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Öğe Algebraic construction of cryptographically good binary linear transformations(Wiley-Hindawi, 2014) Aslan, Bora; Sakalli, Muharrem Tolga[Abstract Not Available]Öğe Binary Finite Field Extensions for Diffusion Matrices over the Finite Field F2m(IEEE, 2021) Pehlivanoglu, Meltem Kurt; Sakalli, Fatma Buyuksaracoglu; Akleylek, Sedat; Sakalli, Muharrem TolgaIn this paper, a new software tool has been developed that computes the corresponding m x m binary matrix over the finite field F-2 of each element which is defined over F-2m (where 3 <= m <= 8) generated by different primitive irreducible polynomials. This extension process is necessary for the optimization of XOR (exclusive OR) counts of diffusion matrices whose elements are defined over the finite field, which are used especially in the diffusion layers of block ciphers. Therefore, the corresponding binary matrices given in this study can be used directly for the construction of new diffusion matrices.Öğe Efficient methods to generate cryptographically significant binary diffusion layers(Inst Engineering Technology-Iet, 2017) Akleylek, Sedat; Rijmen, Vincent; Sakalli, Muharrem Tolga; Ozturk, EmirIn this study, the authors propose new methods using a divide-and-conquer strategy to generate n x n binary matrices ( for composite n) with a high/maximum branch number and the same Hamming weight in each row and column. They 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, they focus on generating 12 x 12, 16 x 16 and 32 x 32 binary matrices with a maximum or maximum achievable branch number and the lowest implementation costs (to the best of their knowledge) to be used in block ciphers. Then, they discuss the implementation properties of binary matrices generated and present experimental results for binary matrices in these sizes. Finally, they apply the proposed methods to larger sizes, i.e. 48 x 48, 64 x 64 and 80 x 80 binary matrices having some applications in secure multi-party computation and fully homomorphic encryption.Öğe Generalisation of Hadamard matrix to generate involutory MDS matrices for lightweight cryptography(Inst Engineering Technology-Iet, 2018) Pehlivanoglu, Meltem Kurt; Sakalli, Muharrem Tolga; Akleylek, Sedat; Duru, Nevcihan; Rijmen, VincentIn this study, the authors generalise Hadamard matrix over F-2m and propose a new form of Hadamard matrix, which they call generalised Hadamard (GHadamard) matrix. Then, they focus on generating lightweight (involutory) maximum distance separable (MDS) matrices. They also extend this idea to any k x k matrix form, where k is not necessarily a power of 2. The new matrix form, GHadamard matrix, is used to generate new 4 x 4 involutory MDS matrices over F-24 and F-28, and 8 x 8 involutory/non- involutory MDS matrices over F-24 by considering the minimum exclusive OR (XOR) count, which is a metric defined to estimate the hardware implementation cost. In this context, they improve the best-known results of XOR counts for 8 x 8 involutory/non-involutory MDS matrices over F-24.Öğe Generating binary diffusion layers with maximum/high branch numbers and low search complexity(Wiley-Hindawi, 2016) Akleylek, Sedat; Sakalli, Muharrem Tolga; Ozturk, Emir; Mesut, Andac Sahin; Tuncay, GokhanIn this paper, we propose a new method to generate n x n binary matrices (for n = k . 2(t) where k and t are positive integers) with a maximum/high of branch numbers and a minimum number of fixed points by using 2(t) x 2(t) Hadamard (almost) maximum distance separable matrices and k x k cyclic binary matrix groups. By using the proposed method, we generate n x 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 x m circulant matrices to generate n x n (for n = k . m) binary matrices with a maximum/high of branch numbers. For this case, some examples for 16 x 16, 48 x 48, and 64 x 64 binary matrices with branch numbers of 8, 15, and 18, respectively, are presented. Copyright (C) 2016 John Wiley & Sons, Ltd.Öğe A new hybrid method combining search and direct based construction ideas to generate all 4 x 4 involutory maximum distance separable (MDS) matrices over binary field extensions(Peerj Inc, 2023) Tuncay, Gokhan; Sakalli, Fatma Buyuksaracoglu; Pehlivanoglu, Meltem Kurt; Yilmazguc, Gulsum Gozde; Akleylek, Sedat; Sakalli, Muharrem TolgaThis article presents a new hybrid method (combining search based methods and direct construction methods) to generate all 4 x 4 involutory maximum distance separable (MDS) matrices over F2m. The proposed method reduces the search space complexity at the level of root n, where n represents the number of all 4 x 4 invertible matrices over F-2m to be searched for. Hence, this enables us to generate all 4 x 4 involutory MDS matrices over F(2)3 and F(2)4. After applying global optimization technique that supports higher Exclusive-OR (XOR) gates (e.g., XOR3, XOR4) to the generated matrices, to the best of our knowledge, we generate the lightest involutory/ non-involutory MDS matrices known over F(2)3, F(2)4 and F(2)8 in terms of XOR count. In this context, we present new 4 x 4 involutory MDS matrices over F(2)3, F(2)4 and F(2)8, which can be implemented by 13 XOR operations with depth 5, 25 XOR operations with depth 5 and 42 XOR operations with depth 4, respectively. Finally, we denote a new property of Hadamard matrix, i.e., (involutory and MDS) Hadamard matrix form is, in fact, a representative matrix form that can be used to generate a small subset of all 2(k) x 2(k) involutory MDS matrices, where k > 1. For k = 1, Hadamard matrix form can be used to generate all involutory MDS matrices.Öğe A new matrix form to generate all 3 x 3 involutory MDS matrices over F2m(Elsevier Science Bv, 2019) Guzel, Gulsum Gozde; Sakalli, Muharrem Tolga; Akleylek, Sedat; Rijmen, Vincent; Cengellenmis, YaseminIn this paper, we propose a new matrix form to generate all 3 x 3 involutory and MDS matrices over F-2(m) and prove that the number of all 3 x 3 involutory and MDS matrices over F-2(m) is (2(m) - 1)(2) . (2(m) - 2) . (2(m) - 4), where m > 2. Moreover, we give 3 x 3 involutory and MDS matrices over F-2(3), F-2(4) and F-2(8) defined by the irreducible polynomials x(3) +x+ 1, x(4) +x + 1 and x(8) + x(7) + x(6) + x + 1, respectively, by considering the minimum XOR count, which is a metric used in the estimation of hardware implementation cost. Finally, we provide the maximum number of 1s in 3 x 3 involutory MDS matrices. (C) 2019 Elsevier B.V. All rights reserved.Öğe A new method to determine algebraic expression of power mapping based S-boxes(Elsevier, 2013) Karaahmetoglu, Osman; Sakalli, Muharrem Tolga; Bulus, Ercan; Tutanescu, IonPower mapping based S-boxes, especially those with finite field inversion, have received significant attention by cryptographers. S-boxes designed by finite field inversion provide good cryptographic properties and are used in most ciphers' design such as Advanced Encryption Standard (AES), Camellia, Shark and others. However, such an S-box consists of a simple algebraic expression, thus the S-box design is completed by adding an affine transformation before the input of the S-box, or after the output of the S-box or both in order to make the overall S-box description more complex in a finite field. In the present study, a new method of computation of the algebraic expression (as a polynomial function over GF(2(8))) of power mapping based S-boxes designed by three different probable cases is described in which the place of the affine transformation differs. The proposed method is compared with the Lagrange interpolation formula with respect to the number of polynomial operations needed. The new method (based on the square-and-multiply technique) is found to reduce time and polynomial operation complexity in the computation of the algebraic expression of S-boxes. (C) 2013 Elsevier B.V. All rights reserved.Öğe On the algebraic construction of cryptographically good 32 x 32 binary linear transformations(Elsevier Science Bv, 2014) Sakalli, Muharrem Tolga; Aslan, BoraBinary linear transformations (also called binary matrices) have matrix representations over GF(2). Binary matrices are used as diffusion layers in block ciphers such as Camellia and ARIA. Also, the 8 x 8 and 16 x 16 binary matrices used in Camellia and ARIA, respectively, have the maximum branch number and therefore are called Maximum Distance Binary Linear (MDBL) codes. In the present study, a new algebraic method to construct cryptographically good 32 x 32 binary linear transformations, which can be used to transform a 256-bit input block to a 256-bit output block, is proposed. When constructing these binary matrices, the two cryptographic properties; the branch number and the number of fixed points are considered. The method proposed is based on 8 x 8 involutory and non-involutory Finite Field Hadamard (FFHadamard) matrices with the elements of GF(2(4)). How to construct 32 x 32 involutory binary matrices of branch number 12, and non-involutory binary matrices of branch number 11 with one fixed point, are described. (C) 2013 Elsevier By. All rights reserved.Öğe On the automorphisms and isomorphisms of MDS matrices and their efficient implementations(Tubitak Scientific & Technological Research Council Turkey, 2020) Sakalli, Muharrem Tolga; Akleylek, Sedat; Akkanat, Kemal; Rijmen, VincentIn this paper, we explicitly define the automorphisms of MDS matrices over the same binary extension field. By extending this idea, we present the isomorphisms between MDS matrices over F-2m and MDS matrices over F-2mt, where t >= 1 and m > 1, which preserves the software implementation properties in view of XOR operations and table lookups of any given MDS matrix over F-2m. Then we propose a novel method to obtain distinct functions related to these automorphisms and isomorphisms to be used in generating isomorphic MDS matrices (new MDS matrices in view of implementation properties) using the existing ones. The comparison with the MDS matrices used in AES, ANUBIS, and subfield-Hadamard construction shows that we generate an involutory 4 x 4 MDS matrix over F-28 (from an involutory 4 x 4 MDS matrix over F-24) whose required number of XOR operations is the same as that of ANUBIS and the subfield-Hadamard construction, and better than that of AES. The proposed method, due to its ground field structure, is intended to be a complementary method for the current construction methods in the literature.Öğe On the Construction of 20 x 20 and 24 x 24 Binary Matrices with Good Implementation Properties for Lightweight Block Ciphers and Hash Functions(Hindawi Ltd, 2014) Sakalli, Muharrem Tolga; Akleylek, Sedat; Aslan, Bora; Bulus, Ercan; Sakalli, Fatma BuyuksaracogluWe present an algebraic construction based on state transform matrix (companion matrix) for n x n (where n + 2(k), k being a positive integer) binary matrices with high branch number and low number of fixed points. We also provide examples for 20 x 20 and 24 x 24 binary matrices having advantages on implementation issues in lightweight block ciphers and hash functions. The powers of the companion matrix for an irreducible polynomial over GF(2) with degree 5 and 4 are used in finite field Hadamard or circulant manner to construct 20 x 20 and 24 x 24 binary matrices, respectively. Moreover, the binary matrices are constructed to have good software and hardware implementation properties. To the best of our knowledge, this is the first study for n x n (where n not equal 2(k), k being a positive integer) binary matrices with high branch number and low number of fixed points.Öğe On the Construction of New Lightweight Involutory MDS Matrices in Generalized Subfield Form(IEEE-Inst Electrical Electronics Engineers Inc, 2023) Pehlivanoglu, Meltem Kurt; Sakalli, Fatma Buyuksaracoglu; Akleylek, Sedat; Sakalli, Muharrem TolgaMaximum Distance Separable (MDS) matrices are used as the main component of diffusion layers in block ciphers. MDS matrices have the optimal diffusion properties and the maximum branch number, which is a criterion to measure diffusion rate and security against linear and differential crypt analysis. However, it is a challenging problem to construct hardware-friendly MDS matrices with optimal or close to optimal circuits, especially for involutory ones. In this paper, we consider the generalized subfield construction method from the global optimization perspective and then give new 4 x 4 involutory MDS matrices over F-2(3) and F-2(5). After that, we present 1,176 (= 28 x 42) new 4 x 4 involutory and MDS diffusion matrices by 33 XORs and depth 3. This new record also improves the previously best-known cost of 38 XOR gates.
