Liljana Babinkostova

On some questions about selective separability

Department of Mathematics, Boise State University, Boise, ID 83725, USAReceived 2 June 2008, revised 10 July 2008, accepted 24 July 2008Published online 1 May 2009Key words Selectively separable, H-separable, Hurewicz property.MSC (2000) Please insertCH implies that selective separability is not preserved by finite powers (solvin g [3, Problems 3.7 and 3.9]).In ZFC, selective separability does not imply H-separability (solving [4, Problem 34]).

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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 (p k )

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{\mathrm {GF}(p^k)}

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( p≥2

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{p\ge 2}

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) 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 [Discrete Appl. Math. 156 (2008), 3139–3149], 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 )

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{\mathrm {GF}(2^k)}

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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 )

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{\mathrm {GF}(p^k)}

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( p≥2