On the Construction of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mn>20</mml:mn><mml:mo>×</mml:mo><mml:mn>20</mml:mn></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mn>2</mml:mn><mml:mn>4</mml:mn><mml:mo>×</mml:mo><mml:mn>24</mml:mn></mml:math>Binary Matrices with Good Implementation Properties for Lightweight Block Ciphers and Hash Functions

Sakallı, Muharrem Tolga; Akleylek, Sedat; Aslan, Bora; Buluş, Ercan; Sakallı, Fatma Büyüksaraçoğlu

doi:10.1155/2014/540253

Cited by 8 publications

(5 citation statements)

References 20 publications

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“…Nevertheless, it seems that their results are not close to the optimal implementation properties (especially on 8 bit platforms) for 32 × 32 binary matrices. In [18], Sakallı et al provide a method using a state transform matrix to generate n × n (where n is not a power of 2) binary matrices with a high branch number and a low number of fixed points. In [12], Albrecht et al suggest using left circulant matrices with a maximum branch number.…”

Section: Previous Studiesmentioning

confidence: 99%

Efficient methods to generate cryptographically significant binary diffusion layers

Akleylek¹,

Rijmen

Sakallı

et al. 2017

IET Information Security

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In this paper, we propose new methods using a divide-and-conquer strategy to generate n × n binary matrices (for composite n) with a high/maximum branch number and the same Hamming weight in each row and column. We introduce new types of binary matrices, namely (BHwC) t,m and (BCwC) q,m types, which are a combination of Hadamard and circulant matrices, and the recursive use of circulant matrices, respectively. With the help of these hybrid structures, the search space to generate a binary matrix with a high/maximum branch number is drastically reduced.By using the proposed methods, we focus on generating 12 × 12, 16 × 16 and 32 × 32 binary matrices with a maximum or maximum achievable branch number and the lowest implementation costs (to the best of our knowledge) to be used in block ciphers. Then, we discuss the implementation properties of binary matrices generated and present experimental results for binary matrices in these sizes. Finally, we apply the proposed methods to larger sizes, i.e., 48 × 48, 64 × 64 and 80 × 80 binary matrices having some applications in secure multi-party computation and fully homomorphic encryption.

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Section: Previous Studiesmentioning

confidence: 99%

Efficient methods to generate cryptographically significant binary diffusion layers

Akleylek¹,

Rijmen

Sakallı

et al. 2017

IET Information Security

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“…The advantages of the present study over the previous studies , , and are to classify 3 × 3 and 4 × 4 cyclic matrix elements in a systematic way and use them in (almost) MDS matrices with special matrix forms. Moreover, the connection between 16 × 16 binary matrices with a branch number of 8 and 4 × 4 MDS matrices over

{double-struckF}_{2^{4}}

in Hadamard form is presented and formalized.…”

Section: Introductionmentioning

confidence: 96%

“…In , 32 × 32 involutory binary matrices with the maximum branch number of 12 were constructed by 8 × 8 matrices over

{double-struckF}_{2^{4}}

(Hadamard matrices with branch numbers of 7 or 8). In , 20 × 20 and 24 × 24 binary matrices with branch numbers of 8 and 10, respectively, were presented by a different interpretation of the method given in and . In , Albrecht et al .…”

Section: Introductionmentioning

confidence: 99%

“…In this study, we combine and extend the strategies of , , and with a different point of view. We first find cyclic binary matrix groups having algebraically suitable properties, and then we construct binary matrices by using the elements of these cyclic groups in 2 × 2, 4 × 4, and 8 × 8 Hadamard matrices and 4 × 4, 12 × 12, and 16 × 16 circulant matrices.…”

Section: Introductionmentioning

confidence: 99%

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Generating binary diffusion layers with maximum/high branch numbers and low search complexity

Akleylek

Sakallı

Öztürk

et al. 2016

Security Comm. Networks

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In this paper, we propose a new method to generate n × n binary matrices (for n = k·2t where k and t are positive integers) with a maximum/high of branch numbers and a minimum number of fixed points by using 2t×2t Hadamard (almost) maximum distance separable matrices and k × k cyclic binary matrix groups. By using the proposed method, we generate n × n (for n = 6, 8, 12, 16, and 32) binary matrices with a maximum of branch numbers, which are efficient in software implementations. The proposed method is also applicable with m × m circulant matrices to generate n × n(for n = k·m) binary matrices with a maximum/high of branch numbers. For this case, some examples for 16 × 16, 48 × 48, and 64 × 64 binary matrices with branch numbers of 8, 15, and 18, respectively, are presented. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Системы импримитивности этой группы сохраняются линейным слоем и слоем нало-жения ключа, поэтому рассеивание блоков импримитивности в алгоритме шифрова-ния может обеспечить только слой s-боксов. Подобные слабости успешно использованы в работах [6,7] по исследованию шифрсистем KHAZAD и PRINT.Обозначим через P = GF (2 n ) конечное поле, n ∈ N. Определение 1 [8,9]. Пусть m ∈ N 0 .…”

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Characterization of linear transformations defined by finite field hadamard matrices and circulant matrices

Volgin¹,

Kryuchkov²

2017

Applied Discrete Mathematics. Supplement

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Приведены общие и криптографические свойства циркулянтных матриц и ли-нейных преобразований, задаваемых матрицами Адамара над конечным полем. Описаны инвариантные подпространства матриц Адамара над конечным полем. Построен класс подпространств, гарантированно являющихся инвариантными для циркулянтных матриц. Наличие инвариантных подпространств приводит к импримитивности группы C(g) [5], порождённой слоем наложения ключа V + n и приводимой матрицей g ∈ GL n (2). Системы импримитивности этой группы сохраняются линейным слоем и слоем нало-жения ключа, поэтому рассеивание блоков импримитивности в алгоритме шифрова-ния может обеспечить только слой s-боксов. Подобные слабости успешно использованы в работах [6,7] по исследованию шифрсистем KHAZAD и PRINT.Обозначим через P = GF (2 n ) конечное поле, n ∈ N. Определение 1 [8,9]. Пусть m ∈ N 0 . Матрица H = (h i,j ) ∈ P 2 m ,2 m называется матрицей Адамара над конечным полем (FFH-матрицей), если для некоторого век-тора (a 0 , . . . , a 2 m −1 ) ∈ P 2 m выполняется соотношение h i,j = a i⊕j . При этом матрицу H будем обозначать H = had(a 0 , . . . , a 2 m −1 ).В работе получены следующие простейшие свойства FFH-матриц.2) H 2 = r 2 E; 3) класс FFH-матриц замкнут относительно операций сложения и умножения мат-риц, умножения матрицы на константу, тензорного произведения матриц; 4) H подобна верхнетреугольной матрице и её характеристический многочлен име-ет вид χ H (x) = (x + r) 2 m ; 5) если H 1 , H 2 ∈ P 2 m ,2 m FFH-матрицы, то H 1 H 2 = H 2 H 1 .Далее опишем инвариантные подпространства матриц Адамара. Теорема 1. Пусть m ∈ N и H = had(a 0 , . . . , a 2 m −1 ) ∈ P 2 m ,2 m . Пусть также

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On the Construction of20×20and24×24Binary Matrices with Good Implementation Properties for Lightweight Block Ciphers and Hash Functions

Cited by 8 publications

References 20 publications

Efficient methods to generate cryptographically significant binary diffusion layers

Efficient methods to generate cryptographically significant binary diffusion layers

Generating binary diffusion layers with maximum/high branch numbers and low search complexity

Characterization of linear transformations defined by finite field hadamard matrices and circulant matrices

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