Abstract: Learning generative models for graph-structured data is challenging because graphs are discrete, combinatorial, and the underlying data distribution is invariant to the ordering of nodes. However, most of the existing generative models for graphs are not invariant to the chosen ordering, which might lead to an undesirable bias in the learned distribution. To address this difficulty, we propose a permutation invariant approach to modeling graphs, using the recent framework of score-based generative modeling. In particular, we design a permutation equivariant, multi-channel graph neural network to model the gradient of the data distribution at the input graph (a.k.a., the score function). This permutation equivariant model of gradients implicitly defines a permutation invariant distribution for graphs. We train this graph neural network with score matching and sample from it with annealed Langevin dynamics. In our experiments, we first demonstrate the capacity of this new architecture in learning discrete graph algorithms. For graph generation, we find that our learning approach achieves better or comparable results to existing models on benchmark datasets.

Abstract: A new color image encryption algorithm is proposed by using chaotic maps. Cipher image is constructed in three phases. In the first phase permutation of digital image is performed with the help of a chaotic map. The second phase uses chaotic substitution box for pixel substitution and finally in the third phase a Boolean operator XOR is used for mixing chaotic logistic based random sequence. Chaotic maps have main role in this encryption. Chaos theory, due to its randomness and unpredictable behaviors, is known as favorite for the purpose of image encryption. The RGB components of image scrambled by permutation-substitution and Boolean operation, show good results for security and performance analysis. Different tests of security analysis like key space, key sensitivity, correlation analysis, entropy, histogram analysis, number of pixel change rate (NPCR) and unified average changing intensity (UACI) tests are employed on the proposed scheme. On the basis of these tests, we believe that proposed scheme is well suited for practical applications.

Abstract: Keyed and unkeyed cryptographic permutations often iterate simple round functions. Substitution-permutation networks (SPNs) are an approach that is popular since the mid 1990s. One of the new directions in the design of these round functions is to reduce the substitution (S-Box) layer from a full one to a partial one, uniformly distributed over all the rounds. LowMC and Zorro are examples of this approach.

Abstract: We propose GM-QAOA, a variation of the Quantum Alternating Operator Ansatz (QAOA) that uses Grover-like selective phase shift mixing operators. GM-QAOA works on any NP optimization problem for which it is possible to efficiently prepare an equal superposition of all feasible solutions; it is designed to perform particularly well for constraint optimization problems, where not all possible variable assignments are feasible solutions. GM-QAOA has the following features: (i) It is not susceptible to Hamiltonian Simulation error (such as Trotterization errors) as its operators can be implemented exactly using standard gate sets and (ii) Solutions with the same objective value are always sampled with the same amplitude. We illustrate the potential of GM-QAOA on several optimization problem classes: for permutation-based optimization problems such as the Traveling Salesperson Problem, we present an efficient algorithm to prepare a superposition of all possible permutations of $n$ numbers, defined on O (n 2) qubits; for the hard constraint k-Vertex-Cover problem, and for an application to Discrete Portfolio Rebalancing, we show that GM-QAOA outperforms existing QAOA approaches.

Abstract: To realize real-time image encryption, a fast color image encryption scheme by combining 3D orthogonal Latin squares (3D-OLSs) with matching matrix is proposed. The 3D-OLSs represent that each plane of two matrices must be Latin square and the corresponding planes of the two matrices must satisfy orthogonality. The matching matrix is to produce a matrix orthogonal with the 3D Latin square. In the permutation process, a new 3D permutation method with 3D-OLSs and matching matrix is devised. The proposed scheme could save encryption time to a certain degree, since the orthogonal Latin squares are defined over integers directly. In the diffusion process, to solve the diffuse problem between two planes in the 3D matrix, some matrices of the diffusion process are changed with three variables. Experimental results and security analyses show that the proposed color image encryption scheme has high security, fast speed and could resist various common attacks.

Abstract: In this paper, we present a new design of cryptosystem characterized by an optimized substitution box (S-box) and random permutation. Our proposed S-box is generated using a modified genetic algorithm. The crossover process is performed with sophisticated research using the best previous population. We use randomness and ergodicity of the logistic map to add complexity and robustness to our proposed method. Many tests proving the nonlinearity of our S-box have been carried out to demonstrate the efficiency of our algorithm. In the second part, we offer a new permutation algorithm based on a chaotic sequence generated from the logistic map. To show the performance of our proposition, we compare our results with previous algorithms. The results of its statistical analysis, like entropy value and correlation between adjacent pixels, show that the proposed image encryption scheme provides security for image encryption. The time speed of the proposed algorithm confirms the possibility of real-time implementation.

Abstract: The Clifford group plays a central role in quantum randomized benchmarking, quantum tomography, and error correction protocols. Here we study the structural properties of this group. We show that any Clifford operator can be uniquely written in the canonical form $F_1HSF_2$, where $H$ is a layer of Hadamard gates, $S$ is a permutation of qubits, and $F_i$ are parameterized Hadamard-free circuits chosen from suitable subgroups of the Clifford group. Our canonical form provides a one-to-one correspondence between Clifford operators and layered quantum circuits. We report a polynomial-time algorithm for computing the canonical form. We employ this canonical form to generate a random uniformly distributed $n$-qubit Clifford operator in runtime $O(n^2)$. The number of random bits consumed by the algorithm matches the information-theoretic lower bound. A surprising connection is highlighted between random uniform Clifford operators and the Mallows distribution on the symmetric group. The variants of the canonical form, one with a short Hadamard-free part and one allowing a circuit depth $9n$ implementation of arbitrary Clifford unitaries in the Linear Nearest Neighbor architecture are also discussed. Finally, we study computational quantum advantage where a classical reversible linear circuit can be implemented more efficiently using Clifford gates, and show an explicit example where such an advantage takes place.

Abstract: We propose GM-QAOA, a variation of the Quantum Alternating Operator Ansatz (QAOA) that uses Grover-like selective phase shift mixing operators. GM-QAOA works on any NP optimization problem for which it is possible to efficiently prepare an equal superposition of all feasible solutions; it is designed to perform particularly well for constraint optimization problems, where not all possible variable assignments are feasible solutions. GM-QAOA has the following features: (i) It is not susceptible to Hamiltonian Simulation error (such as Trotterization errors) as its operators can be implemented exactly using standard gate sets and (ii) Solutions with the same objective value are always sampled with the same amplitude. We illustrate the potential of GM-QAOA on several optimization problem classes: for permutation-based optimization problems such as the Traveling Salesperson Problem, we present an efficient algorithm to prepare a superposition of all possible permutations of $n$ numbers, defined on $O(n^2)$ qubits; for the hard constraint $k$-Vertex-Cover problem, and for an application to Discrete Portfolio Rebalancing, we show that GM-QAOA outperforms existing QAOA approaches.

Abstract: To ensure the security of digital images during transmission and storage, an efficient and secure chaos-based color image encryption scheme using bit-level permutation is proposed. Our proposed image encryption algorithm belongs to symmetric cryptography. Here, we process three color components simultaneously instead of individually, and consider the correlation between them. We propose a novel bit-level permutation algorithm that contains three parts: a plain-image related rows and columns substitution, a pixel-level roll shift part, and a bit-level cyclic shift part. In the plain-related rows and columns substitution part, we involve the plain-image information to generate a control sequence by using a skew tent system. This process ensures that the correlation between three color components can be totally broken, and our cryptosystem has enough plain-image sensitivity to resist the differential attack. In the pixel-level roll shift part and bit-level cyclic shift part, we have a fully bit-level permutation controlled by two sequences using a Rucklidge system. The simulation and some common security analyses are given. Test results show that our proposed scheme has good security performance and a speed advantage compared to other works.

Abstract: The success of AES encryption standard created challenges for the cryptographers to construct strong substitution-boxes using different underlying approaches. It is because they are solely responsible to decide the robustness of cryptosystem against linear and differential cryptanalyses. With an aim to fulfill the mentioned requirement of robustness, a novel group theoretic and graphical method is proposed to construct S-box with optimal features. Firstly, a strong S-box is generated with the help of orbits of coset graphs and the action of proposed powerful permutation of symmetric group S 256 . In addition, a specific group is designed the action of whose pairs of permutations has the ability to generate as many as 462422016 strong S-boxes. Few of such proposed S-boxes are reported and assessed against standard performance parameters to validate the effectiveness of proposed findings. The features of proposed S-boxes are compared with most of the recent S-boxes to validate the superior performance. Moreover, they are also applied for image encryption to demonstrate their suitability for multimedia security applications.

Abstract: In this era of the information age with digitalization, the transmission of sensitive real-time image information over insecure channels is highly-likely to be accessed or even attacked by an adversary. To prevent such unauthorized access, cryptography is being used to convert sensitive information in real-time images into unintelligible data. Most of the time, schemes are proposed with a high level of security. However, the challenge always remains the slower speeds due to their high complexity which makes them unusable in the applications of real-time images. In this paper, an efficient image encryption algorithm has been developed and tested for real-time images. The proposed scheme makes use of encryption with an efficient permutation technique based on a modular logistic map to bring down the size of the chaotic value vector, required to permute real-time image. We show that an efficient permutation is obtained using only $$\sqrt{N}$$ chaotic numbers for a square image with 3N pixels (N Pixels in each color bit plane). The algorithm makes use of a 192-bit key; divided into smaller blocks and each block selected chaotically to diffuse the pixel using multiple XOR operations. The experimental analysis reveals that the proposed algorithm is immune to various statistical and differential attacks such as entropy, histogram analysis, spectral characteristic analysis, etc. A comparison of the proposed scheme with some state-of-the-art techniques show that it performs better, and as such, can be utilized for efficient real-time image encryption.

Abstract: Recent work has shown that convolutional neural network classifiers overly rely on texture at the expense of shape cues. We make a similar but different distinction between shape and local image cues, on the one hand, and global image statistics, on the other. Our method, called Permuted Adaptive Instance Normalization (pAdaIN), reduces the representation of global statistics in the hidden layers of image classifiers. pAdaIN samples a random permutation $\pi$ that rearranges the samples in a given batch. Adaptive Instance Normalization (AdaIN) is then applied between the activations of each (non-permuted) sample $i$ and the corresponding activations of the sample $\pi(i)$, thus swapping statistics between the samples of the batch. Since the global image statistics are distorted, this swapping procedure causes the network to rely on cues, such as shape or texture. By choosing the random permutation with probability $p$ and the identity permutation otherwise, one can control the effect's strength. With the correct choice of $p$, fixed apriori for all experiments and selected without considering the test data, our method consistently outperforms baselines in multiple settings. In image classification, our method improves on both CIFAR100 and ImageNet using multiple architectures. In the setting of robustness, our method improves on both ImageNet-C and Cifar-100-C for multiple architectures. In the setting of domain adaptation and domain generalization, our method achieves state of the art results on the transfer learning task from GTAV to Cityscapes and on the PACS benchmark.

Abstract: Crossover is one of the most important operators in a genetic algorithm by which offspring production for the next generation is performed. There are a number of crossover operators for each type of chromosome representation of solutions that are closely related to different types of optimisation problems. Crossover operation in genetic algorithms, aimed at solving permutation-based combinatorial optimisation problems, is more computationally expensive compared to other cases. This is mainly caused by the fact that no duplicate numbers are allowed in a chromosome and therefore offspring legalisation is needed after each substring exchange. Under these conditions, the time required for performing crossover operation increases significantly with increasing chromosome size, which may deeply affect the efficiency of these genetic algorithms. In this paper, a genetic algorithm that uses path representation for chromosomes and benefits from an alternative form of the well-known partially mapped crossover is proposed. The results of numerical experiments performed on a set of benchmark problems clearly show that the use of this crossover operator can significantly increase the efficiency of permutation-based genetic algorithms and also help in producing good quality solutions.

Abstract: Substitution boxes (S-box) with strong and secure cryptographic properties are widely used for providing the key property of nonlinearity in block ciphers. This is critical to be resistant to a standard attack including linear and differential cryptanalysis. The ability to create a cryptographically strong S-box depends on its construction technique. This work aims to design and develop a cryptographically strong 8 × 8 S-box for block ciphers. In this work, the construction of the S-box is based on the linear fractional transformation and permutation function. Three steps involved in producing the S-box. In step one, an irreducible polynomial of degree eight is chosen, and all roots of the primitive irreducible polynomial are calculated. In step two, algebraic properties of linear fractional transformation are applied in Galois Field GF (28). Finally, the produced matrix is permuted to add randomness to the S-box. The strength of the S-box is measured by calculating its potency to create confusion. To analyze the security properties of the S-box, some well-known and commonly used algebraic attacks are used. The proposed S-box is analyzed by nonlinearity test, algebraic degree, differential uniformity, and strict avalanche criterion which are the avalanche effect test, completeness test, and strong S-box test. S-box analysis is done before and after the application of the permutation function and the analysis result shows that the S-box with permutation function has reached the optimal properties as a secure S-box.

Abstract: Stochastic unary computing provides low-area circuits. However, the required area consuming stochastic number generators (SNGs) in these circuits can diminish their overall gain in area, particularly if several SNGs are required. We propose area-efficient SNGs by sharing the permuted output of one linear feedback shift register (LFSR) among several SNGs. With no hardware overhead, the proposed architecture generates stochastic bit streams with minimum stochastic computing correlation (SCC). Compared to the circular shifting approach presented in prior work, our approach produces stochastic bit streams with 67% less average SCC when a 10-bit LFSR is shared between two SNGs. To generalize our approach, we propose an algorithm to find a set of $m$ permutations ( $n > m > 2$ ) with a minimum pairwise SCC, for an $n$ -bit LFSR. The search space for finding permutations with an exact minimum SCC grows rapidly when $n$ increases and it is intractable to perform a search algorithm using accurately calculated pairwise SCC values, for $n > 9$ . We propose a similarity function that can be used in the proposed search algorithm to quickly find a set of permutations with SCC values close to the minimum one. We evaluate our approach for several applications. The results show that, compared to prior work, it achieves lower mean-squared error (MSE) with the same (or even lower) area. Additionally, based on simulation results, we show that replacing the comparator component of an SNG circuit with a weighted binary generator can reduce SCC.

Abstract: For the secure transmission of data through the medium of internet, images have significant importance. Image encryption provides secure transmission of images by converting recognizable form of image into an unrecognizable form. Chaos is considered as a natural required ingredient for cryptography applications, by providing unpredictability, sensitivity of initial state and erogodicity. Therefore from the last decade, a number of chaos-based cryptosystems have been developed for the protection of transmitted images’ content. In this paper, a chaos based algorithm is developed and experimented on six different standard empirical images. The proposed cryptosystem is based on substitution-permutation network (SPN) with cipher block chaining (CBC) mode of operation. A novel algorithm is proposed for the construction of substitution box by using chaotic sine map, which is applied on a block-input of bytes, followed by a permutation based on discretized Henon map, which is applied on a block-input of bits instead of bytes. The hyper chaotic Lu system, which is nonlinear and produces discrete values with long orbits, is used as pseudorandom generator to set new values to control parameters of discretized Henon map for bit-permutation for each block. Moreover, proposed bit-permutation is applied by a matrix formulation which accelerates the bit permutation process for a block-input. Security analysis and results obtained from simulations show that cryptosystem is good resistant to various well-known attacks and have good key space therefore is reliable for secure transmission of images.

Abstract: In this study, the authors settle the feasibility of mixed integer linear programming (MILP)-aided bit-based division property for ciphers with non-bit-permutation linear layers. First, they transform the complicated linear layers to their primitive representations. Then, the original Copy and exclusive OR models are generalised, and these models are exploited to depict the primitive representations. Accord- ingly, the MILP-aided bit-based division property can be applied to much more primitives with complicated linear layers. As an illus- tration, they rst evaluate the bit-based division properties of some word-oriented block ciphers. For Midori64, they obtain a 7-round integral distinguisher, which achieves one more round than the previous results. At the same time, the data requirements of some existing distinguishers are also reduced. They decrease the data complexities of 4-round and 5-round distinguishers for LED and Joltik-BC by half. Then, the bit-based division properties of some bit-oriented ciphers such as Serpent and Noekeon are considered. The data complexities of their distinguishers for short rounds are reduced. Besides, they evaluate the bit-based division properties of the internal permutations in some hash functions. An 18-round zero-sum distinguisher for SPONGENT-88 is proposed, which achieves four more rounds than the previous ones. Some integral distinguishers for PHOTON permutations are improved.

Abstract: In this paper, we study the resolution of a permutation flow shop problem with sequence-independent setup time. The objective is to minimize the maximum of job completion time, also called the makespan. In this contribution, we propose three methods of resolution, a mixed-integer linear programming (MILP) model; two heuristics, the first based on Johnson’s rule and the second based on the NEH algorithm; and finally two metaheuristics, the iterative local search algorithm and the iterated greedy algorithm. A set of test problems is simulated numerically to validate the effectiveness of our resolution approaches. For relatively small-size problems, it has been revealed that the adapted NEH heuristic has the best performance than that of the Johnson-based heuristic. For the relatively medium and large problems, the comparative study between the two metaheuristics based on the exploration of the neighborhood shows that the iterated greedy algorithm records the best performances.

Abstract: We introduce the Sparkle family of permutations operating on 256, 384 and 512 bits. These are combined with the Beetle mode to construct a family of authenticated ciphers, Schwaemm, with security levels ranging from 120 to 250 bits. We also use them to build new sponge-based hash functions, Esch256 and Esch384. Our permutations are among those with the lowest footprint in software, without sacrificing throughput. These properties are allowed by our use of an ARX component (the Alzette S-box) as well as a carefully chosen number of rounds. The corresponding analysis is enabled by the long trail strategy which gives us the tools we need to efficiently bound the probability of all the differential and linear trails for an arbitrary number of rounds. We also present a new application of this approach where the only trails considered are those mapping the rate to the outer part of the internal state, such trails being the only relevant trails for instance in a differential collision attack. To further decrease the number of rounds without compromising security, we modify the message injection in the classical sponge construction to break the alignment between the rate and our S-box layer.

Abstract: The boomerang attack, introduced by Wagner in 1999, is a cryptanalysis technique against block ciphers based on differential cryptanalysis. In particular it takes into consideration two differentials, one for the upper part of the cipher and one for the lower part, and it exploits the dependency of these two differentials. At Eurocrypt’18, Cid et al. introduced a new tool, called the Boomerang Connectivity Table (BCT), that permits to simplify this analysis. Next, Boura and Canteaut introduced an important parameter for cryptographic S-boxes called boomerang uniformity, that is the maximum value in the BCT. Very recently, the boomerang uniformity of some classes of permutations (in particular quadratic functions) have been studied by Li, Qu, Sun and Li, and by Mesnager, Tang and Xiong. In this paper we further study the boomerang uniformity of some non-quadratic differentially 4-uniform functions. In particular, we consider the case of the Bracken-Leander cubic function and three classes of 4-uniform functions constructed by Li, Wang and Yu, obtained from modifying the inverse functions.

Abstract: With the advancement of computer and communication technologies in data processing, storage, and increasing bandwidth of data transmission, the amount of multimedia data generation and sharing has increased exponentially Therefore, the security of multimedia data becomes more important in transmission, processing, and storing the day-to-day Cryptography is one of the important mechanisms in preserving the confidentiality, integrity, and availability of multimedia data such as images In this research, a digital encryption method is presented based on Galois fields, consisting of two main stages of diffusion and permutation At the Diffusion stage, using matrix multiplication operations in the GF (256), the overlapping rows and columns of image pixels are mixing The permutation stage changes image pixels position by using The 2D chaotic map in GF (2n) or The 3D chaotic map in the GF field (2k) The proposed method, with a maximum of two rounds repetition of the main steps, reaches the optimal values of the parameters of the performance evaluation By performing standard security tests, the proposed method is resistant to differential and statistical attacks and has yielded relatively good performance compared to similar digital image encryption methods

Abstract: A novel image encryption scheme combining the 5D hyper chaotic system and DNA technology is proposed in this paper. The proposed scheme is related to the plaintext and external secret key, which does not need to manage the huge amounts of dynamic secret keys and does not to design synchronization method as the one-time-pad encryption scheme. The proposed scheme consists of four parts: pixel-level diffusion, pixel-level permutation, DNA-level diffusion and second permutation. In pixel-level diffusion process, chaotic sequences iterated by 5D hyper chaotic system with initial values (which are set as secret keys) are used to rewrite the pixel values of plaintext image and they are also used to generate second permutation rule. Then the pixel-level permutation rules are obtained by chaotic system with modified initial values that are related to the plaintext image and external secret key. In this case, the permutation rules are different when the plaintext images are different. In the DNA-level diffusion process, we select a part of pixel values of the pixel-level permutated image and external secret key to generate key streams used in DNA-level diffusion process. In this case, the decryption part can obtain the selected pixel values during the decryption process, which avoids transmitting huge secret keys and synchronizing them with plaintext images. In the second permutation, we rearrange the position of the selected pixel values, which can enhance the security of the proposed scheme. Finally, experimental results and security analysis verify that the proposed scheme can achieve good encryption effect and resist various attacks.

Abstract: Generating additive secret shares of a shuffled dataset - such that neither party knows the order in which it is permuted - is a fundamental building block in many protocols, such as secure collaborative filtering, oblivious sorting, and secure function evaluation on set intersection. Traditional approaches to this problem either involve expensive public-key based crypto or using symmetric crypto on permutation networks. While public-key-based solutions are bandwidth efficient, they are computation-heavy. On the other hand, constructions based on permutation networks are communication-bound, especially when the dataset contains large elements, for e.g., feature vectors in an ML context.

Abstract: At Eurocrypt 2018, Cid et al. introduced the Boomerang Connectivity Table (BCT), a tool to compute the probability of the middle round of a boomerang distinguisher from the description of the cipher’s Sbox(es). Their new table and the following works led to a refined understanding of boomerangs, and resulted in a series of improved attacks. Still, these works only addressed the case of Substitution Permutation Networks, and completely left out the case of ciphers following a Feistel construction. In this article, we address this lack by introducing the FBCT, the Feistel counterpart of the BCT. We show that the coefficient at row Δi, ∇o corresponds to the number of times the second order derivative at points Δi, ∇o) cancels out. We explore the properties of the FBCT and compare it to what is known on the BCT. Taking matters further, we show how to compute the probability of a boomerang switch over multiple rounds with a generic formula.