Hankel Rhotrices and Constructions of Maximum Distance Separable Rhotrices over Finite Fields

Many block ciphers in cryptography use Maximum Distance Separable (MDS) matrices to strengthen the diffusion layer. Rhotrices are represented by coupled matrices. Therefore, use of rhotrices in the cryptographic ciphers doubled the security of the cryptosystem. We define Hankel rhotrix and further construct the maximum distance separable rhotrices over finite fields

Invariant subspaces in SPN block cipher

Исследуется рассеивание подпространств, инвариантных относительно нелинейного преобразования XSL-шифра, линейным преобразованием. Приведён конструктивный способ поиска подпространств, инвариантных относительно одной итерации XSL-шифра. Показано, что подпространства, инвариантные относительно нелинейных преобразований из некоторых классов, не сохраняются любой матрицей, построенной из ненулевых элементов расширения поля F2. На основании теоретико-графового и группового подходов доказан ряд свойств множеств специального вида, инвариантных относительно раундовой функции XSL-шифра

Real-time Partial Encryption of Oigital Video using Symmetric Dynamic Dual Keys Algorithm (SDD)

In recent years, as digital video technologies have been broadly used in TV,communication and multimedia. Security and privacy issues of the transmitted data have become an important concern in multimedia technology. Digital video stream is quite different from traditional textual data because interframe dependencies exist in digital video. Special digital video encryption algorithms are required because of their special characteristics, such as coding structure, large amount of data and real-time constraints. This paper presents a real-time partial encryption to digital video technique based on Symmetric Dynamic Dual (SDD) keys algorithm which is fast enough to meet the realtime requirements with high level of security. This approach uses dual key for encryption with variable (dynamic) block bits size,each block bits size (3 or 4 bits ) are interpreted as an element of a finite field.. The first key is called control key determines the length of bits block (3 or 4 bits block) size to encrypt, and the second key is used for encryption by using a equation: Y = X. A+ BWhereX is bits block, AandB are the encryption keys. The mathematical operations addition and multiplication in this equation are based on mathematical theory of Galois field GF(2 ). In this technique the I-frame (Intra-frame ) of the digital video scene is extracted and decomposed the color picture into its three color channels:luma channel (Y) and two chrominance channels Cb and Cr,with note that the frames of digital video is in YCbCr color system, the SDD algorithm is applied to the Y channel. The encryption algorithm achieves best timing results, and it provides high level of security by its great resistant against brute force attacks, because it uses dual key and dynamic block cipher, hence it will be very difficult to guess the key. To decrypt the ciphertext with 128 bits, the attacker needs 8.86569157e+188 of possibilities of keys as minimum and 7.91569097e+253 as maximum

On the Construction of Lightweight Orthogonal MDS Matrices

In present paper, we investigate 4 problems. Firstly, it is known that, a matrix is MDS if and only if all sub-matrices of this matrix of degree from 1 to $n$ are full rank. In this paper, we propose a theorem that an orthogonal matrix is MDS if and only if all sub-matrices of this orthogonal matrix of degree from 1 to $\lfloor\frac{n}{2}\rfloor$ are full rank. With this theorem, calculation of constructing orthogonal MDS matrices is reduced largely. Secondly, Although it has been proven that the $2^d\times2^d$ circulant orthogonal matrix does not exist over the finite field, we discover that it also does not exist over a bigger set. Thirdly, previous algorithms have to continually change entries of the matrix to construct a lot of candidates. Unfortunately, in these candidates, only very few candidates are orthogonal matrices. With the matrix polynomial residue ring and the minimum polynomials of lightweight element-matrices, we propose an extremely efficient algorithm for constructing $4\times4$ circulant orthogonal MDS matrices. In this algorithm, every candidate must be an circulant orthogonal matrix. Finally, we use this algorithm to construct a lot of lightweight results, and some of them are constructed first time

On the Construction of Lightweight Circulant Involutory MDS Matrices

In the present paper, we investigate the problem of constructing MDS matrices with as few bit XOR operations as possible. The key contribution of the present paper is constructing MDS matrices with entries in the set of $m\times m$ non-singular matrices over $\mathbb{F}_2$ directly, and the linear transformations we used to construct MDS matrices are not assumed pairwise commutative. With this method, it is shown that circulant involutory MDS matrices, which have been proved do not exist over the finite field $\mathbb{F}_{2^m}$, can be constructed by using non-commutative entries. Some constructions of $4\times4$ and $5\times5$ circulant involutory MDS matrices are given when $m=4,8$. To the best of our knowledge, it is the first time that circulant involutory MDS matrices have been constructed. Furthermore, some lower bounds on XORs that required to evaluate one row of circulant and Hadamard MDS matrices of order 4 are given when $m=4,8$. Some constructions achieving the bound are also given, which have fewer XORs than previous constructions

On Constructions of a Sort of MDS Block Diffusion Matrices for Block Ciphers and Hash Functions

Many modern block ciphers use maximum distance separate (MDS) matrices as their diffusion layers. In this paper, we propose a new method to verify a sort of MDS diffusion block matrices whose blocks are all polynomials in a certain primitive block over the finite field $\mathbb F_2$. And then we discover a new kind of transformations that can retain MDS property of diffusion matrices and generate a series of new MDS matrices from a given one. Moreover, we get an equivalence relation from this kind of transformation. And MDS property is an invariant with respect to this equivalence relation which can greatly reduce the amount of computation when we search for MDS matrices. The minimal polynomials of matrices play an important role in our strategy. To avoid being too theoretical, we list a series of MDS diffusion matrices obtained from our method for some specific parameters. Furthermore, we talk about MDS recursive diffusion layers with our method and extend the corresponding work of M. Sajadieh et al. published on FSE 2012 and the work of S. Wu published on SAC 2012

Lightweight MDS Involution Matrices

In this article, we provide new methods to look for lightweight MDS matrices, and in particular involutory ones. By proving many new properties and equivalence classes for various MDS matrices constructions such as circulant, Hadamard, Cauchy and Hadamard-Cauchy, we exhibit new search algorithms that greatly reduce the search space and make lightweight MDS matrices of rather high dimension possible to find. We also explain why the choice of the irreducible polynomial might have a significant impact on the lightweightness, and in contrary to the classical belief, we show that the Hamming weight has no direct impact. Even though we focused our studies on involutory MDS matrices, we also obtained results for non-involutory MDS matrices. Overall, using Hadamard or Hadamard-Cauchy constructions, we provide the (involutory or non-involutory) MDS matrices with the least possible XOR gates for the classical dimensions 4x4, 8x8, 16x16 and 32x32 in GF(2^4) and GF(2^8). Compared to the best known matrices, some of our new candidates save up to 50% on the amount of XOR gates required for an hardware implementation. Finally, our work indicates that involutory MDS matrices are really interesting building blocks for designers as they can be implemented with almost the same number of XOR gates as non-involutory MDS matrices, the latter being usually non-lightweight when the inverse matrix is required

Proceedings of AUTOMATA 2011 : 17th International Workshop on Cellular Automata and Discrete Complex Systems

International audienceThe proceedings contain full (reviewed) papers and short (non reviewed) papers that were presented at the workshop