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

Aragona, Riccardo; Caranti, A.; Volta, Francesca Dalla; Sala, Massimiliano

doi:10.1016/j.ffa.2013.10.005

Cited by 12 publications

(23 citation statements)

References 17 publications

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“…If A = {0}, then A = C = D = B = {0}, which contradicts that U is non-trivial. If A is a proper subgroup of (F 2 ) n , then by (5) and since both a + aϕ 2 and aϕ belong to A we have…”

Section: By Lemma 44 and Bymentioning

confidence: 99%

“…More relevant in [28], Paterson built a DES-like cipher, resistant to both linear and differential cryptanalysis, whose encryption functions generate an imprimitive group and showed how the knowledge of this trapdoor can be turned into an efficient attack to the cipher. For this reason, a branch of research in Symmetric Cryptography is focused on showing that the group generated by the encryption functions of a given cipher is primitive and not of affine type (see [4,5,6,11,17,18,31,33,34,35]). In this sense, our purpose is to give sufficient conditions for the primitivity of the group generated by the round functions of a wave cipher.…”

Section: Introductionmentioning

confidence: 99%

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Wave-shaped round functions and primitive groups

Aragona¹,

Calderini²,

Civino³

et al. 2019

Advances in Mathematics of Communications

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Round functions used as building blocks for iterated block ciphers, both in the case of Substitution-Permutation Networks (SPN) and Feistel Networks (FN), are often obtained as the composition of different layers which provide confusion and diffusion, and key additions. The bijectivity of any encryption function, crucial in order to make the decryption possible, is guaranteed by the use of invertible layers or by the Feistel structure. In this work a new family of ciphers, called wave ciphers, is introduced. In wave ciphers, round functions feature wave functions, which are vectorial Boolean functions obtained as the composition of non-invertible layers, where the confusion layer enlarges the message which returns to its original size after the diffusion layer is applied. This is motivated by the fact that relaxing the requirement that all the layers are invertible allows to consider more functions which are optimal with regard to non-linearity. In particular it allows to consider injective APN S-boxes. In order to guarantee efficient decryption we propose to use wave functions in Feistel Networks. With regard to security, the immunity from some group-theoretical attacks is investigated. In particular, it is shown how to avoid that the group generated by the round functions acts imprimitively, which represents a serious flaw for the cipher. The primitivity of this group is derived as a consequence of a more general result, which allows to reduce the problem of proving that a given FN generates a primitive group to the one of proving that an SPN, directly related to the given FN, generates a primitive group. Finally, a concrete instance of real-world size wave cipher is proposed as an example, and its resistance against differential and linear cryptanalysis is also established.

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“…If A = {0}, then A = C = D = B = {0}, which contradicts that U is non-trivial. If A is a proper subgroup of (F 2 ) n , then by (5) and since both a + aϕ 2 and aϕ belong to A we have…”

Section: By Lemma 44 and Bymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wave-shaped round functions and primitive groups

Aragona¹,

Calderini²,

Civino³

et al. 2019

Advances in Mathematics of Communications

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show abstract

“…• the bit of value 1 in position (13,4) is contained in the submatrices (Λ PRESENT ) (3,1)∶ (16,1) and (Λ PRESENT ) (4,1)∶ (16,2) ;…”

Section: Mixing Layer Of Presentmentioning

confidence: 99%

Type-preserving matrices and security of block ciphers

Aragona¹,

Meneghetti²

2019

Advances in Mathematics of Communications

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We introduce a new property for mixing layers which guarantees protection against algebraic attacks based on the imprimitivity of the group generated by the round functions. Mixing layers satisfying this property are called non-type-preserving. Our main result is to characterize such mixing layers by providing a list of necessary and sufficient conditions on the structure of their underlying binary matrices. Then we show how several families of linear maps are non-type-preserving, including the mixing layers of AES, GOST and PRESENT. Finally we prove that the group generated by the round functions of an SPN cipher with addition modulo 2 n as key mixing function is primitive if its mixing layer satisfies this property.

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“…To better understand the structure of Γ ∞ , we refer to translation based ciphers [11,3]. This is a class of iterated block ciphers including some well-known ciphers, as for instance AES [14] and SERPENT [1].…”

Section: Translation Based Ciphersmentioning

confidence: 99%

“…We now collect together some results which can be found in [3] (see Lemma 3.4, Lemma 3.5 and Proposition 3.6). Lemma 2.3.…”

Section: Translation Based Ciphersmentioning

confidence: 99%

Primitivity of PRESENT and other lightweight ciphers

et al. 2018

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We provide two sufficient conditions to guarantee that the round functions of a translation based cipher generate a primitive group. Furthermore, under the same hypotheses, and assuming that a round of the cipher is strongly proper and consists of m-bit S-Boxes, with m = 3, 4 or 5, we prove that such a group is the alternating group. As an immediate consequence, we deduce that the round functions of some lightweight translation based ciphers, such as the PRESENT cipher, generate the alternating group.

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

Cited by 12 publications

References 17 publications

Wave-shaped round functions and primitive groups

Wave-shaped round functions and primitive groups

Type-preserving matrices and security of block ciphers

Primitivity of PRESENT and other lightweight ciphers

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