Abstract: Entropy is a powerful tool for the analysis of time series, as it allows describing the probability distributions of the possible state of a system, and therefore the information encoded in it. Nevertheless, important information may be codified also in the temporal dynamics, an aspect which is not usually taken into account. The idea of calculating entropy based on permutation patterns (that is, permutations defined by the order relations among values of a time series) has received a lot of attention in the last years, especially for the understanding of complex and chaotic systems. Permutation entropy directly accounts for the temporal information contained in the time series; furthermore, it has the quality of simplicity, robustness and very low computational cost. To celebrate the tenth anniversary of the original work, here we analyze the theoretical foundations of the permutation entropy, as well as the main recent applications to the analysis of economical markets and to the understanding of biomedical systems.

Abstract: We consider the online stochastic matching problem proposed by Feldman et al. [Feldman J, Mehta A, Mirrokni VS, Muthukrishnan S (2009) Online stochastic matching: Beating 1-1/e. Annual IEEE Sympos. Foundations Comput. Sci. 117--126] as a model of display ad allocation. We are given a bipartite graph; one side of the graph corresponds to a fixed set of bins, and the other side represents the set of possible ball types. At each time step, a ball is sampled independently from the given distribution and it needs to be matched upon its arrival to an empty bin. The goal is to maximize the number of allocations. We present an online algorithm for this problem with a competitive ratio of 0.702. Before our result, algorithms with a competitive ratio better than 1-1/e were known under the assumption that the expected number of arriving balls of each type is integral. A key idea of the algorithm is to collect statistics about the decisions of the optimum offline solution using Monte Carlo sampling and use those statistics to guide the decisions of the online algorithm. We also show that our algorithm achieves a competitive ratio of 0.705 when the rates are integral. On the hardness side, we prove that no online algorithm can have a competitive ratio better than 0.823 under the known distribution model (and henceforth under the permutation model). This improves upon the 5/6 hardness result proved by Goel and Mehta [Goel G, Mehta A (2008) Online budgeted matching in random input models with applications to adwords. ACM-SIAM Symposium Discrete Algorithms 982--991] for the permutation model.

Abstract: The fourth-corner problem entails estimation and statistical testing of the relationship between species traits and environmental variables from the analysis of three data tables. In a 2008 paper, S. Dray and P. Legendre proposed and evaluated five permutation methods for statistical significance testing, including a new two-step testing procedure. However, none of these attained the correct type I error in all cases of interest. We solve this problem by showing that a small modification of their two-step procedure controls the type I error in all cases. The modification consists of adjusting the significance level from to α or, equivalently, of reporting the maximum of the individual P values as the final one. The test is also applicable to the three-table ordination method RLQ.

Abstract: An efficient image encryption algorithm using the generalized Arnold map is proposed. The algorithm is composed of two stages, i.e., permutation and diffusion. First, a total circular function, rather than the traditional periodic position permutation, is used in the permutation stage. It can substantially reduce the correlation between adjacent pixels. Then, in the stage of diffusion, double diffusion functions, i.e., positive and opposite module, are utilized with a novel generation of the keystream. As the keystream depends on the processed image, the proposed method can resist known- and chosen-plaintext attacks. Experimental results and theoretical analysis indicate the effectiveness of our method. An extension of the proposed algorithm to other chaotic systems is also discussed.

Abstract: Estimation of distribution algorithms (EDAs) are a set of algorithms that belong to the field of Evolutionary Computation. Characterized by the use of probabilistic models to represent the solutions and the dependencies between the variables of the problem, these algorithms have been applied to a wide set of academic and real-world optimization problems, achieving competitive results in most scenarios. Nevertheless, there are some optimization problems, whose solutions can be naturally represented as permutations, for which EDAs have not been extensively developed. Although some work has been carried out in this direction, most of the approaches are adaptations of EDAs designed for problems based on integer or real domains, and only a few algorithms have been specifically designed to deal with permutation-based problems. In order to set the basis for a development of EDAs in permutation-based problems similar to that which occurred in other optimization fields (integer and real-value problems), in this paper we carry out a thorough review of state-of-the-art EDAs applied to permutation-based problems. Furthermore, we provide some ideas on probabilistic modeling over permutation spaces that could inspire the researchers of EDAs to design new approaches for these kinds of problems.

Abstract: This paper considers--for the first time--the concept of key-alternating ciphers in a provable security setting. Key-alternating ciphers can be seen as a generalization of a construction proposed by Even and Mansour in 1991. This construction builds a block cipher PX from an n-bit permutation P and two n-bit keys k0 and k1, setting PX{k0,k1} (x) = k1 ⊕ P(x ⊕ k0). Here we consider a (natural) extension of the Even-Mansour construction with t permutations P1,…,Pt and t+1 keys, k0,…, kt. We demonstrate in a formal model that such a cipher is secure in the sense that an attacker needs to make at least 22n/3 queries to the underlying permutations to be able to distinguish the construction from random. We argue further that the bound is tight for t=2 but there is a gap in the bounds for t>2, which is left as an open and interesting problem. Additionally, in terms of statistical attacks, we show that the distribution of Fourier coefficients for the cipher over all keys is close to ideal. Lastly, we define a practical instance of the construction with t=2 using AES referred to as AES2. Any attack on AES2 with complexity below 285 will have to make use of AES with a fixed known key in a non-black box manner. However, we conjecture its security is 2128.

Abstract: We analyze the security of the iterated Even-Mansour cipher (a.k.a. key-alternating cipher), a very simple and natural construction of a blockcipher in the random permutation model. This construction, first considered by Even and Mansour (J. Cryptology, 1997) with a single permutation, was recently generalized to use t permutations in the work of Bogdanov et al. (EUROCRYPT 2012). They proved that the construction is secure up to $ \mathcal{O} (N^{2/3})$ queries (where N is the domain size of the permutations), as soon as the number t of rounds is 2 or more. This is tight for t=2, however in the general case the best known attack requires Ω(Nt/(t+1)) queries. In this paper, we give asymptotically tight security proofs for two types of adversaries: 1 for non-adaptive chosen-plaintext adversaries, we prove that the construction achieves an optimal security bound of $ \mathcal{O} (N^{t/(t+1)})$ queries; 2 for adaptive chosen-plaintext and ciphertext adversaries, we prove that the construction achieves security up to $ \mathcal{O} (N^{t/(t+2)})$ queries (for t even). This improves previous results for t≥6. Our proof crucially relies on the use of a coupling to upper-bound the statistical distance of the outputs of the iterated Even-Mansour cipher to the uniform distribution.

Abstract: Recently, a colour image encryption algorithm based on chaos was proposed by cascading two position permutation operations and one substitution operation, which are all determined by some pseudo-random number sequences generated by iterating the Logistic map. This paper evaluates the security level of the encryption algorithm and finds that the position permutation-only part and the substitution part can be separately broken with only $\lceil (\log_2(3MN))/8 \rceil$ and 2 chosen plain-images, respectively, where $MN$ is the size of the plain-image. Concise theoretical analyses are provided to support the chosen-plaintext attack, which are verified by experimental results also.

Abstract: This paper considers—for the first time—the concept of key- alternating ciphers in a provable security setting. Key-alternating ciphers can be seen as a generalization of a construction proposed by Even and Mansour in 1991. This construction builds a block cipher PX from an n-bit permutation P and two n-bit keys k0 and k1, setting PXk0,k1 (x )= k1 ⊕ P (x ⊕ k0). Here we consider a (natural) extension of the Even- Mansour construction with t permutations P1,...,Pt and t +1 keys, k0,...,kt. We demonstrate in a formal model that such a cipher is secure in the sense that an attacker needs to make at least 2 2n/3 queries to the underlying permutations to be able to distinguish the construction from random. We argue further that the bound is tight for t = 2 but there is a gap in the bounds for t> 2, which is left as an open and interesting problem. Additionally, in terms of statistical attacks, we show that the distribution of Fourier coefficients for the cipher over all keys is close to ideal. Lastly, we define a practical instance of the construction with t =2 using AES referred to as AES 2 . Any attack on AES 2 with complexity

Abstract: Multivariate distance matrix regression (MDMR) analysis is a statistical technique that allows researchers to relate P variables to an additional M factors collected on N individuals, where P>>N. The technique can be applied to a number of research settings involving high dimensional data types such as DNA sequence data, gene expression microarray data and imaging data. MDMR analysis involves computing the distance between all pairs of individuals with respect to P variables of interest and constructing an N x N matrix whose elements reflect these distances. Permutation tests can be used to test linear hypotheses that consider whether or not the M additional factors collected on the individuals can explain variation in the observed distances between and among the N individuals as reflected in the matrix. MDMR analysis is an excellent complement to cluster analysis and other traditional multivariate analysis techniques. Despite its appeal and utility, properties of the statistics used in MDMR analysis have not been explored in detail. In this paper we consider the level accuracy and power of MDMR analysis assuming different distance measures and analysis settings. We also describe the utility of MDMR analysis in assessing hypotheses about the appropriate number of clusters arising from a cluster analysis.

Abstract: This paper considers—for the first time—the concept of keyalternating ciphers in a provable security setting. Key-alternating ciphers can be seen as a generalization of a construction proposed by Even and Mansour in 1991. This construction builds a block cipher PX from an n-bit permutation P and two n-bit keys k0 and k1, setting PXk0,k1(x) = k1 ⊕ P (x ⊕ k0). Here we consider a (natural) extension of the EvenMansour construction with t permutations P1, . . . , Pt and t + 1 keys, k0, . . . , kt. We demonstrate in a formal model that such a cipher is secure in the sense that an attacker needs to make at least 2 queries to the underlying permutations to be able to distinguish the construction from random. We argue further that the bound is tight for t = 2 but there is a gap in the bounds for t > 2, which is left as an open and interesting problem. Additionally, in terms of statistical attacks, we show that the distribution of Fourier coefficients for the cipher over all keys is close to ideal. Lastly, we define a practical instance of the construction with t = 2 using AES referred to as AES. Any attack on AES with complexity below 2 will have to make use of AES with a fixed known key in a non-black box manner. However, we conjecture its security is 2.

Abstract: Many case-control tests of rare variation are implemented in statistical frameworks that make correction for confounders like population stratification difficult. Simple permutation of disease status is unacceptable for resolving this issue because the replicate data sets do not have the same confounding as the original data set. These limitations make it difficult to apply rare-variant tests to samples in which confounding most likely exists, e.g., samples collected from admixed populations. To enable the use of such rare-variant methods in structured samples, as well as to facilitate permutation tests for any situation in which case-control tests require adjustment for confounding covariates, we propose to establish the significance of a rare-variant test via a modified permutation procedure. Our procedure uses Fisher's noncentral hypergeometric distribution to generate permuted data sets with the same structure present in the actual data set such that inference is valid in the presence of confounding factors. We use simulated sequence data based on coalescent models to show that our permutation strategy corrects for confounding due to population stratification that, if ignored, would otherwise inflate the size of a rare-variant test. We further illustrate the approach by using sequence data from the Dallas Heart Study of energy metabolism traits. Researchers can implement our permutation approach by using the R package BiasedUrn.

Abstract: In this paper, we present a method for the construction of 8x8 substitution boxes used in the area of cryptography. A symmetric group permutation S"8 is applied on Galois field elements that originally belong to GF(2^8), and as a consequence, 40320 new substitution boxes are synthesized. The Liu J substitution box is used as a seed in the creation process of the new algebraically complex nonlinear components. The core design of this new algorithm relies on the symmetric group permutation operation which is embedded in the algebraic structure of the new substitution box. We study the characteristics of the newly created substitution boxes and highlight the improved performance parameters and their usefulness in practical applications. In particular, we focus on the nonlinear properties and the behavior of input/output bits and determine the suitability of a particular substitution box for a specific type of encryption application. A comparison with some of the prevailing and popular substitution boxes is presented.

Abstract: Compared to binary low-density parity-check (LDPC) codes, nonbinary LDPC codes have better error performance when the code length is moderate. This paper presents an efficient layered decoder architecture for nonbinary quasi-cyclic (QC) LDPC codes using the proposed barrel-shifter-based permutation network and minimum value filter which is used to determine the first few smallest values from a given set. Through the permutation network, the decoding operations related to the multiplications over finite fields can be efficiently handled in the check-node operations, which simplifies the permutations in the variable-node operations and, hence, enables the layered decoder to be realized efficiently. In order to increase the throughput, we utilize the proposed permutation network and the minimum value filter to devise a selective-input min-max decoder architecture. Using a 90-nm CMOS process, we implemented three nonbinary decoders to demonstrate the proposed techniques.

Abstract: Keccak is one of the five hash functions selected for the final round of the SHA-3 competition, and its inner primitive is a permutation called Keccak-f. In this paper, we observe that for the inverse of the only nonlinear transformation in Keccak-f, the algebraic degree of any output coordinate and the one of the product of any two output coordinates are both 3, which is 2 less than its size of 5. Combining this observation with a proposition on the upper bound of the degree of iterated permutations, we improve the zero-sum distinguisher for the Keccak-f permutation with full 24 rounds by lowering the size of the zero-sum partition from 21590 to 21575.

Abstract: A method for encoding data bits includes computing checksum parity bits based on the data bits. A set of equations satisfied by the data bits and the checksum parity bits corresponds to a dense parity-check matrix. The dense parity-check matrix comprises sums of permutation sub-matrices.

Abstract: This paper considers—for the first time—the concept of key- alternating ciphers in a provable security setting. Key-alternating ciphers can be seen as a generalization of a construction proposed by Even and Mansour in 1991. This construction builds a block cipher PX from an n-bit permutation P and two n-bit keys k0 and k1, setting PXk0,k1 (x )= k1 ⊕ P (x ⊕ k0). Here we consider a (natural) extension of the Even- Mansour construction with t permutations P1,...,Pt and t +1 keys, k0,...,kt. We demonstrate in a formal model that such a cipher is secure in the sense that an attacker needs to make at least 2 2n/3 queries to the underlying permutations to be able to distinguish the construction from random. We argue further that the bound is tight for t = 2 but there is a gap in the bounds for t> 2, which is left as an open and interesting problem. Additionally, in terms of statistical attacks, we show that the distribution of Fourier coefficients for the cipher over all keys is close to ideal. Lastly, we define a practical instance of the construction with t =2 using AES referred to as AES 2 . Any attack on AES 2 with complexity

Abstract: A bit interleaving method involves applying a bit permutation process to a QC LDPC codeword made up of N cyclic blocks each including Q bits, and dividing the codeword, after the bit permutation process, into a plurality of constellation words each imade up of M bits, the codeword being divided into N/M sections, each constellation word being associated with one of the N/M sections, and the bit permutation process being performed such that each of the constellation words includes one bit from each of M different cyclic blocks associated with a given section.

Abstract: Zero-difference balanced functions introduced recently are an interesting subject of study, as they unify difference sets, permutation polynomials, perfect nonlinear functions, planar functions, and semifields, and have applications in combinatorics, coding theory, cryptography, and finite geometry. In this article, we give a well-rounded treatment of zero-difference balanced functions. We survey known zero-difference balanced functions, construct new ones, and summarize some of their applications.

Abstract: Each length $k$ pattern occurs equally often in the set $S_n$ of all permutations of length $n$, but the same is not true in general for a proper subset of $S_n$. Miklos Bona recently proved that if we consider the set of $n$-permutations avoiding the pattern 132, all other non-monotone patterns of length 3 are equally common. In this paper we focus on the set $\operatorname{Av}_n (123)$ of $n$-permutations avoiding $123$, and give exact formulae for the occurrences of each length 3 pattern. While this set does not have the same symmetries as $\operatorname{Av}_n (132)$, we find several similarities between the two and prove that the number of 231 patterns is the same in each.

Abstract: We use cyclotomy to design new classes of permutation polynomials over finite fields. This allows us to generate many classes of permutation polynomials in an algorithmic way. Many of them are permutation polynomials of large indices.

Abstract: This paper introduces a propositional encoding for recursive path orders (RPO), in connection with dependency pairs. Hence, we capture in a uniform setting all common instances of RPO, i.e., lexicographic path orders (LPO), multiset path orders (MPO), and lexicographic path orders with status (LPOS). This facilitates the application of SAT solvers for termination analysis of term rewrite systems (TRSs). We address four main inter-related issues and show how to encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (A) the lexicographic comparison w.r.t. a permutation of the arguments; (B) the multiset extension of a base order; (C) the combined search for a path order together with an argument filter to orient a set of inequalities; and (D) how the choice of the argument filter influences the set of inequalities that have to be oriented (so-called usable rules). We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power.