Group theoretic properties of Rijndael-like ciphers

Abstract

We provide conditions for which the round functions of an -bit Rijndael-like block cipher generate the alternating group on the set {0, 1} . These conditions show that the class of Rijndael-like ciphers whose round functions generate the alternating group on their message space is large, and includes both the actual Rijndael and the block cipher used by the compression function of the WHIRLPOOL hash function. The result indicates that there is no trapdoor design for a Rijndael-like cipher based on the imprimitivity of the group action of its proper round functions which is difficult to detect.

On the group generated by the round functions of translation based ciphers over arbitrary finite fields

“…AES [8], SERPENT [1], DES [15], IDEA [14]) this appears to be a difficult problem. However, more manageable overgroups of Γ have been investigated (see [20,11,21,19]), such as the ones that we now define.…”

A note on some algebraic trapdoors for block ciphers

“…However, this group depends, strongly, on the key-schedule used to create the roundkeys. For this reason, usually, we study the properties of the group Γ ∞ (C) (see for instance [4,13,19,20,21]). But, as we showed above, we could have that even if the group Γ ∞ (C) is considered secure, the group Γ(C) may be not secure with respect to the trapdoors considered here.…”

“…Another requested property for a cipher is that the group generated by the encryption functions is not "small" (see for instance [15]). Usually, this property is investigated for the group of the round functions ( [4,13,19]). With an ad hoc proof, in [21] and [20] Wernsdorf proved respectively that Γ ∞ (AES) = Alt(V ) and Γ ∞ (SERP EN T ) = Alt(V ), where Alt(V ) is the alternating group.…”

The round functions of KASUMI generate the alternating group

“…[2,10,16]). For the DES [4], AES [22], and other ciphers, several results on the cyclic and group theoretic structure of their components have already been found (see [3,5,8,13,21,23,24]). …”