Arbitrary Permutation Research Articles

A Reversible Logical Circuit Synthesis Algorithm Based on Decomposition of Cycle Representations of Permutations

A reversible function is isomorphic to a permutation and an arbitrary permutation can be represented by a series of cycles. A new synthesis algorithm for 3-qubit reversible circuits was presented. It consists of two parts, the first part used the Number of reversible function’s Different Bits (NDBs) to decide whether the NOT gate should be added to decrease the Hamming distance of the input and output vectors; the second part was based on the idea of exploring properties of the cycle representation of permutations, decomposed the cycles to make the permutation closer to the identity permutation and finally turn into the identity permutation, it was realized by using totally controlled Toffoli gates with positive and negative controls.

Implementation of new hybrid lightweight cryptosystem

Embedded systems, Internet of Things (IoT) and mobile computing devices are used in various domains which include public-private infrastructure, industrial installation and critical environment. Generally, information handled by these devices is private and critical. Therefore, it must be appropriately secured from different attacks and hackers. Lightweight cryptography is an aspiring field which investigates the implementation of cryptographic primitives and algorithms for resource constrained devices. In this paper, a new compact hybrid lightweight encryption technique has been proposed. Proposed technique uses the fastest bit permutation instruction PERMS with S-box of PRESENT block cipher for non-linearity. An arbitrary n-bit permutation is performed using PERMS instruction in less than log (n) number of instructions. This new hybrid system has been analyzed for software performance on Advanced RISC Machine (ARM) and Intel processor whereas Cadens tool is used to analyze the hardware performance. The result of the proposed technique is improved by the factor of eight as compared to the PRESENT-GRP hybrid block cipher. Moreover, PERMS instruction bit permutation properties result a very good avalanche effect and compact implementation in the both hardware and software environment.

Controllability of symmetric spin networks

We consider a network of n spin 12 systems which are pairwise interacting via Ising interaction and are controlled by the same electro-magnetic control field. Such a system presents symmetries since the Hamiltonian is unchanged if we permute two spins. This prevents full (operator) controllability, in that not every unitary evolution can be obtained. We prove however that controllability is verified if we restrict ourselves to unitary evolutions which preserve the above permutation invariance. For low dimensional cases, n = 2 and n = 3, we provide an analysis of the Lie group of available evolutions and give explicit control laws to transfer between two arbitrary permutation invariant states. This class of states includes highly entangled states such as Greenberger-Horne-Zeilinger (GHZ) states and W states, which are of interest in quantum information.

Coleman automorphisms of permutational wreath products II

ABSTRACTLet K be an arbitrary permutation group on a finite set Ω. Let G = H≀K be the corresponding permutational wreath product of a group H by K. It is proved that every Coleman automorphism of G is inner whenever H is either an almost simple group or a p-constrained group with for some prime p. In particular, the normalizer conjecture holds for such groups G. Other positive results regarding the normalizer conjecture are also obtained. Our results extend some known ones.

On the Order of Random Permutation with Cycle Weights

Let $\operatorname{Ord}(\tau)$ be the order of an element $\tau$ in the group $S_n$ of permutations of an $n$-element set $X$. The present paper is concerned with the so-called general parametric model of a random permutation; according to this model an arbitrary fixed permutation $\tau$ from $S_n$ is observed with the probability $\theta_1^{u_1}\dotsb\theta_n^{u_n}/H(n)$, where $u_i$ is the number of cycles of length $i$ of the permutation $\tau$, $\{\theta_i,\ i\in \mathbf{N}\}$ are some nonnegative parameters (the weights of cycles of length $i$ of the permutation $\tau$), and $H(n)$ is the corresponding normalizing factor. We assume that an arbitrary permutation $\tau_n$ has such a distribution. The function $p(n)=H(n)/n!$ is assumed to be $\mathrm{RO}$-varying at infinity with the lower index exceeding $-1$ (in particular, it can vary regularly), and the sequence $\{\theta_i,\ i\in \mathbf N\}$ is bounded. Under these assumptions it is shown that the random variable $\ln\operatorname{Ord}(\tau_n)$ is asymptotically normal with mean $\sum_{k=1}^n\theta_k\ln (k)/k$ and variance $\sum_{k=1}^n\theta_k\ln^2(k)/k$. In particular, this scheme subsumes the class of random $A$-permutations (i.e., when $\theta_i=\chi\{i\in A\}$), where $A$ is an arbitrary fixed subset of the positive integers. This scheme also includes the Ewens model of random permutation, where $\theta_i\equiv\theta>0$ for any $i\in\mathbf N$. The limit theorem we prove here extends some previous results for these schemes. In particular, with $\theta_i\equiv1$ for any $i\in\mathbf N$, the result just mentioned implies the well-known Erdös--Turán limit theorem.

Particle computation: complexity, algorithms, and logic

We investigate algorithmic control of a large swarm of mobile particles (such as robots, sensors, or building material) that move in a 2D workspace using a global input signal (such as gravity or a magnetic field). Upon activation of the field, each particle moves maximally in the same direction until forward progress is blocked by a stationary obstacle or another stationary particle. In an open workspace, this system model is of limited use because it has only two controllable degrees of freedom—all particles receive the same inputs and move uniformly. We show that adding a maze of obstacles to the environment can make the system drastically more complex but also more useful. We provide a wide range of results for a wide range of questions. These can be subdivided into external algorithmic problems, in which particle configurations serve as input for computations that are performed elsewhere, and internal logic problems, in which the particle configurations themselves are used for carrying out computations. For external algorithms, we give both negative and positive results. If we are given a set of stationary obstacles, we prove that it is NP-hard to decide whether a given initial configuration of unit-sized particles can be transformed into a desired target configuration. Moreover, we show that finding a control sequence of minimum length is PSPACE-complete. We also work on the inverse problem, providing constructive algorithms to design workspaces that efficiently implement arbitrary permutations between different configurations. For internal logic, we investigate how arbitrary computations can be implemented. We demonstrate how to encode dual-rail logic to build a universal logic gate that concurrently evaluates and, nand, nor, and or operations. Using many of these gates and appropriate interconnects, we can evaluate any logical expression. However, we establish that simulating the full range of complex interactions present in arbitrary digital circuits encounters a fundamental difficulty: a fan-out gate cannot be generated. We resolve this missing component with the help of 2 × 1 particles, which can create fan-out gates that produce multiple copies of the inputs. Using these gates we provide rules for replicating arbitrary digital circuits.

Polynomial measure of coherence

Coherence, the superposition of orthogonal quantum states, is indispensable in various quantum processes. Inspired by the polynomial invariant for classifying and quantifying entanglement, we first define polynomial coherence measure and systematically investigate its properties. Except for the qubit case, we show that there is no polynomial coherence measure satisfying the criterion that its value takes zero if and only if for incoherent states. Then, we release this strict criterion and obtain a necessary condition for polynomial coherence measure. Furthermore, we give a typical example of polynomial coherence measure for pure states and extend it to mixed states via a convex-roof construction. Analytical formula of our convex-roof polynomial coherence measure is obtained for symmetric states which are invariant under arbitrary basis permutation. Consequently, for general mixed states, we give a lower bound of our coherence measure.

Enhancement of multimedia security using random permutation with wavelet function

Digital data include financial documents and medical records, where security is not guaranteed. Unfortunately, cybercrime has developed for several multimedia applications. To overcome cybercrime, forensic analysis is applied. However, an increasing amount of malware is embedded in video payloads, and to minimize this amount, pseudo arbitrary permutation of movable Haar wavelet coefficients (PAP-MHWC) was developed. Pseudo arbitrary permutation using a secret key is permutated to decrease the probability of malware and reduce the distortion rate. Forensic security collects log file information as verification. Haar wavelet coefficients are a sequence of square-shaped functions used in a Fourier analysis to identify cyber forensic regions in video files. Such coefficients enable working with increasingly complex files by decomposing them into various positions and scales. Video frame overlapping is removed using a translation-invariance wavelet transform, thereby improving the forensic security rate. Experiment results show that the proposed PAP-MHWC method achieves a better performance in terms of malware detection accuracy, false-positive ratio, malware detection time, and malware crime probability rate than previous methods.

DEEP ATTRACTOR NETWORK FOR SINGLE-MICROPHONE SPEAKER SEPARATION.

Despite the overwhelming success of deep learning in various speech processing tasks, the problem of separating simultaneous speakers in a mixture remains challenging. Two major difficulties in such systems are the arbitrary source permutation and unknown number of sources in the mixture. We propose a novel deep learning framework for single channel speech separation by creating attractor points in high dimensional embedding space of the acoustic signals which pull together the time-frequency bins corresponding to each source. Attractor points in this study are created by finding the centroids of the sources in the embedding space, which are subsequently used to determine the similarity of each bin in the mixture to each source. The network is then trained to minimize the reconstruction error of each source by optimizing the embeddings. The proposed model is different from prior works in that it implements an end-to-end training, and it does not depend on the number of sources in the mixture. Two strategies are explored in the test time, K-means and fixed attractor points, where the latter requires no post-processing and can be implemented in real-time. We evaluated our system on Wall Street Journal dataset and show 5.49% improvement over the previous state-of-the-art methods.

Special Transpositions of Permutation Elements and Properties of Their Composition

We propose a strategy to solve the problem of optimization of a linear function on the set of cyclic permutations. The strategy is based on the properties of transpositions of special kind. The properties of this class of transpositions are investigated. Assertions about the impact of compositions of such transpositions on an arbitrary permutation are proved. For the approximate solutions obtained using the above strategy, estimation is substantiated.

Class-preserving coleman automorphisms of permutational wreath products with 2-closed base groups

Let [Formula: see text] be a nontrivial [Formula: see text]-closed group and let [Formula: see text] be an arbitrary permutation group on a finite set [Formula: see text]. Let [Formula: see text] be the corresponding permutational wreath product of [Formula: see text] by [Formula: see text]. It is shown that every class-preserving Coleman automorphism of [Formula: see text]-power order of [Formula: see text] is inner. As a direct consequence, it is obtained that the normalizer property holds for [Formula: see text]. Further, it is shown that every class-preserving Coleman automorphism of [Formula: see text] is inner whenever [Formula: see text] is nilpotent. Our results generalize some known ones.

Joint Independent Subspace Analysis Using Second-Order Statistics

This paper deals with a novel generalization of classical blind source separation (BSS) in two directions. First, relaxing the constraint that the latent sources must be statistically independent. This generalization is well-known and sometimes termed independent subspace analysis (ISA). Second, jointly analyzing several ISA problems, where the link is due to statistical dependence among corresponding sources in different mixtures. When the data are one-dimensional, i.e., multiple classical BSS problems, this model, known as independent vector analysis (IVA), has already been studied. In this paper, we combine IVA with ISA and term this new model joint independent subspace analysis (JISA). We provide full performance analysis of JISA, including closed-form expressions for minimal mean square error (MSE), Fisher information and Cramer–Rao lower bound, in the separation of Gaussian data. The derived MSE applies also for non-Gaussian data, when only second-order statistics are used. We generalize previously known results on IVA, including its ability to uniquely resolve instantaneous mixtures of real Gaussian stationary data, and having the same arbitrary permutation at all mixtures. Numerical experiments validate our theoretical results and show the gain with respect to two competing approaches that either use a finer block partition or a different norm.

L2 discrepancy of symmetrized generalized Hammersley point sets in base b

Two popular and often applied methods to obtain two-dimensional point sets with the optimal order of Lp discrepancy are digit scrambling and symmetrization. In this paper we combine these two techniques and symmetrize b-adic Hammersley point sets scrambled with arbitrary permutations. It is already known that these modifications indeed assure that the Lp discrepancy is of optimal order O(log⁡N/N) for p∈[1,∞) in contrast to the classical Hammersley point set. We prove an exact formula for the L2 discrepancy of these point sets for special permutations. We also present the permutations which lead to the lowest L2 discrepancy for every base b∈{2,…,27} by employing computer search algorithms.

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 ...

Double Derangement Permutations

Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that a permutation a is a double derangement with respect to x and y if and for each . In this paper, we give an explicit formula for , the number of double derangements with respect to x and y. Let and let and be two subsets of with and . Suppose that denotes the number of derangements x such that . As the main result, we show that if and z is a permutation such that for and for , then where .

A Modular-Arithmetic-Based Encryption Scheme

This paper considers a scenario in which a sender who holds a k -bit to k -bit trapdoor permutation f wants to transmit a message x to a receiver who holds the inverse permutation f-1; with the condition that encryption should require just one computation of f , decryption should require just one computation of f , the length of the enciphered text should be precisely k and the length n of the text x that can be encrypted is close to k . Our scheme takes the encryption of x to be f (rx) , which is a simple probabilistic encoding of x . Assuming an ideal hash function and an arbitrary trapdoor permutation, we describe and prove secure a simple invertible enmesh scheme that is bit-optimal in that the length of the string x that can be encrypted by f (rx) is almost k . Our scheme achieves semantic security, which implies chosen-cipher text security and non-malleability. Keywords: Asymmetric encryption

Reversible Markov structures on divisible set partitions

We studyk-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integerk= 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, fork> 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeablek-divisible partitions that are consistent under random deletion. We further introduce the notion ofMarkovian partition structures, which are ensembles of exchangeable Markov chains onk-divisible partitions that are consistent under a random process ofMarkovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

Reversible Markov structures on divisible set partitions

We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k > 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeable k-divisible partitions that are consistent under random deletion. We further introduce the notion of Markovian partition structures, which are ensembles of exchangeable Markov chains on k-divisible partitions that are consistent under a random process of Markovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

Optimality of entanglement witnesses constructed from arbitrary permutations

The optimality of a class of entanglement witnesses constructed from any permutation $$\pi $$? for any $$n\otimes n$$n?n bipartite systems is investigated. A necessary and sufficient condition is presented for such entanglement witnesses to be optimal, and thus, a class of optimal entanglement witnesses constructed by permutations is obtained.

On some universal construction of minimal topological generating sets for inverse limits of iterated wreath products of non-Abelian finite simple groups

Let $$(H_i)_{i\ge 1}$$(Hi)i?1 be an arbitrary sequence of non-Abelian finite simple transitive permutation groups. By using the combinatorial language of time-varying automata, we provide an explicit and naturally defined construction of a two-element set which generates a dense subgroup in the inverse limit $$\ldots \wr H_2\wr H_1$$??H2?H1 of iterated permutational wreath products of the groups $$H_i$$Hi. The corresponding automaton is equipped with three states, one of which is neutral and the semigroup generated by the other two states is free. We derive other algebraic and geometric properties of the group generated by this automaton. By using the notion of a Mealy automaton, we obtain the analogous construction for the infinite permutational wreath power of an arbitrary non-Abelian finite simple transitive permutation group $$H$$H on a set $$X$$X. We show that the wreath power $$\ldots \wr H\wr H$$??H?H contains a dense 2-generated not finitely presented amenable subgroup of exponential growth, which is generated by a 3-state Mealy automaton over the alphabet $$X$$X. The self-similar group generated by this automaton is self-replicating, contracting and regular weakly branch over the commutator subgroup.