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ABSTRACT Analysis of variance and linear models is undoubtedly one of the most useful statistical contributions to experimental and observational science. With the ability to characterize a system through multivariate responses, these methods have emerged to be general tools regardless of response dimensionality. Contemporary methods for establishing statistical inference, such as ANOVA simultaneous component analysis (ASCA), are based on Monte Carlo sampling; however, a flat uniform resampling scheme may violate the structure of the uncertainty for unbalanced designs as well as for observational data. In this work, we provide permutation strategies for inferential testing for unbalanced designs including interaction models and establish nonuniform randomization based on the concept of propensity score matching. Lastly, we provide a general method for modelling continuous covariates based on kernel smoothers. All methods are characterized on their ability to provide unbiased Type I error results. [ABSTRACT FROM AUTHOR]

The security of biometric data in biometric‐based authentication systems is a significant concern. Cancellable biometrics aim to generate templates that can be replaced by new templates if compromised. We propose a new approach for generating cancellable biometric templates based on linear regression with random permutation. Our approach generates a virtual image for every biometric image by applying linear regression. In the next step, the cancellable biometric template is produced by randomly permuting each virtual image depending on a key assigned to each individual. If the template is compromised, it can be cancelled, and a new template can be generated by altering the key. Our method has shown superior performance compared to existing random permutation‐based methods in terms of authentication accuracy across six databases, encompassing face, iris, and ear, even when dealing with low‐resolution images. It performed well on challenging databases like UBIRIS and Georgia Tech, demonstrating the robustness of the proposed approach. [ABSTRACT FROM AUTHOR]

A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing trajectories. When normalized the latter is just a random distribution function supported by the unit interval. We establish that its expectations under various weighted probability measures defined on the subsets of Sn are quasihypergeometric distribution functions. Their limits as n → ∞ cover the class of two-parameter beta distributions. It is shown that, under appropriate conditions, the convergence rate is of the negative power of n order. That opens a new possibility to model the beta distributions using divisors of permutations. [ABSTRACT FROM AUTHOR]

With the development of modern data collection techniques, researchers often encounter high-dimensional data across various research fields. An important problem is to determine whether several groups of these high-dimensional data originate from the same population. To address this, this paper presents a novel k-sample test for equal distributions for high-dimensional data, utilizing the Maximum Mean Discrepancy (MMD). The test statistic is constructed using a V-statistic-based estimator of the squared MMD derived for several samples. The asymptotic null and alternative distributions of the test statistic are derived. To approximate the null distribution accurately, three simple methods are described. To evaluate the performance of the proposed test, two simulation studies and a real data example are presented, demonstrating the effectiveness and reliability of the test in practical applications. [ABSTRACT FROM AUTHOR]

How many random transpositions (meaning that we swap given pairs of elements with given probabilities independently) are needed to ensure that each element of [n] is uniformly distributed--in the sense that the probability that i is mapped to j is 1/n for all i and j And what if we insist that each pair is uniformly distributed In this paper we show that the minimum for the first problem is about 1 2nlog2 n, with this being exact when n is a power of 2. For the second problem, we show that, rather surprisingly, the answer is not quadratic: O(nlog2 n) random transpositions suffice. We also show that if we ask only that the pair (1, 2) is uniformly distributed, then the answer is 2n 3. This proves a conjecture of Groenland, Johnston, Radcliffe, and Scott. [ABSTRACT FROM AUTHOR]

A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing trajectories. When normalized the latter is just a random distribution function supported by the unit interval. We establish that its expectations under various weighted probability measures defined on the subsets of Sn are quasihypergeometric distribution functions. Their limits as n -> 1 cover the class of two-parameter beta distributions. It is shown that, under appropriate conditions, the convergence rate is of the negative power of n order.

We solve the secretary problem in the case that the ranked items arrive in a statistically biased order rather than in uniformly random order. The bias is given by the left-to-right minimum exponentially tilted distribution with parameter q ∈ (0, ∞). That is, for σ ∈ Sn, the probability of σ is proportional to q LR− n (σ), where the left-to-right minimum statistic LR− n is defined by LR− n (σ) = |{j ∈ [n] : σj = min{σi : 1 ≤ i ≤ j}}|, σ ∈ Sn. For q ∈ (0, 1), higher ranked items tend to arrive earlier than in the case of the uniform distribution, and for q ∈ (1, ∞), they tend to arrive later, where the highest ranked item is denoted by 1 and the lowest ranked item is denoted by n. In the classical problem, the asymptotically optimal strategy is to reject the first M∗ n items, where M∗ n ∼ n e, and then to select the first item ranked higher than any of the first M∗ n items (if such an item exists). This yields e −1 as the limiting probability of success. With the above bias on arrivals, and for the parameter q = qn depending on n, we calculate the asymptotic behavior of the optimal strategy M∗ n and the corresponding limiting probability of success, for all regimes of {qn}∞ n=1. In particular, if the leading order asymptotic behavior of {qn}∞ n=1 is at least 1 log n, and if also its order is no more than o(n), then the limiting probability of success when using an asymptotically optimal strategy is e −1; otherwise, this limiting probability of success is greater than e −1. Also, the limiting fraction of numbers, limn→∞ M∗ n n, that are summarily rejected by an asymptotically optimal strategy lies in (0, 1) if and only if limn→∞ qn ∈ (0, ∞). [ABSTRACT FROM AUTHOR]

Using the language of runs and patterns, a peak in a sequence of integers can be interpreted as observing a fall (or descent) immediately after a rise (or ascent). In this paper, we obtain the exact distribution of the number of peaks in a permutation by using the nonhomogeneous finite Markov chain imbedding technique and an insertion procedure. As a byproduct, we also obtain the Euler numbers, which are a sequence of the number of alternating permutations. The method is extended to obtaining the joint distribution of the number of peaks and the number of falls. Several numerical examples are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

With the development of modern data collection techniques, researchers often encounter high-dimensional data across various research fields. An important problem is to determine whether several groups of these high-dimensional data originate from the same population. To address this, this paper presents a novel k-sample test for equal distributions for high-dimensional data, utilizing the Maximum Mean Discrepancy (MMD). The test statistic is constructed using a V-statistic-based estimator of the squared MMD derived for several samples. The asymptotic null and alternative distributions of the test statistic are derived. To approximate the null distribution accurately, three simple methods are described. To evaluate the performance of the proposed test, two simulation studies and a real data example are presented, demonstrating the effectiveness and reliability of the test in practical applications.

Let πn be a uniformly chosen random permutation on [n]. Using an analysis of the probability that two overlapping consecutive k-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns of all lengths k ∈ {1, 2,..., n} in πn is n 2 2 (1 - o (1)) as n → ∞. This exhibits the fact that random permutations pack consecutive patterns near-perfectly. [ABSTRACT FROM AUTHOR]

As one of the most successful industrial realizations of Internet of Things, a smart grid is a smart IoT system that deploys widespread smart meters to capture fine-grained data on residential power usage. Unfortunately, it always suffers diverse privacy attacks, which seriously increases the risk of violating the privacy of customers. Although some solutions have been proposed to address this privacy issue, most of them mainly rely on a trusted party and focus on the sanitization of metering masurements. Moreover, these solutions are vulnerable to advanced attacks. In this paper, we propose a decentralized mechanism for privacy-preserving computation in smart grid called DDP, which leaverages the differential privacy and extends the data sanitization from the value domain to the time domain. Specifically, we inject Laplace noise to the measurements at the end of each customer in a distributed manner, and then use a random permutation algorithm to shuffle the power measurement sequence, thereby enforcing differential privacy after aggregation and preventing the sensitive power usage mode informaton of the customers from being inferred by other parties. Extensive experiments demonstrate that DDP shows an outstanding performance in terms of privacy from the non-intrusive load monitoring (NILM) attacks and utility by using two different error analysis. [ABSTRACT FROM AUTHOR]

We develop a new statistical model to analyse time‐varying ranking data. The model can be used with a large number of ranked items, accommodates exogenous time‐varying covariates and partial rankings, and is estimated via the maximum likelihood in a straightforward manner. Rankings are modelled using the Plackett–Luce distribution with time‐varying worth parameters that follow a mean‐reverting time series process. To capture the dependence of the worth parameters on past rankings, we utilise the conditional score in the fashion of the generalised autoregressive score models. Simulation experiments show that the small‐sample properties of the maximum‐likelihood estimator improve rapidly with the length of the time series and suggest that statistical inference relying on conventional Hessian‐based standard errors is usable even for medium‐sized samples. In an empirical study, we apply the model to the results of the Ice Hockey World Championships. We also discuss applications to rankings based on underlying indices, repeated surveys and non‐parametric efficiency analysis. [ABSTRACT FROM AUTHOR]

Hiding countermeasure is among the best-known secure implementation techniques designed to counteract side-channel attacks. It uses a permutation algorithm to shuffle data. In today's Post-Quantum Cryptography (PQC), hiding countermeasure has earned the limelight for its "shufflability" in lattice-based, and code-based, cryptographic algorithms. In this narrative, most importantly, as a rule, fast generation of permutations is paramount to both efficacy and security of an algorithm. The Fisher–Yates (FY) shuffling method has long been a popular choice for this purpose: the FY method generates randomly shuffled (finite) indices. However, despite its theoretical verity, with the FY method we anticipate the following risks of misuse, which can lead to biased shuffling sequences: (i) incorrect implementation, (ii) poor random source, and (iii) the chosen random number being too small. In this paper, we introduce a new statistical test called "approximate permutation criterion" ("APC"). We use it to examine some known cases of misused FY shuffling (i–iii). APC takes into consideration the fact that the super-exponential rate of growth of the factorial function N ! , which represents the number of permutations of N indices, defies any meaningful form of statistical tests. With APC one can verify whether the output permutations are biased or not with much lower testing cost. Mathematically, in this paper we introduce the so-called " k th order permutation verification", the underpinning notion upon which APC is based. We also compare APC with full sample space to demonstrate how well it encapsulates the statistical randomness of random permutations. We thereby provide a new method that identifies a bias that exists in the output permutations when implementing FY Shuffling through a visual ratio test and the chi-square (χ 2) distribution test. [Display omitted] [ABSTRACT FROM AUTHOR]

In recent years, reversible data hiding (RDH) in encrypted images got much attention due to its wide applications in the areas such as cloud computing, military image transmission, medical image transmission, etc. This paper introduces a new solution for reversible data hiding in encrypted images. One of the main challenges while designing a reversible data hiding scheme in an encrypted image is the embedding rate and bit error rate during image recovery. The scheme proposed in this manuscript ensures a good embedding rate and the lossless recovery of the original image. The key idea behind the proposed technique is that the encrypted image will be partitioned into non-overlapping blocks, and the pixels in each block will be categorized into white pixels and black pixels based on a predefined pattern. The black pixels will be mapped into a new pixel value based on the two bits from the secret message that is to be embedded into the selected image block. For mapping purposes, we generate four different random permutations of all the possible gray-scale values (0 to 255). At the receiver side, corresponding to each block in the image we have to generate four different weighted mesh graphs. The image recovery and data extraction are carried out by analyzing the total edge weight of these mesh graphs. The results obtained from the experimental study are much better while comparing with a few of the well-known recently introduced reversible data hiding schemes in encrypted images. [ABSTRACT FROM AUTHOR]

It remains challenging how to find existing but undiscovered genome sequence mutations or predict potential genome sequence mutations based on real sequence data. Motivated by this, we develop approaches to detect new, undiscovered genome sequences. Because discovering new genome sequences through biological experiments is resource-intensive, we want to achieve the new genome sequence detection task mathematically. However, little literature tells us how to detect new, undiscovered genome sequence mutations mathematically. We form a new framework based on natural vector convex hull method that conducts alignment-free sequence analysis. Our newly developed two approaches, Random-permutation Algorithm with Penalty (RAP) and Random-permutation Algorithm with Penalty and COstrained Search (RAPCOS), use the geometry properties captured by natural vectors. In our experiment, we discover a mathematically new human immunodeficiency virus (HIV) genome sequence using some real HIV genome sequences. Significantly, the proposed methods are applicable to solve the new genome sequence detection challenge and have many good properties, such as robustness, rapid convergence, and fast computation. [ABSTRACT FROM AUTHOR]

For η ∈ S 3 , let S n av (η) denote the set of permutations in S n that avoid the pattern η , and let E n av (η) denote the expectation with respect to the uniform probability measure on S n av (η). For n ≥ k ≥ 2 and τ ∈ S k av (η) , let N n (k) (σ) denote the number of occurrences of k consecutive numbers appearing in k consecutive positions in σ ∈ S n av (η) , and let N n (k ; τ) (σ) denote the number of such occurrences for which the order of the appearance of the k numbers is the pattern τ. We obtain explicit formulas for E n av (η) N n (k ; τ) and E n av (η) N n (k) , for all 2 ≤ k ≤ n , all η ∈ S 3 and all τ ∈ S k av (η). These exact formulas then yield asymptotic formulas as n → ∞ with k fixed, and as n → ∞ with k = k n → ∞. We also obtain analogous results for S n av (η 1 , ⋯ , η r) , the subset of S n consisting of permutations avoiding the patterns { η i } i = 1 r , where η i ∈ S m i , in the case that { η i } i = 1 n are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to r = 2 , η 1 = 2413 , η 2 = 3142. [ABSTRACT FROM AUTHOR]

Analysis and utilization of massive meter data can help decision-makers provide reasonable decisions. Therefore, multi-functional meter data processing has received considerable attention in recent years. Nevertheless, it might compromise users’ privacy, such as releasing users’ lifestyles and habits. In this paper, we propose an efficient and privacy-preserving massive data process for smart grids. The presented protocol utilizes the Paillier homomorphic encryption and Horner’s Rule to achieve a privacy-preserving two-level random permutation method, making large-scale meter data permuted randomly and sufficiently in a privacy-preserving way. As a result, the analysis center can simultaneously implement various data processing functions (such as variance, comparing, linear regression analysis), and it does not know the source of data. The security analysis shows that our protocol can realize data confidentiality and data source anonymity. The detailed analyses demonstrate that our protocol is efficient in terms of computational and communication costs. Furthermore, it can support fault tolerance of entity failures and has flexible system scalability.

Given a probability distribution p := {pk}k=1∞ on the positive integers, there are two natural ways to construct a random permutation in Sn or a random permutation of N from IID samples from p. One is called the p-biased construction and the other the p-shifted construction. In the first part of the paper we consider the case that the distribution p is the geometric distribution with parameter 1 - q ∈ (0, 1). In this case, the p-shifted random permutation has the Mallows distribution with parameter q. Let Pnb;Geo(1-q) and Pns;Geo(1-q) denote the biased and the shifted distributions on Sn. The expected number of inversions of a permutation under Pns;Geo(1-q) is greater than under Pnb;Geo(1-q), and under either of these distributions, a permutation tends to have many fewer inversions than it would have under the uniform distribution. For fixed n, both Pnb;Geo(1-q) and Pns;Geo(1-q) converge weakly as q → 1 to the uniform distribution on Sn. We compare the biased and the shifted distributions by studying the inversion statistic under ... and ... for various rates of convergence of qn to 1. In the second part of the paper we consider p-biased and p-shifted permutations for the case that the distribution p is itself random and distributed as a GEM(θ)-distribution. In particular, in both the GEM(θ)-biased and the GEM(θ)-shifted cases, the expected number of inversions behaves asymptotically as it does under the Geo(1 - q)-shifted distribution with θ = q / 1-q . This allows one to consider the GEM(θ)-shifted case as the random counterpart of the Geo(q)-shifted case. We also consider another p-biased distribution with random p for which the expected number of inversions behaves asymptotically as it does under the Geo(1 - q)-biased case with θ and q as above, and with θ → ∞ and q → 1 [ABSTRACT FROM AUTHOR]

In principle, the image encryption algorithm produces an encrypted image. The encrypted image is composed of arbitrary patterns that do not provide any clues about the plain image and its cipher key. Ideally, the encrypted image is entirely independent of its plain image. Many functions can be used to achieve this goal. Based on the functions used, image encryption techniques are categorized into: (1) Block-based; (2) Chaotic-based; (3) Transformation-based; (4) Conventional-based; and (5) Miscellaneous based. This study proposes a magic cube puzzle approach to encrypt an 8-bit grayscale image. This approach transforms a plain image into a particular size magic cube puzzle, which is consists of a set of blocks. The magic cube puzzle algorithm will diffuse the pixels of the plain image as in a Rubik’s Cube game, by rotating each block in a particular direction called the transposition orientation. The block’s transposition orientation is used as the key seed, while the generation of the cipher key uses a random permutation of the key seed with a certain key length. Several performance metrics have been used to assess the goals, and the results have been compared to several standard encryption methods. This study showed that the proposed method was better than the other methods, except for entropy metrics. For further studies, modification of the method will be carried out in such a way as to be able to increase its entropy value to very close to 8 and its application to true color images. In essence, the magic cube puzzle approach has a large space for pixel diffusion that is possibly supposed to get bigger as a series of data has transformed into several magic cubes. Then, each magic cube has transposed with a different technique. This proposed approach is expected to add to a wealth of knowledge in the field of data encryption.

Let Pnsep denote the uniform probability measure on the set of separable permutations in Sn. Let ℕ∗=ℕ∪{∞} with an appropriate metric and denote by S(ℕ,ℕ∗) the compact metric space consisting of functions σ={σi}i=1∞ from ℕ to ℕ∗ which are injections when restricted to σ−1(ℕ); that is, if σi=σj, i ≠ j, then σi=∞. Extending permutations σ∈Sn by defining σj=j, for j > n, we have Sn⊂S(ℕ,ℕ∗). We show that {Pnsep}n=1∞ converges weakly on S(ℕ,ℕ∗) to a limiting distribution of regenerative type, which we calculate explicitly. [ABSTRACT FROM AUTHOR]

In the recent decade, chaos‐based image encryption algorithms gained attention due to their pros and cons. The authors suggest one such algorithm of image encryption for different colour space using feed forward initial condition to pursue random permutation by combining 1D logistic map with a series of tent maps. The algorithm adds one more novel step to protect and encrypt binary image by concealing it within a cover grey image using bit‐plane decomposition methods. Sensitivity towards initial condition is shown using block division of image by applying a stream of random initial conditions to each block for encryption along with XOR operation which leads the key space high (≃2484). For better performance analysis, the proposed scheme has been tested with many colour space images including a binary image. The proposed technique is more efficient with respect to time complexity and resistance to vulnerability aspects.

We examine the variance of a linear statistic defined on the symmetric group endowed with the Ewens probability. Despite the dependence of the summands, it can be bounded from above by a constant multiple of the sum of variances of the summands. We find the exact value of this constant. The analysis of the appearing quadratic forms and eigenvalue search is built upon the exponential matrices and discrete Hahn's polynomials. [ABSTRACT FROM AUTHOR]

We consider permutations avoiding a pattern of length three under the family of Mallows distributions. In particular, for any pattern τ∈S3∖{321}, we obtain rather precise results on the asymptotic probability as n → ∞ that a permutation σ∈Sn under the Mallows distribution with parameter q ∈ (0, 1) avoids the pattern. By a duality between the parameters q and 1q, we also obtain rather precise results on the above probability for q > 1 and any pattern τ∈S3∖{123}. [ABSTRACT FROM AUTHOR]

Homomorphic encryption became popular and powerful cryptographic primitive for various cloud computing applications. In the recent decades several developments has been made. Few schemes based on coding theory have been proposed but none of them support unlimited operations with security. We propose a modified Reed-Muller Code based symmetric key fully homomorphic encryption to improve its security by using message expansion technique. Message expansion with prepended random fixed length string provides one-to-many mapping between message and codeword, thus one-to many mapping between plaintext and ciphertext. The proposed scheme supports both (MOD 2) additive and multiplication operations unlimitedly. We make an effort to prove the security of the scheme under indistinguishability under chosen-plaintext attack (IND-CPA) through a game-based security proof. The security proof gives a mathematical analysis and its complexity of hardness. Also, it presents security analysis against all the known attacks with respect to the message expansion and homomorphic operations.

Face-based biometric recognition is widely used nowadays, where substantial face images are commonly stored on third-party servers. Since the sensitive information of an individual is contained in facial image such as the age and health condition, it is necessary to protect its privacy and security. This paper investigates a cancelable color face template protection algorithm. To make full use of quaternion representation, the structural information including local variance and gradient is respectively served as the real part. To achieve revocability and ability to redistribute, the strategy of random permutation with binary matrix is adopted. Afterwards, the quaternion-based two-dimensional principal component analysis is employed to extract features. With them, the extreme learning machine can be trained and used for recognition. Experimental results performed on four different color face datasets have demonstrated that the fusion of structural information can greatly improve the accuracy. More importantly, the random permutation not only does not reduce the recognition accuracy, but also guarantees the security and revocation of face template. [ABSTRACT FROM AUTHOR]

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A locally uniform random permutation is generated by sampling n points independently from some absolutely continuous distribution ρ on the plane and interpreting them as a permutation by the rule that i maps to j if the i th point from the left is the j th point from below. As n tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences gives rise to a surface. In a recent paper by the author it was shown that, for any r ≥ 0 , under the correct scaling as n tends to infinity, the surface of the largest union of ⌊ r n ⌋ decreasing subsequences approaches a limit in the sense that it will come close to a maximizer of a specific variational integral (and, under reasonable assumptions, that the maximizer is essentially unique). In the present paper we show that there exists a continuous maximizer, provided that ρ has bounded density and support. The key ingredient in the proof is a new theorem about real functions of two variables that are increasing in both variables: We show that, for any constant C , any such function can be made continuous without increasing the diameter of its image or decreasing anywhere the product of its partial derivatives clipped by C , that is the minimum of the product and C. [ABSTRACT FROM AUTHOR]

Previous approaches to background subtraction usually approximate the distribution of pixels with artificial models. In this paper, we focus on automatically learning the distribution, using a novel background subtraction model named Dynamic Deep Pixel Distribution Learning (D-DPDL). In our D-DPDL model, a distribution descriptor named Random Permutation of Temporal Pixels (RPoTP) is dynamically generated as the input to a convolutional neural network for learning the statistical distribution, and a Bayesian refinement model is tailored to handle the random noise introduced by the random permutation. Because the temporal pixels are randomly permutated to guarantee that only statistical information is retained in RPoTP features, the network is forced to learn the pixel distribution. Moreover, since the noise is random, the Bayesian theorem is naturally selected to propose an empirical model as a compensation based on the similarity between pixels. Evaluations using standard benchmark demonstrates the superiority of the proposed approach compared with the state-of-the-art, including traditional methods as well as deep learning methods. [ABSTRACT FROM AUTHOR]

In principle, the image encryption algorithm produces an encrypted image. The encrypted image is composed of arbitrary patterns that do not provide any clues about the plain image and its cipher key. Ideally, the encrypted image is entirely independent of its plain image. Many functions can be used to achieve this goal. Based on the functions used, image encryption techniques are categorized into: (1) Block-based; (2) Chaotic-based; (3) Transformation-based; (4) Conventional-based; and (5) Miscellaneous based. This study proposes a magic cube puzzle approach to encrypt an 8-bit grayscale image. This approach transforms a plain image into a particular size magic cube puzzle, which is consists of a set of blocks. The magic cube puzzle algorithm will diffuse the pixels of the plain image as in a Rubik's Cube game, by rotating each block in a particular direction called the transposition orientation. The block's transposition orientation is used as the key seed, while the generation of the cipher key uses a random permutation of the key seed with a certain key length. Several performance metrics have been used to assess the goals, and the results have been compared to several standard encryption methods. This study showed that the proposed method was better than the other methods, except for entropy metrics. For further studies, modification of the method will be carried out in such a way as to be able to increase its entropy value to very close to 8 and its application to true color images. In essence, the magic cube puzzle approach has a large space for pixel diffusion that is possibly supposed to get bigger as a series of data has transformed into several magic cubes. Then, each magic cube has transposed with a different technique. This proposed approach is expected to add to a wealth of knowledge in the field of data encryption. [ABSTRACT FROM AUTHOR]

In the recent decade, chaos‐based image encryption algorithms gained attention due to their pros and cons. The authors suggest one such algorithm of image encryption for different colour space using feed forward initial condition to pursue random permutation by combining 1D logistic map with a series of tent maps. The algorithm adds one more novel step to protect and encrypt binary image by concealing it within a cover grey image using bit‐plane decomposition methods. Sensitivity towards initial condition is shown using block division of image by applying a stream of random initial conditions to each block for encryption along with XOR operation which leads the key space high (≃2484). For better performance analysis, the proposed scheme has been tested with many colour space images including a binary image. The proposed technique is more efficient with respect to time complexity and resistance to vulnerability aspects. [ABSTRACT FROM AUTHOR]