Inverse Permutation Research Articles

Double-image encryption and authentication scheme based on compressed sensing and double random phase encoding

To enhance the security of image information, a double-image encryption and authentication scheme combining compressive sensing (CS) and double random phase encoding (DRPE) is proposed. First, two plaintext images are taken as real and imaginary parts to form a complex-valued image, which is then encoded using DRPE. Next, extract the phase of the encrypted complex-valued image and encode it into authentication information. Simultaneously, the complex-valued image is sampled using the measurement matrix, which is optimized by Schmidt orthogonalization, and then quantized to form a compressed image. Subsequently, the authentication information is embedded into the compressed image, and by applying the permutation and diffusion algorithm to it, the ciphertext image is obtained. At the receiving end, the ciphertext is decrypted using inverse permutation and inverse diffusion algorithms, and the authentication information is decoded to acquire the authentication image. Finally, synchronous authentication of the two reconstructed images is achieved using a nonlinear cross-correlation method. Additionally, the keys in the proposed scheme are generated using high-dimensional chaotic systems, effectively reducing the required storage space and transmission bandwidth. Simulation results demonstrate that the proposed scheme has high image reconstruction performance at different compression ratios and possesses outstanding security and authentication capabilities.

Multiplexed All‐Optical Permutation Operations Using a Reconfigurable Diffractive Optical Network

AbstractLarge‐scale and high‐dimensional permutation operations are important for various applications in, for example, telecommunications and encryption. Here, all‐optical diffractive computing is used to execute a set of high‐dimensional permutation operations between an input and output field‐of‐view through layer rotations in a diffractive optical network. In this reconfigurable multiplexed design , every diffractive layer has four orientations: , , , and . Each unique combination of these layers represents a distinct rotation state, tailored for a specific permutation operation. Therefore, a K‐layer rotatable diffractive design can all‐optically perform up to independent permutation operations. The original input information can be decrypted by applying the specific inverse permutation matrix to output patterns. The feasibility of this reconfigurable multiplexed diffractive design is demonstrated by approximating 256 randomly selected permutation matrices using = 4 rotatable diffractive layers. To further enhance its multiplexing capability, input polarization diversity is also utilized. Additionally, this reconfigurable diffractive design is experimentally validated using terahertz radiation and 3D‐printed diffractive layers, providing a decent match to numerical results. The presented rotation‐multiplexed diffractive processor is particularly useful due to its mechanical reconfigurability, offering multifunctional representation through a single fabrication process.

Accurate diabatization based on combined-hyperbolic-inverse-power-representation: 1,2 2A' states of BeH2.

A diabatic potential energy matrix (DPEM) for the two lowest states of BeH2+ has been constructed using the combined-hyperbolic-inverse-power-representation (CHIPR) method. By imposing symmetry constraints on the coefficients of polynomials, the complete nuclear permutation inversion symmetry is correctly preserved in the CHIPR functional form. The symmetrized CHIPR functional form is then used in the diabatization by ansatz procedure. The abinitio energies are reproduced with satisfactory accuracy. In addition, the CHIPR-based DPEM also reproduces the local topology of a conical intersection. Future work will focus on a complete four-state diabatic representation with emphasis on the long-range interactions and spin-orbit couplings, which will enable accurate quantum scattering calculations for the Be+(2P) + H2 → BeH+(X1Σ+) + H(2S) reaction.

On the Two-sided Permutation Inversion Problem

In the permutation inversion problem, the task is to find the preimage of some challenge value, given oracle access to the permutation. This fundamental problem in query complexity appears in many contexts, particularly cryptography. In this work, we examine the setting in which the oracle allows for quantum queries to both the forward and the inverse direction of the permutation—except that the challenge value cannot be submitted to the latter. Within that setting, we consider three options for the inversion algorithm: whether it can get quantum advice about the permutation, whether the query algorithm can restrict the distribution with which the challenge input is sampled, and whether it must produce the entire preimage (search) or only the first bit (decision). We prove several theorems connecting the hardness of the resulting variations of the permutation inversion problem and establish lower bounds for them. Our results show that, perhaps surprisingly, the permutation inversion problem does not become significantly easier when the adversary is granted oracle access to the inverse—provided it cannot query the challenge itself.

On the universality and isotopy-isomorphy of (r,s,t)-inverse quasigroups and loops with applications to cryptography

This paper introduced a condition called $\mathcal{R}$-condition under which $(r,s,t)$-inverse quasigroups are universal. Middle isotopic $(r,s,t)$-inverse loops, satisfying the $\mathcal{R}$-condition and possessing a trivial set of $r$-weak inverse permutations were shown to be isomorphic; isotopy-isomorphy for $(r,s,t)$-inverse loops. Isotopy-isomorphy for $(r,s,t)$-inverse loops was generally characterized. With the $\mathcal{R}$-condition, it was shown that for positive integers $r$, $s$ and $t$, if there is a $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd(k,r+s+t)>1$, then there exists an $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd\big(k(r+s+t), (r+s+t)^2\big)$. The procedure of application of such $(r,s,t)$-inverse quasigroups to cryptography was described and explained, while the feasibility of such $(r,s,t)$-inverse quasigroups was illustrated with sample values of $k,r,s$ and $t$.

Accurate Full-Dimensional Global Diabatic Potential Energy Matrix for the Two Lowest-Lying Electronic States of the H + O2 ↔ HO + O Reaction.

A new and more accurate diabatic potential energy matrix (DPEM) is developed for the two lowest-lying electronic states of HO2, covering both the strong interaction region and reaction asymptotes. The ab initio calculations were performed at the Davidson corrected multireference configuration interaction level with the augmented correlation-consistent polarized valence quintuple-zeta basis set (MRCI+Q/AV5Z). The accuracy of the electronic structure calculations is validated by excellent agreement with the experimental HO2 equilibrium geometry, fundamental vibrational frequencies, and H + O2 ↔ OH + O reaction energy. Through the combination of an electronic angular momentum-method and a configuration interaction vector-based method, the mixing angle between the first two 2A″ states of HO2 was successfully determined. Elements of the 2×2 DPEM were fit to neural networks with a proper account of the complete nuclear permutation inversion symmetry of HO2. The DPEM correctly predicted the properties of conical intersection seams at linear and T-shape geometries, thus providing a reliable platform for studying both the spectroscopy of HO2 and the nonadiabatic dynamics for the H + O2 ↔ OH + O reaction.

Permutation graphs and the weak Bruhat order

Permutation graphs are graphs associated with permutations where edges represent inversions. We study different classes of permutation graphs and  isomorphic permutation graphs. A complete answer of a basic isomorphism question is given for trees. A connection to the majorization order for integer partitions is established, in connection with the notion of inversion vectors. Finally, we study special transitivity properties related to inversions of permutations.

Design of inverse permutations derived by compositions of permutations

Design of inverse permutations derived by compositions of permutations

The Jordan-Hölder property and Grothendieck monoids of exact categories

We investigate the Jordan-Hölder property (JHP) in exact categories. First, we show that (JHP) holds in an exact category if and only if the Grothendieck monoid introduced by Berenstein and Greenstein is free. Moreover, we give a criterion for this which only uses the Grothendieck group and the number of simple objects. Next, we apply these results to the representation theory of artin algebras. For a large class of exact categories including functorially finite torsion(-free) classes, (JHP) holds precisely when the number of indecomposable projectives is equal to that of simples. We study torsion-free classes in a quiver of type A in detail using the combinatorics of symmetric groups. We introduce Bruhat inversions of permutations and show that simples in a torsion-free class are in bijection with Bruhat inversions of the corresponding c-sortable element. We use this to give a combinatorial criterion for (JHP).

A new scheme to enhance the performance of permutation index\u2013differential chaos shift keying communications system

In this paper, a new scheme based on permutation index–differential chaos shift keying is proposed, modeled, and evaluated in AWGN channel environment. Data is sent by frames, and each frame is headed by a single reference signal and followed by some information-bearing signals. Modulation is performed through permutations of a reference signal according to the mapped data. At the receiver, each incoming information-bearing signal undergoes all inverse permutation possibilities to perform a correlation with the delayed and stored version of the received reference signal. To decode the information bits, the detector selects the highest correlator outputs. The proposed scheme named single reference–permutation index–differential chaos shift keying is an enhanced version of PI-DCSK, and uses a single reference signal for multiple information-bearing ones. Hence, the energy requirement is saved by almost a half. The bit error performance is studied using the baseband system model and analytically tested using Gaussian approximation method. Results show the BER performance outperforms other standard and recently developed differentially coherent chaos systems, including Permutation Index–DCSK by an average of 2.25 dB. Moreover, the analytical form which is developed to predict the bit error rate (BER) is validated by simulation. Results demonstrate the performance in AWGN is closely matching with the simulation results, particularly at high SNR.

The number of inversions of permutations with fixed shape

The Robinson-Schensted correspondence can be viewed as a map from permutations to partitions. In this work, we study the number of inversions of permutations corresponding to a fixed partition $\lambda$ under this map. Hohlweg characterized permutations having shape $\lambda$ with the minimum number of inversions. Here, we give the first results in this direction for higher numbers of inversions. We give explicit conjectures for both the structure and the number of permutations associated to $\lambda$ where the extra number of inversions is less than the length of the smallest column of $\lambda$. We prove the result when $\lambda$ has two columns.

Accurate quantum dynamics simulation of the photodetachment spectrum of the nitrate anion (NO3 -) based on an artificial neural network diabatic potential model.

The photodetachment spectrum of the nitrate anion (NO3 -) is simulated from first principles using wavepacket quantum dynamics propagation and a newly developed accurate full-dimensional fully coupled five state diabatic potential model. This model utilizes the recently proposed complete nuclear permutation inversion invariant artificial neural network diabatization technique [D. M. G. Williams and W. Eisfeld, J. Phys. Chem. A 124, 7608 (2020)]. The quantum dynamics simulations are designed such that temperature effects and the impact of near threshold detachment are taken into account. Thus, the two available experiments at high temperature and at cryogenic temperature using the slow electron velocity-map imaging technique can be reproduced in very good agreement. These results clearly show the relevance of hot bands and vibronic coupling between the X̃ 2A2 ' ground state and the B̃ 2E' excited state of the neutral radical. This together with the recent experiment at low temperature gives further support for the proper assignment of the ν3 fundamental, which has been debated for many years. An assignment of a not yet discussed hot band line is also proposed.

Self-Supervised Learning of Audio Representations From Permutations With Differentiable Ranking

Self-supervised pre-training using so-called "pretext" tasks has recently shown impressive performance across a wide range of modalities. In this work, we advance self-supervised learning from permutations, by pre-training a model to reorder shuffled parts of the spectrogram of an audio signal, to improve downstream classification performance. We make two main contributions. First, we overcome the main challenges of integrating permutation inversions into an end-to-end training scheme, using recent advances in differentiable ranking. This was heretofore sidestepped by casting the reordering task as classification, fundamentally reducing the space of permutations that can be exploited. Our experiments validate that learning from all possible permutations improves the quality of the pre-trained representations over using a limited, fixed set. Second, we show that inverting permutations is a meaningful pretext task for learning audio representations in an unsupervised fashion. In particular, we improve instrument classification and pitch estimation of musical notes by reordering spectrogram patches in the time-frequency space.

Neural Network Based Quasi-diabatic Representation for S0 and S1 States of Formaldehyde.

A neural network based quasi-diabatic potential energy matrix Hd that describes the photodissociation of formaldehyde involving the two lowest singlet states S0 and S1 is constructed. It has strict complete nuclear permutation inversion symmetry encoded and can reproduce high level ab initio electronic structure data, including energies, energy gradients, and derivative couplings, with excellent accuracy. It has been fully saturated in the configuration space to cover all possible reaction pathways with a trajectory-guided point sampling approach. This Hd will not only enable the accurate full-dimensional dynamic simulations of the photodissociation of formaldehyde involving S0 and S1 but also provide a crucial ingredient for incorporating spin-orbit couplings into a diabatic framework, thus ultimately enabling the study of both internal conversion and intersystem crossing in formaldehyde on the same footing.

Significant role of the specific prime number p = 257 in the improvement of cryptosystems

Cryptology is the significant science which is inseparable from the means of communication of secrets. In a safe manner, it has the main objective of transmitting (potentially sensitive) information between two interlocutors. One distinguishes mainly two “dual” disciplines within cryptology: (a) cryptography, which is interested in the security of information. (b) cryptanalysis, which seeks to attack it. One have a starting set of 256 elements, we add a new element to this set to form a set of 257 elements. In this paper, we consider a finite field that contains 257 elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers modulo p when p is a prime number. For our case ℤ/pℤ, p = 257. We apply it to affine ciphers and show that this cipher looks like a permutation cipher. The idea based on this result, is to use the affine ciphers with the modulo 257 (as an initial permutation) in any specific algorithm of ciphering. Besides, one finishes with the decryption affine with the modulo 257 like an inverse permutation. This is to significantly increase the security of the specific encryption algorithm and to lengthen the 16-bits encryption key.

Exclusive Neural Network Representation of the Quasi-Diabatic Hamiltonians Including Conical Intersections.

We propose a numerically simple and straightforward, yet accurate and efficient neural networks-based fitting strategy to construct coupled potential energy surfaces (PESs) in a quasi-diabatic representation. The fundamental invariants are incorporated to account for the complete nuclear permutation inversion symmetry. Instead of derivative couplings or interstate couplings, a so-called modified derivative coupling term is fitted by neural networks, resulting in accurate description of near degeneracy points, such as the conical intersections. The adiabatic energies, energy gradients, and derivative couplings are well reproduced, and the vanishing of derivative couplings as well as the isotropic topography of adiabatic and diabatic energies in asymptotic regions are automatically satisfied. All of these features of the coupled global PESs are requisite for accurate dynamics simulations. Our approach is expected to be very useful in developing highly accurate coupled PESs in a quasi-diabatic representation in an efficient machine learning-based way.

Complete Nuclear Permutation Inversion Invariant Artificial Neural Network (CNPI-ANN) Diabatization for the Accurate Treatment of Vibronic Coupling Problems.

A recently developed scheme to produce accurate high-dimensional coupled diabatic potential energy surfaces (PESs) based on artificial neural networks (ANNs) [ J. Chem. Phys. 2018, 149, 204106 and J. Chem. Phys. 2019, 151, 164118] is modified to account for the proper complete nuclear permutation inversion (CNPI) invariance. This new approach cures the problem intrinsic to the highly flexible ANN representation of diabatic PESs to account for the proper molecular symmetry accurately. It turns out that the use of CNPI invariants as coordinates for the input layer of the ANN leads to a much more compact and thus more efficient representation of the diabatic PES model without any loss of accuracy. In connection with a properly symmetrized vibronic coupling reference model, which is modified by the output neurons of the CNPI-ANN, the resulting adiabatic PESs show perfect symmetry and high accuracy. In the present paper, the new approach will be described and thoroughly tested. The test case is the representation and corresponding vibrational/vibronic nuclear dynamics of the low-lying electronic states of planar NO3 for which a large number of ab initio data is available. Thus, the present results can be compared directly with the previous studies.

Encryption of RGB Images by Means of a Novel Cryptosystem using Elliptic Curves and Chaos

A new symmetric cryptosystem is proposed in this paper which will be used to carry out image encryption without compression. The proposed system has been developed to encrypt images in BMP format. A significant contribution of this investigation is the use of one solution point of a zero valued constant elliptic curve in order to generate a set of 15 encryption keys. On the other hand, using a differential equation that produce chaotic numbers a permutation and its inverse permutation are generated. Every encryption key and both permutations are the same size as the image. The proposed cryptosystem is tested for the purpose of verifying its resistance against several cryptographic attacks as well as the randomness and quality of its ciphering. Results are showed and compared with some recent scientific articles published between 2017 and 2019.

Bidirectional Backpropagation

We extend backpropagation (BP) learning from ordinary unidirectional training to bidirectional training of deep multilayer neural networks. This gives a form of backward chaining or inverse inference from an observed network output to a candidate input that produced the output. The trained network learns a bidirectional mapping and can apply to some inverse problems. A bidirectional multilayer neural network can exactly represent some invertible functions. We prove that a fixed three-layer network can always exactly represent any finite permutation function and its inverse. The forward pass computes the permutation function value. The backward pass computes the inverse permutation with the same weights and hidden neurons. A joint forward–backward error function allows BP learning in both directions without overwriting learning in either direction. The learning applies to classification and regression. The algorithms do not require that the underlying sampled function has an inverse. A trained regression network tends to map an output back to the centroid of its preimage set.

Cryptanalysis of PRIMATEs

PRIMATEs is a family of authenticated encryption design submitted to competition for authenticated encryption: security, applicability, and robustness. The three modes of operation in PRIMATEs family are: APE, HANUMAN, GIBBON with security levels: 80, 120 bits. APE is robust despite the nonce misusing. In this study, we revise the algebraic model and find new integral distinguishers for both PRIMATE permutation and its inverse permutation. Moreover, we construct a zero-sum distinguisher for full 12-round PRIMATE-80/120 permutation with the $2^{100}/2^{105}$ complexity, improving over previous work. We also perform an integral attack on 8-round finalization of APE-80/120 with $2^{30}$ chosen messages. The key recovery process is optimized using the FFT technique presented by Todo and Aoki. Our work is the best attack against APE, demonstrating the practical attack on 8-round finalization of APE-80. The new integral distinguishers apply to create forgeries on 5/6-round finalization of APE and HANUMAN that require $2^{15}/2^{30}$ chosen messages, which is the first forgery attack against APE and HANUMAN.