8 research outputs found
Construction of New Families of ‎MDS‎ Diffusion Layers
Diffusion layers are crucial components of symmetric ciphers‎. ‎These components‎, ‎along with suitable Sboxes‎, ‎can make symmetric ciphers resistant against statistical attacks like linear and differential cryptanalysis‎. ‎Conventional ‎‎MDS diffusion layers, which are defined as matrices over finite fields, have been used in symmetric ciphers such as AES‎, ‎Twofish and SNOW‎. ‎In this paper‎, ‎we study linear, linearized and nonlinear MDS diffusion layers‎. We investigate linearized diffusion layers, ‎which are a generalization of conventional diffusion layers‎; t‎hese diffusion layers are used in symmetric ciphers like SMS4‎, ‎Loiss and ZUC‎. W‎e introduce some ‎new ‎families of linearized MDS diffusion layers ‎and as a consequence, ‎we ‎present a‎ ‎method ‎for ‎construction of ‎‎‎‎randomized linear ‎‎‎‎‎diffusion ‎layers over a finite field. Nonlinear MDS diffusion layers are introduced in Klimov\u27s thesis; we investigate nonlinear MDS diffusion layers theoretically, and we present a new family of nonlinear MDS diffusion layers. We show that these nonlinear diffusion layers can be made randomized with a low ‎implementatio‎n cost. An important fact about linearized and nonlinear diffusion layers is that they are more resistant against algebraic attacks in comparison to conventional diffusion layers. A ‎special case of diffusion layers are ‎‎‎(0,1)‎-‎diffusion layers. This type of diffusion layers are used in symmetric ciphers like ARIA‎. ‎W‎e examine (0,1)‎-‎diffusion layers and prove a theorem about them‎. ‎At last‎, ‎we study linearized MDS diffusion layers of symmetric ciphers Loiss, SMS4 and ZUC‎, from the mathematical viewpoint
Cryptographic Properties of Addition Modulo
The operation of modular addition modulo a power of two is one of the most applied operations in symmetric cryptography.
For example, modular addition is used in RC6, MARS and Twofish block ciphers and RC4, Bluetooth and Rabbit stream ciphers.
In this paper, we study statistical and algebraic properties of modular addition modulo a power of two. We obtain probability
distribution of modular addition carry bits along with conditional probability distribution of these carry bits. Using these probability
distributions and Markovity of modular addition carry bits, we compute the joint probability distribution of arbitrary number
of modular addition carry bits. Then, we examine algebraic properties of modular addition with a constant and obtain the number
of terms as well as algebraic degrees of component Boolean functions of modular addition with a constant. Finally, we present another
formula for the ANF of the component Boolean functions of modular addition modulo a power of two. This formula
contains more information than representations which are presented in cryptographic literature, up to now
Watching systems of triangular graphs
A watching system in a graph is a set , where and is a subset of closed neighborhood of
such that the sets are non-empty and distinct, for any . In this
paper, we study the watching systems of line graph which is
called triangular graph and denoted by . The minimum size of a
watching system of is denoted by . We show that
The annihilating-ideal graph of is weakly perfect
A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let be a commutative ring with identity and be the set of ideals with non-zero annihilator. The annihilating-ideal graph of is defined as the graph with the vertex set and two distinct vertices and are adjacent if and only if . In this paper, we show that the graph , for every positive integer , is weakly perfect. Moreover, the exact value of the clique number of is given and it is proved that is class 1 for every positive integer
On Double-Star Decomposition of Graphs
A tree containing exactly two non-pendant vertices is called a double-star. A double-star with degree sequence (k1 + 1, k2 + 1, 1, . . . , 1) is denoted by Sk1,k2. We study the edge-decomposition of graphs into double-stars. It was proved that every double-star of size k decomposes every 2k-regular graph. In this paper, we extend this result by showing that every graph in which every vertex has degree 2k + 1 or 2k + 2 and containing a 2-factor is decomposed into Sk1,k2 and Sk1−1,k2, for all positive integers k1 and k2 such that k1 + k2 = k