Permutation Properties Research Articles

Second-order nonlinear optical response of graphene

Although massless Dirac fermions in graphene constitute a centrosymmetric medium for in-plane excitations, their second-order nonlinear optical response is nonzero if the effects of spatial dispersion are taken into account. Here we present a rigorous quantum-mechanical theory of the second-order nonlinear response of graphene beyond the electric dipole approximation, which includes both intraband and interband transitions. The resulting nonlinear susceptibility tensor satisfies all symmetry and permutation properties, and can be applied to all three-wave mixing processes. We obtain useful analytic expressions in the limit of a degenerate electron distribution, which reveal quite strong second-order nonlinearity at long wavelengths, Fermi-edge resonances, and unusual polarization properties.

Plane Permutations and Applications to a Result of Zagier--Stanley and Distances of Permutations

In this paper, we introduce plane permutations, i.e., pairs $\mathfrak{p}=(s,\pi)$, where $s$ is an $n$-cycle and $\pi$ is an arbitrary permutation, represented as a two-row array. Accordingly, a plane permutation gives rise to three distinct permutations: the permutation induced by the upper horizontal ($s$), the vertical ($\pi$), and the diagonal ($D_{\mathfrak{p}}$) of the array. The latter can also be viewed as the three permutations of a hypermap. In particular, a map corresponds to a plane permutation, in which the diagonal is a fixed-point free involution. We study the transposition action on plane permutations obtained by permuting their diagonal blocks. We establish basic properties of plane permutations and analyze transpositions and exceedances, deriving various enumerative results. In particular we prove a recurrence for the number of plane permutations having a fixed diagonal and $k$ cycles in the vertical, generalizing a recursion of Chapuy for maps filtered by the genus. As applications of ...

Differential and linear properties analysis of the ciphers rijndael, serpent, threefish with 16-bit inputs and outputs

The results of the differential and linear properties analysis of the ciphers Rijndael, Serpent, Threefish with 16-bit input and output data blocks, indicating that all the ciphers repeat the properties of random permutations in the considered application mode are presented. The main provisions of a new methodology for accelerated cryptanalysis of block symmetric ciphers to the linear and differential cryptanalysis attacks consisting in the fact that all modern block ciphers after a certain number of cycles under the laws of transition distribution of XOR difference tables (total differentials) and the laws of bias of XOR difference tables (total differentials) repeat the corresponding indexes of random permutations are confirmed. The testing of randomness indexes of large ciphers can be accomplished through the development and subsequent analysis of randomness indexes of reduced models that allow carrying out real-time computational experiments. The results are further evidence that the full-scale ciphers and their small versions asymptotically repeat properties of random permutations of the corresponding degree.

A comparison of local search algorithms with population-based algorithms in hybrid flow shop scheduling problems with realistic characteristics

In this research, a bi-criteria group scheduling problem is investigated in a hybrid flow shop (HFS) environment, wherein the parallel machines are unrelated. The objective of the problem is to minimize a linear combination of the total weighted completion time being mindful of the producer and the total weighted tardiness being mindful of the customers. The underlying assumption is that the jobs are released into the system in dynamic times. The machine ready times are considered to be dynamic as well. Sequence-dependent setup times are required for changing the process between groups of jobs. The runtimes of jobs are assumed to be decreasing as the workers learn how to process similar jobs. A mixed-integer linear programing model is developed for the problem. However, since the problem is non-deterministic polynomial-time hard (NP-hard), it may not be solved to optimality within a reasonable time. This research comprehensively addresses the question of what type of meta-heuristic algorithm is more appropriate for solving these problems. In particular, local search algorithms are compared to the population-based algorithms with respect to the permutation or non-permutation properties of the optimal solution. Three algorithms based on tabu search as well as three algorithms based on simulated annealing are developed to represent the local search algorithms. Two other algorithms are developed based on genetic algorithm (GA) to exemplify the population-based algorithms. The results of a comprehensive experimental design reveal that GA-based algorithms have greater potential for identifying better quality solutions in HFS scheduling problems compared to the local search algorithms.

Information Leakage of a Kind of Block Cipher Structure

The block cipher New Structure II was designed at eprint 2011, it could be treated as a generalized Feistel structure. By checking the property of the linear permutation, we find a full round differential in New Structure II with probability 1. Since this differential can not distinguish the correct unknown key from the wrong keys, we fail in launching a key-recovery attack on this structure. However, we may guess 1/4 part of the plaintext without calculating the key. And we notice this weakness may cause insecurity in some special environments.

On permutation complexity of fixed points of some uniform binary morphisms

Combinatorics We study properties of infinite permutations generated by fixed points of some uniform binary morphisms, and find the formula for their complexity.

Quantum image encryption based on restricted geometric and color transformations

A novel encryption scheme for quantum images based on restricted geometric and color transformations is proposed. The new strategy comprises efficient permutation and diffusion properties for quantum image encryption. The core idea of the permutation stage is to scramble the codes of the pixel positions through restricted geometric transformations. Then, a new quantum diffusion operation is implemented on the permutated quantum image based on restricted color transformations. The encryption keys of the two stages are generated by two sensitive chaotic maps, which can ensure the security of the scheme. The final step, measurement, is built by the probabilistic model. Experiments conducted on statistical analysis demonstrate that significant improvements in the results are in favor of the proposed approach.

Some classes of monomial complete permutation polynomials over finite fields of characteristic two

In this paper, four classes of complete permutation polynomials over finite fields of characteristic two are presented. To consider the permutation property of the first three classes, Dickson polynomials play a key role. The fourth class is a generalization of a known result. In addition, we also calculate the inverses of these bijective monomials.

Dickson Polynomials of the Second Kind that Permute $\mathbb{Z}_m$

In this paper, we investigate the permutation property of the Dickson polynomials $E_n(x, a)$ of the second kind over $\mathbb{Z}_m$. Due to a known result, it suffices to consider permutation polynomials $E_n(x, a)$ over $\mathbb{Z}_{p^t}$, where $p$ is a prime and $t$ is a positive integer. We identify all permutation polynomials of $E_n(x, a)$ over $\mathbb{Z}_{p^t}$ for (I) $p=2$ and (II) $p$ is odd and $a$ is a square over $\mathbb{Z}_p$. For odd $p$ and nonsquares $a$ in $\mathbb{Z}_p$, we determine a large class (if not all) of permutation polynomials $E_n(x, a)$ over $\mathbb{Z}_{p^t}$. A conjecture is also presented in this paper. If this conjecture is true, then all Dickson permutation polynomials $E_n(x, a)$ of the second kind over $\mathbb{Z}_m$ are determined.

Separation Probabilities for Products of Permutations

We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements 1,2,. . .,kare in distinct cycles of the random permutation of {1,2,. . .,n} obtained as a product of two uniformly randomn-cycles.

Gelfand-Kirillov dimension of some primitive abundant semigroups

In this paper, the growth and Gelfand-Kirillov dimension of some primitive abundant semigroups are investigated. It is shown that for certain primitive abundant (regular) semigroup S, S as well as the semigroup algebra K [S] has polynomial growth if and only if all of its cancellative submonoids (subgroups) T as well as K[T] have polynomial growth. As applications, it is shown that if S is a finitely generated primitive inverse monoid having the permutational property, then clK dim K[S] = GK dim K[S] = rk(S).

SPONGENT: The Design Space of Lightweight Cryptographic Hashing

The design of secure yet efficiently implementable cryptographic algorithms is a fundamental problem of cryptography. Lately, lightweight cryptography--optimizing the algorithms to fit the most constrained environments--has received a great deal of attention, the recent research being mainly focused on building block ciphers. As opposed to that, the design of lightweight hash functions is still far from being well investigated with only few proposals in the public domain. In this paper, we aim to address this gap by exploring the design space of lightweight hash functions based on the sponge construction instantiated with present-type permutations. The resulting family of hash functions is called spongent. We propose 13 spongent variants--or different levels of collision and (second) preimage resistance as well as for various implementation constraints. For each of them, we provide several ASIC hardware implementations--ranging from the lowest area to the highest throughput. We make efforts to address the fairness of comparison with other designs in the field by providing an exhaustive hardware evaluation on various technologies, including an open core library. We also prove essential differential properties of spongent permutations, give a security analysis in terms of collision and preimage resistance, as well as study in detail dedicated linear distinguishers.

Permuting data on random-access block storage

Permutation is a fundamental operator for array data, with applications in, for example, changing matrix layouts and reorganizing data cubes. We consider the problem of permuting large quantities of data stored on secondary storage that supports fast random block accesses, such as solid state drives and distributed key-value stores. Faster random accesses open up interesting new opportunities for permutation. While external merge sort has often been used for permutation, it is an overkill that fails to exploit the property of permutation fully and carries unnecessary overhead in storing and comparing keys. We propose faster algorithms with lower memory requirements for a large, useful class of permutations. We also tackle practical challenges that traditional permutation algorithms have not dealt with, such as exploiting random block accesses more aggressively, considering the cost asymmetry between reads and writes, and handling arbitrary data dimension sizes (as opposed to perfect powers often assumed by previous work). As a result, our algorithms are faster and more broadly applicable.

Discrete symmetries and model-independent patterns of lepton mixing

In the context of discrete flavor symmetries, we elaborate a method that allows one to obtain relations between the mixing parameters in a model-independent way. Under very general conditions, we show that flavor groups of the von Dyck type, that are not necessarily finite, determine the absolute values of the entries of one column of the mixing matrix. We apply our formalism to finite subgroups of the infinite von Dyck groups, such as the modular groups, and find cases that yield an excellent agreement with the best fit values for the mixing angles. We explore the Klein group as the residual symmetry of the neutrino sector and explain the permutation property that appears between the elements of the mixing matrix in this case.

Different partners, different places: mathematics applied to the construction of four-couple folk dances

We describe the solution to a mathematical question that arises in the context of constructing four-couple dances in which there is no repetition in the positions and partnerships formed by the dancers during the intermediate stages of the dance. Our description makes use of various properties of permutations and cycle notation which are well known to mathematicians but probably less so more broadly. An implementation of the mathematical solution as an actual dance is discussed. We also consider generalizations of the original problem and explain a connection with the theory of orthogonal Latin squares.

Ramsey Properties of Permutations

The age of each countable homogeneous permutation forms a Ramsey class.Thus, there are five countably infinite Ramsey classes of permutations.

Anti-cropping digital image watermarking using Sudoku

Many digital image watermarking schemes have been developed to embed copyright information into an image. However, an attacker may reuse parts of a watermarked image by cropping out unwanted parts. Several techniques have been designed to overcome this attack but due to their limited redundancy approach, some section of the images can still be retrieved without detectable watermark. In this paper, a new watermarking scheme that is robust against severe cropping using Sudoku is proposed. It is based on Sudoku's permutation property that allows evenly distributed copies of watermark pieces in all parts of the cover image. A valid Sudoku solution is used during the embedding as well as during the detection of the watermark. Using classic 9 × 9 Sudoku, the scheme demonstrated robustness of up to 94% of random cropping.

Information Leakage of VGF2 Structure

The block cipher VGF2 was designed at Inscrypt 2009. It was based on a new variant of generalized Feistel structure. By checking the property of the linear permutation, we find a full round differential in VGF2 with probability 1. Since this differential cannot distinguish the correct unknown key from the wrong keys, we fail in launching a key-recovery attack on this structure. However, we may guess half part of the plaintext without calculating the key. And we notice that this weakness may cause insecurity in some special environments.

Large ciphers - the random permutation. comparison of indicators statistical security block symmetric ciphers before the ukrainian competition

This paper represents additional materials to prove the hypothesis that modern ciphers are asymptotically random substitutions. Here we compare statistics between real modern ciphers and their reduced models. The following indicators are considered: to be avalanche effect, completeness, degree of avalanche effect and degree of the strict avalanche criterion. We describe methods for determining these indicators and analyze the results of applying them to Ukrainian block cipher competitors and also for Rijndael and GOST ciphers. The presented results clearly confirm that the reduced models have practically the same statistics as their cipher prototypes discussed in this paper (Rijndael, GOST and Ukrainian block cipher competitors – Muhomor, Kalina, ADE, Labirint). This means we can build reduced models with randomness indicators that correspond to their big counterparts, and, therefore, the conclusions we get for 'small' ciphers can be applied to the real ciphers. In particular, it refers to the conclusion that both 'small' and real ciphers asymptotically acquire properties of random permutations. Correlative indicators in dynamics of their changes are generally the same as the results obtained from studying differential and linear properties for 'small' and real ciphers. We get additional evidence and arguments in favor of proposed method for measuring block ciphers security by studying their reduced models. As the additional result of the research, it was concluded that three of the four Ukrainian block cipher competitors have better statistic indicators than well-known Rijndael and thus are superior.

Canonical Decompositions of Affine Permutations, Affine Codes, and Split $k$-Schur Functions

We develop a new perspective on the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, implicit in work of Thomas Lam. This decomposition is closely related to the affine code, which generalizes the $k$-bounded partition associated to Grassmannian elements. We also prove that the affine code readily encodes a number of basic combinatorial properties of an affine permutation. As an application, we prove a new special case of the Littlewood-Richardson Rule for $k$-Schur functions, using the canonical decomposition to control for which permutations appear in the expansion of the $k$-Schur function in noncommuting variables over the affine nil-Coxeter algebra.