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    Classifying 8-bit to 8-bit S-boxes based on power mappings from the point of DDT and LAT distributions
    (Springer-Verlag Berlin, 2008) Aslan, Bora; Sakallı, Muharrem Tolga; Buluş, Ercan
    S-boxes are vital elements in the design of symmetric ciphers. To date, the techniques for the construction of S-boxes have included pseudo-random generation, finite field inversion, power mappings and heuristic techniques. From these techniques, the use of finite field inversion in the construction of an S-box is so popular because it presents good cryptographic properties. On the other hand, while S-boxes such as AES, Shark, Square and Hierocrypt that are based on inversion mapping over GF(2(n)) use an affine transformation after the output of the S-box, in some ciphers like Camellia, an additional affine transformation is used before the input. In this paper, we classify 8-bit to 8-bit S-boxes based on power mappings into classes according to DDT and LAT distributions. Moreover, a formula is given for the calculation of the number of terms in the algebraic expression for a power mapping based S-box according to the given three probable cases.
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    On the algebraic expression of the AES S-box like S-boxes
    (2010) Sakallı, Muharrem Tolga; Aslan, Bora; Buluş, Ercan; Mesut, Andaç Şahin; Büyüksaraço?lu, Fatma; Karaahmeto?lu, Osman
    In the literature, there are several proposed block ciphers like AES, Square, Shark and Hierocrypt which use S-boxes that are based on inversion mapping over a finite field. Because of the simple algebraic structure of S-boxes generated in this way, these ciphers usually use a bitwise affine transformation after the inversion mapping. In some ciphers like Camellia, an additional affine transformation is used before the input of the S-box as well. In this paper, we study algebraic expressions of S-boxes based on power mappings with the aid of finite field theory and show that the number of terms in the algebraic expression of an S-box based on power mappings changes according to the place an affine transformation is added. Moreover, a new method is presented to resolve the algebraic expression of the AES S-box like S-boxes according to the given three probable cases. © 2010 Springer-Verlag.
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    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) Sakallı, Muharrem Tolga; Akleylek, Sedat; Aslan, Bora; Buluş, Ercan; Büyüksaraçoğlu Sakallı, Fatma
    We 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.