimitivity

KW - Rijndael cipher

KW - Symmetric groups

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EP - 54

JO - Groups, Complexity, Cryptology

JF - Groups, Complexity, Cryptology

IS - 1

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Algebraic properties of generalized Rijndael-like ciphers

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Keywords

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