Abstract: Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature so far. In this paper we present a number of permutation trinomials over finite fields, which are of different forms.

Abstract: In relational databases, relations between objects, represented by binary matrices or tensors, may be arbitrarily complex. In practice however, there are recurring relational patterns such as transitive, permutation, and sequential relationships, that have a regular structure which is not captured by the classical notion of matrix rank or tensor rank. In this paper, we show that factorizing the relational tensor using a logistic or hinge loss instead of the more standard squared loss is more appropriate because it can accurately model many common relations with a fixed-size embedding (depends sub-linearly on the number of entities in the knowledge base). We illustrate this fact empirically by being able to efficiently predict missing links in several synthetic and real-world experiments. Further, we provide theoretical justification for logistic loss by studying its connection to a complexity measure from the field of information complexity called sign rank. Sign rank is a more appropriate complexity measure as it is low for transitive, permutation, or sequential relationships, while being suitably large, with a high probability, for uniformly sampled binary matrices/tensors.

Abstract: Two dimensional conformal field theories with large central charge and a sparse low-lying spectrum are expected to admit a classical string holographic dual. We construct a large class of such theories employing permutation orbifold technology. In particular, we describe the group theoretic constraints on permutation groups to ensure a (stringy) holographic CFT. The primary result we uncover is that in order for the degeneracy of states to be finite in the large central charge limit, the groups of interest are the so-called oligomorphic permutation groups. Further requiring that the low-lying spectrum be sparse enough puts a bound on the number of orbits of these groups (on finite element subsets). Along the way we also study familiar cyclic and symmetric orbifolds to build intuition. We also demonstrate how holographic spectral properties are tied to the geometry of covering spaces for permutation orbifolds.

Abstract: We study how to construct efficient tweakable block ciphers in the Random Permutation model, where all parties have access to public random permutation oracles. We propose a construction that combines, more efficiently than by mere black-box composition, the CLRW construction (which turns a traditional block cipher into a tweakable block cipher) of Landecker et al. (CRYPTO 2012) and the iterated Even-Mansour construction (which turns a tuple of public permutations into a traditional block cipher) that has received considerable attention since the work of Bogdanov et al. (EUROCRYPT 2012). More concretely, we introduce the (one-round) tweakable Even-Mansour (TEM) cipher, constructed from a single n-bit permutation P and a uniform and almost XOR-universal family of hash functions \((H_k)\) from some tweak space to \(\{0,1\}^n\), and defined as \((k,t,x)\mapsto H_k(t)\oplus P(H_k(t)\oplus x)\), where k is the key, t is the tweak, and x is the n-bit message, as well as its generalization obtained by cascading r independently keyed rounds of this construction. Our main result is a security bound up to approximately \(2^{2n/3}\) adversarial queries against adaptive chosen-plaintext and ciphertext distinguishers for the two-round TEM construction, using Patarin’s H-coefficients technique. We also provide an analysis based on the coupling technique showing that asymptotically, as the number of rounds r grows, the security provided by the r-round TEM construction approaches the information-theoretic bound of \(2^n\) adversarial queries.

Abstract: We present a novel algorithm for significant pattern mining, Westfall-Young light. The target patterns are statistically significantly enriched in one of two classes of objects. Our method corrects for multiple hypothesis testing and correlations between patterns via the Westfall-Young permutation procedure, which empirically estimates the null distribution of pattern frequencies in each class via permutations.In our experiments, Westfall-Young light dramatically outperforms the current state-of-the-art approach, both in terms of runtime and memory efficiency on popular real-world benchmark datasets for pattern mining. The key to this efficiency is that, unlike all existing methods, our algorithm does not need to solve the underlying frequent pattern mining problem anew for each permutation and does not need to store the occurrence list of all frequent patterns. Westfall-Young light opens the door to significant pattern mining on large datasets that previously involved prohibitive runtime or memory costs.Our code is available from http://www.bsse.ethz.ch/mlcb/research/machine-learning/wylight.html

Abstract: Abstract In recent years, the operation efficiency of chaos-based image cryptosystems has drawn much more concerns. However, the workload arised from floating point arithmetic in chaotic map iteration prevents the efficiency promotion of these cryptosystems. In this paper, we present a novel image encryption scheme using Gray code based permutation approach. The new permutation strategy takes full advantage of (n, p, k)-Gray-code achievements, and is performed with high efficiency. A plain pixel-related image diffusion scheme is introduced to compose a complete cryptosystem. Simulations and extensive security analyses have been carried out and the results demonstrate the high security and operation efficiency of the proposed scheme.

Abstract: In this paper, a new selective encryption scheme based on DWT, AES S-box and chaotic permutation is proposed. The new scheme is composed of six steps: Image decomposition, Block permutation, DWT decomposition, substitution phase, chaotic permutation phase and reconstruction phase. Firstly, it generates four subbands, namely cAP, cVP, cHP and cDP, and encrypts only cAP subband, which contains the meaningful part of data. The proposed cryptosystem is evaluated using various security and statistical analysis. The performance tests show that the proposed scheme is secure against statistical and differential attacks.

Abstract: We study space-filling properties of good lattice point sets and obtain some general theoretical results. We show that linear level permutation does not decrease the minimum distance for good lattice point sets, and we identify several classes of such sets with large minimum distance. Based on good lattice point sets, some maximin distance designs are also constructed.

Abstract: Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying interpretation that relates a vast array of combinatorial structures. In this paper, we introduce the notion of patterns in inversion sequences. A sequence $(e_1,e_2,\ldots,e_n)$ is an inversion sequence if $0 \leq e_i \pi_i \}|$. This correspondence makes it a natural extension to study patterns in inversion sequences much in the same way that patterns have been studied in permutations. This paper, the first of two on patterns in inversion sequences, focuses on the enumeration of inversion sequences that avoid words of length three. Our results connect patterns in inversion sequences to a number of well-known numerical sequences including Fibonacci numbers, Bell numbers, Schr\"oder numbers, and Euler up/down numbers.

Abstract: We survey permutation-based methods for approximate k-nearest neighbor search. In these methods, every data point is represented by a ranked list of pivots sorted by the distance to this point. Such ranked lists are called permutations. The underpinning assumption is that, for both metric and non-metric spaces, the distance between permutations is a good proxy for the distance between original points. Thus, it should be possible to efficiently retrieve most true nearest neighbors by examining only a tiny subset of data points whose permutations are similar to the permutation of a query. We further test this assumption by carrying out an extensive experimental evaluation where permutation methods are pitted against state-of-the art benchmarks (the multi-probe LSH, the VP-tree, and proximity-graph based retrieval) on a variety of realistically large data set from the image and textual domain. The focus is on the high-accuracy retrieval methods for generic spaces. Additionally, we assume that both data and indices are stored in main memory. We find permutation methods to be reasonably efficient and describe a setup where these methods are most useful. To ease reproducibility, we make our software and data sets publicly available.

Abstract: The rank modulation scheme has been proposed for efficient writing and storing data in nonvolatile memory storage. Error correction in the rank modulation scheme is done by considering permutation codes. In this paper, we consider codes in the set of all permutations on $n$ elements, $S_{n}$ , using the Kendall $\tau $ -metric. The main goal of this paper is to derive new bounds on the size of such codes. For this purpose, we also consider perfect codes, diameter perfect codes, and the size of optimal anticodes in the Kendall $\tau $ -metric, structures which have their own considerable interest. We prove that there are no perfect single-error-correcting codes in $S_{n}$ , where $n>4$ is a prime or $4\leq n\leq 10$ . We present lower bounds on the size of optimal anticodes with odd diameter. As a consequence, we obtain a new upper bound on the size of codes in $S_{n}$ with even minimum Kendall $\tau $ -distance. We present larger single-error-correcting codes than the known ones in $S_{5}$ and $S_{7}$ .

Abstract: This article discusses problems with and solutions to performing valid permutation tests for quantitative trait loci in the presence of polygenic effects. Although permutation testing is a popular approach for determining statistical significance of a test statistic with an unknown distribution--for instance, the maximum of multiple correlated statistics or some omnibus test statistic for a gene, gene-set, or pathway--naive application of permutations may result in an invalid test. The risk of performing an invalid permutation test is particularly acute in complex trait mapping where polygenicity may combine with a structured population resulting from the presence of families, cryptic relatedness, admixture, or population stratification. I give both analytical derivations and a conceptual understanding of why typical permutation procedures fail and suggest an alternative permutation-based algorithm, MVNpermute, that succeeds. In particular, I examine the case where a linear mixed model is used to analyze a quantitative trait and show that both phenotype and genotype permutations may result in an invalid permutation test. I provide a formula that predicts the amount of inflation of the type 1 error rate depending on the degree of misspecification of the covariance structure of the polygenic effect and the heritability of the trait. I validate this formula by doing simulations, showing that the permutation distribution matches the theoretical expectation, and that my suggested permutation-based test obtains the correct null distribution. Finally, I discuss situations where naive permutations of the phenotype or genotype are valid and the applicability of the results to other test statistics.

Abstract: Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than κ ≈ 2.20557 (a specific algebraic integer at which infinite antichains first appear). Using language- and order-theoretic methods, we prove that the substitution closures of geometric grid classes are well partially ordered, finitely based, and that all their subclasses have algebraic generating functions. We go on to show that the inflation of a geometric grid class by a strongly rational class is well partially ordered, and that all its subclasses have rational generating functions. This latter fact allows us to conclude that every permutation class with growth rate less than κ has a rational generating function. This bound is tight as there are permutation classes with growth rate κ which have nonrational generating functions.

Abstract: We prove the existence of an algorithm $A$ for computing 2-d or 3-d convex hulls that is optimal for every point set in the following sense: for every sequence $\sigma$ of $n$ points and for every algorithm $A'$ in a certain class $\mathcal{A}$, the running time of $A$ on input $\sigma$ is at most a constant factor times the maximum running time of $A'$ on the worst possible permutation of $\sigma$ for $A'$. We establish a stronger property: for every sequence $\sigma$ of points and every algorithm $A'$, the running time of $A$ on $\sigma$ is at most a constant factor times the average running time of $A'$ over all permutations of $\sigma$. We call algorithms satisfying these properties instance-optimal in the order-oblivious and random-order setting. Such instance-optimal algorithms simultaneously subsume output-sensitive algorithms and distribution-dependent average-case algorithms, and all algorithms that do not take advantage of the order of the input or that assume the input is given in a random order. The class $\mathcal{A}$ under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments. To establish an instance-specific lower bound, we deviate from traditional Ben-Or-style proofs and adopt a new adversary argument. For 2-d convex hulls, we prove that a version of the well known algorithm by Kirkpatrick and Seidel (1986) or Chan, Snoeyink, and Yap (1995) already attains this lower bound. For 3-d convex hulls, we propose a new algorithm. We further obtain instance-optimal results for a few other standard problems in computational geometry. Our framework also reveals connection to distribution-sensitive data structures and yields new results as a byproduct, for example, on on-line orthogonal range searching in 2-d and on-line halfspace range reporting in 2-d and 3-d.

Abstract: Sliding Token is a natural reconfiguration problem in which vertices of independent sets are iteratively replaced by neighbors. We develop techniques that may be useful in answering the conjecture that Sliding Token is polynomial-time decidable on bipartite graphs. Along the way, we give efficient algorithms for Sliding Token on bipartite permutation and bipartite distance-hereditary graphs.

Abstract: We observe that for a and b relatively prime, the "abacus construction" identifies the set of simultaneous (a,b)-core partitions with lattice points in a rational simplex. Furthermore, many statistics on (a,b)-cores are piecewise polynomial functions on this simplex. We apply these results to rational Catalan combinatorics. Using Ehrhart theory, we reprove Anderson's theorem that there are (a+b-1)!/a!b! simultaneous (a,b)-cores, and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an (a,b)-core is (a+b+1)(a-1)(b-1)/24. Our methods also give new derivations of analogous formulas for the number and average size of self-conjugate (a,b)-cores. We conjecture a unimodality result for q rational Catalan numbers, and make preliminary investigations in applying these methods to the (q,t)-symmetry and specialization conjectures. We prove these conjectures for low degree terms and when a=3, connecting them to the Catalan hyperplane arrangement and quadratic permutation statistics.

Abstract: For a number of problems in the theory of online algorithms, it is known that the assumption that elements arrive in uniformly random order enables the design of algorithms with much better performance guarantees than under worst-case assumptions. The quintessential example of this phenomenon is the secretary problem, in which an algorithm attempts to stop a sequence at the moment it observes the maximum value in the sequence. As is well known, if the sequence is presented in uniformly random order there is an algorithm that succeeds with probability 1/e, whereas no non-trivial performance guarantee is possible if the elements arrive in worst-case order.In many of the applications of online algorithms, it is reasonable to assume there is some randomness in the input sequence, but unreasonable to assume that the arrival ordering is uniformly random. This work initiates an investigation into relaxations of the random-ordering hypothesis in online algorithms, by focusing on the secretary problem and asking what performance guarantees one can prove under relaxed assumptions. Toward this end, we present two sets of properties of distributions over permutations as sufficient conditions, called the (p,q,δ)-block-independence property} and (k,δ)-uniform-induced-ordering property}. We show these two are asymptotically equivalent by borrowing some techniques from the celebrated approximation theory. Moreover, we show they both imply the existence of secretary algorithms with constant probability of correct selection, approaching the optimal constant 1/e as the related parameters of the property tend towards their extreme values. Both of these properties are significantly weaker than the usual assumption of uniform randomness; we substantiate this by providing several constructions of distributions that satisfy (p,q,δ)-block-independence. As one application of our investigation, we prove that Θ(log log n) is the minimum entropy of any permutation distribution that permits constant probability of correct selection in the secretary problem with $n$ elements. While our block-independence condition is sufficient for constant probability of correct selection, it is not necessary; however, we present complexity-theoretic evidence that no simple necessary and sufficient criterion exists. Finally, we explore the extent to which the performance guarantees of other algorithms are preserved when one relaxes the uniform random ordering assumption to (p,q,δ)-block-independence, obtaining a negative result for the weighted bipartite matching algorithm of Korula and Pal.

Abstract: Consider the process of random transpositions on the complete graph Kn. We use representation theory to give an exact, simple formula for the expected number of cycles of size k at time t, in terms of an incomplete Beta function. Using this we show that the expected number of cycles of size k jumps from 0 to its equilibrium value, 1/k, at the time where the giant component of the associated random graph first exceeds k. Consequently we deduce a new and simple proof of Schramm’s theorem on random transpositions, that giant cycles emerge at the same time as the giant component in the random graph. We also calculate the “window” for this transition and find that it is quite thin. Finally, we give a new proof of a result by the first author and Durrett that the random transposition process exhibits a certain slowdown transition. The proof makes use of a recent formula for the character decomposition of the number of cycles of a given size in a permutation, and the Frobenius formula for the character ratios.

Abstract: Recent advances in block-cipher theory deliver security analyses in models where one or more underlying components e.g., a function or a permutation are ideal i.e., randomly chosen. This paper addresses the question of finding new constructions achieving the highest possible security level under minimal assumptions in such ideal models. We present a new block-cipher construction, derived from the Swap-or-Not construction by Hoang et al. CRYPTO '12. With n-bit block length, our construction is a secure pseudorandom permutation PRP against attackers making $$2^{n - O\log n}$$ block-cipher queries, and $$2^{n - O1}$$ queries to the underlying component which has itself domain size roughly n. This security level is nearly optimal. So far, only key-alternating ciphers have been known to achieve comparable security using On independent random permutations. In contrast, we only use a singlefunction or permutation, and still achieve similar efficiency. Our second contribution is a generic method to enhance a block cipher, initially only secure as a PRP, to additionally withstand related-key attacks without substantial loss in terms of concrete security.

Abstract: We propose a genetic algorithm GA to search for plateaued boolean functions, which represent suitable candidates for the design of stream ciphers due to their good cryptographic properties. Using the spectral inversion technique introduced by Clark, Jacob, Maitra and Stanica, our GA encodes the chromosome of a candidate solution as a permutation of a three-valued Walsh spectrum. Additionally, we design specialized crossover and mutation operators so that the swapped positions in the offspring chromosomes correspond to different values in the resulting Walsh spectra. Some tests performed on the set of pseudoboolean functions of $$n=6$$ and $$n=7$$ variables show that in the former case our GA outperforms Clark et al.'s simulated annealing algorithm with respect to the ratio of generated plateaued boolean functions per number of optimization runs.

Abstract: We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least \begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*} Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromatic K4 is minimized. We show that all the extremal colourings must contain monochromatic K4 only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

Abstract: We present a novel algorithm, Westfall-Young light, for detecting patterns, such as itemsets and subgraphs, which are statistically significantly enriched in one of two classes. Our method corrects rigorously for multiple hypothesis testing and correlations between patterns through the Westfall-Young permutation procedure, which empirically estimates the null distribution of pattern frequencies in each class via permutations. In our experiments, Westfall-Young light dramatically outperforms the current state-of-the-art approach in terms of both runtime and memory efficiency on popular real-world benchmark datasets for pattern mining. The key to this efficiency is that unlike all existing methods, our algorithm neither needs to solve the underlying frequent itemset mining problem anew for each permutation nor needs to store the occurrence list of all frequent patterns. Westfall-Young light opens the door to significant pattern mining on large datasets that previously led to prohibitive runtime or memory costs.

Abstract: Using the recently developed notion of permutation limits this paper derives the limiting distribution of the number of fixed points and cycle structure for any convergent sequence of random permutations, under mild regularity conditions. In particular this covers random permutations generated from Mallows Model with Kendall's Tau, $\mu$ random permutations introduced in [11], as well as a class of exponential families introduced in [15].

Abstract: In this paper we develop two permutation theorems on argument increasing functions of a multivariate random vector and a real parameter vector. We use the unified approach of our two theorems to provide some important theoretical results on the capital allocation in actuarial science, the deductible and upper limit allocations in insurance policy, and portfolio allocation in financial engineering. Our results successfully improve or extend the corresponding works in the literature.