RETRACTED ARTICLE: Construction of new S-boxes based on triangle groups and its applications in copyright protection

Region of interest-based medical image encryption technique based on chaotic S-boxes

Block cipher has been a standout amongst the most reliable option by which data security is accomplished. Block cipher strength against various attacks relies on substitution boxes. In literature, extensively algebraic structures, and chaotic systems-based techniques are available to design the cryptographic substitution boxes. Although, algebraic and chaotic systems-based approaches have favorable characteristics for the design of substitution boxes, but on the other side researchers have also pointed weaknesses in these approaches. First-time multilevel information fusion is introduced to construct the substitution boxes, having four layers; Multi Sources, Multi Features, Nonlinear Multi Features Whitening and Substitution Boxes Construction. Our proposed design does not hold the weakness of algebraic structures and chaotic systems because our novel s-box construction relies on the strength of true random numbers. In our proposed method true random numbers are generated from the inevitable random noise of medical imaging. The proposed design passes all the substitution box security evaluation criteria including Nonlinearity, Bit Independence Criterion (BIC), Strict Avalanche Criterion (SAC), Differential Approximation Probability (DP), Linear Approximation Probability (LP), and statistical tests, including resistance to Differential Attack, Correlation Analysis, 2D, 3D histogram analysis. The outcomes of the evaluation criteria validate that the proposed substitution boxes are effective for block ciphers; furthermore, the proposed substitution boxes attain better cryptographic strength as compared to very recent state-of-the-art techniques.

In block ciphers, the substitution boxes (S-boxes) are used with the purpose to induce confusion in cryptosystems. For the last three decades most of algebraic and chaotic S-boxes are based on the associative algebras of Galois fields. However, recently a little but remarkable work on the designing of S-boxes over some local associative algebras have been published, which is in fact a mock of the associative algebras of Galois fields. The main purpose of these establishments is to hide data and improve the security levels of crypto algorithms. In this study we redesign S-boxes over the elements of inverse property loop. The fascinating features of this structure are; it’s non-associativity and the existence of the inverse of zero element. These properties not only increase the availability of the number of structures but also provide easy approach in constructing S-boxes. The newly designed S-boxes are assessed by various analyses, which include nonlinearity, strict avalanche criterion, bit independence criterion, linear approximation probabilities, and differential approximation probabilities. Majority logic criterion is also performed to evaluate its application in various encryption systems.

Differential cryptanalysis and linear cryptanalysis are two classic analytical methods for block ciphers, and they are of great significance for evaluating the security of block ciphers. In practical security evaluations, a commonly used approach is to study lower bounds on the number of active F-functions in multi-round differential characteristics and linear approximations of the cipher structure, thereby establishing upper bounds on the maximum differential characteristic probability and the maximum linear approximation probability. If these upper bounds are sufficiently small, the cipher structure is considered to have resistance against differential and linear cryptanalysis. Based on this, this paper conducts a detailed study on the resistance of a class of four-branch extended generalized Feistel networks against differential and linear cryptanalysis. Under the condition that the F-function is bijective, it is proven that the lower bound on the number of active F-functions in \(k\)-round (\(k \ge 1\)) differential characteristics or linear approximations is \(k - 1\). Thus, if the maximum differential probability and the maximum linear approximation probability of the F-function are denoted as \(p\) and \(q\), respectively, the upper bounds on the \(k\)-round differential characteristic probability and linear approximation probability are \({p}^{k - 1}\) and \({q}^{k - 1}\), respectively.

Algebraic structures play a vital role in securing important data. These structures are utilized to construct the non-linear components of block ciphers. Since constructing non-linear components through algebraic structures is crucial for the confusion aspects of encryption schemes, relying solely on these structures can result in limited key spaces. To address this issue, a complex non-chain ring with combination of Henon map based algorithm is presented. This proposed work aims to enhance both security and key space by utilizing complex algebraic structures. In this study, we investigate the use of unit elements from non-chain rings, establishing a mapping from the group of units to Galois fields to enhance the algebraic robustness of the scheme. A substitution box is constructed from Galois field and is applied in image processing. For pixel mixing, the Henon map is employed, and the exclusive OR operation is performed using a key generated from image pixels and the substitution box. The algorithm’s performance is evaluated using standard parameters such as histograms, Structural Similarity Index Measure (SSIM), correlation analysis, and National Institute of Standards and Technology (NIST) tests. For S-box analysis, we apply the strict avalanche criterion (SAC), bit independence criterion (BIC), linear approximation probability (LAP), and differential approximation probability (DAP). This algebraic approach utilizes a robust and state-of-the-art image encryption system that is highly effective against potential attackers.

A substitution box is used to induce nonlinearity in plaintext for encryption systems. Recently, the application of chaotic maps to encryption applications has resulted in some interesting nonlinear transformations. In this paper, we propose an efficient method to design nonlinear components for block ciphers that are based on TD-ERCS chaotic sequence. The new substitution box is analyzed for nonlinearity, bit independence, strict avalanche criterion, generalized majority logic criterion, and differential and linear approximation probabilities. The results show high resistance to differential and linear cryptanalysis in comparison to some recently proposed chaotic substitution boxes.

Over binary filed F2 there are 16 primitive irreducible polynomials of degree 8, and hence one can construct 16 Galois field extensions of order 256. In this paper, we provide a novel technique to design 16 different robust 8 × 8 substitution boxes (S-boxes) over the elements these 16 Galois fields. For the purpose, on these Galois fields we define 16 linear fractional transformations as: z ⟼ (az + b)/(cz + d), where z is any arbitrary element in any of Galois fields and a, b, c, d are fixed elements from any Galois field GF(28). Accordingly for fixed parameters a, b, c, d, we obtained 16 distinct S-boxes. The algebraic strength of the proposed S-boxes is analyzed by Nonlinearity test, Strict Avalanche Criterion (SAC), Linear Approximation Probability (LP), Bit Independent Criterion (BIC), and Differential Approximation Probability (DP). As an application, by the majority logic criterion (MLC), entropy, correlation, contrast, energy and homogeneity of a plain image and its encrypted image through newly proposed S-box are assessed. Further, to fix the rank of proposed S-boxes, a comparison of these analyses is given with AES S-box, APA S-box, Residue Prime S-box, Gray S-box, Xyi S-box, Skipjack S-box and S8 AES S-box.

A strong substitution-box is main ingredient in cryptography. Many encryption schemes have been proposed since 1970’s such as DES, AES and IDEA. In this paper we construct S-boxes using a new technique, our proposed algorithm relies on modified Pascal’s triangle and elliptic curve. The substitution-boxes are analyzed by non-linearity, strict avalanche criterion, bit independence criterion, differential approximation probability and linear approximation probability. Comparison is also made with some existing S-boxes such as AES, APA, Gray, $$S_{8}$$ AES, Skipjack, Xyi and residue prime. We use our proposed substitution-boxes for image encryption and noise removal.

Differential cryptanalysis is one of the fundamental cryptanalysis techniques to evaluate the security of the block cipher. In many cases, resistance to differential cryptanalysis is proven through the upper bound of the differential characteristic probability, not the differential probability. Since the attacker uses a differential rather than a differential characteristic, resistance based on a differential characteristic tends to overestimate the security level of the block cipher. Such an overestimation is notably observed in lightweight block ciphers SKINNY, Midori, and CRAFT. In this paper, we examine the gap between the differential characteristics and the differential probability of lightweight block ciphers. We present practical methods for computing differential probability using a multistage graph. Using these methods, we count the exact number of maximum differential characteristics with fixed plaintext/ciphertext difference and activity pattern. By the exact number of maximum differential characteristics, we can calculate the probability that is closer to the real differential probability. In addition, by modifying the method, we compute a more accurate differential probability by considering the characteristics of the lower probability. We find differential distinguishers of 9-round Midori64 with probability 2−61.58, 9-round SKINNY64 with 2−58.67 and 14-round CRAFT with 2−60.32. Furthermore, we find a related-tweakey differential distinguisher of 11-round SKINNY64-64 with 2−55.93 and a related-tweak differential distinguisher of 17-round CRAFT with probability 2−63.37. Finally, we explain why these gaps are notable in Midori64, SKINNY64 and CRAFT by relating the S-box differential distribution table.

The Substitution Box (S-box) is a fundamental nonlinear transformation utilized within cryptographic algorithms to map an input bit string into an output bit string. Due to the complexity and variability of cryptographic attacks, constructing S-boxes using traditional methods has proven challenging. However, recent advancements in optimization algorithms have provided new ideas for S-box construction. To address the difficulty of constructing high-performance S-boxes through traditional means, this paper proposes a design for an 8×8 S-box based on the improved sine-cosine algorithm. In this study, genetic algorithm's mutation and crossover operations are incorporated into the sine-cosine algorithm to optimize the initial S-box constructed using the improved Newton iterative method. The resulting S-box is then evaluated based on cryptographic criteria such as bijection, nonlinearity, strict avalanche criterion, bit-independent criterion, linear probability, and differential probability. Comparative analysis with several previous S-box generation methods demonstrates that the proposed S-box design exhibits superior cryptographic properties.

Cryptography deals with designing practical mathematical algorithms having the two primitive elements of confusion and diffusion. The security of encrypted data is highly dependent on these two primitive elements and a key. S-box is the nonlinear component present in a symmetric encryption algorithm that provides confusion. A cryptographically strong bijective S-box structure in cryptosystem ensures near-optimal resistance against cryptanalytic attacks. It provides uncertainty and nonlinearity that ensures high confidentiality and security against cryptanalysis attacks. The nonlinearity of an S-box is highly dependent on the dispersal of input data using an S-box. Cryptographic performance criteria of chaos-based S-boxes are worse than algebraic S-box design methods, especially differential probability. This article reports a novel approach to design an 8 × 8 S-box using chaos and randomization using dispersion property to S-box cryptographic properties, especially differential probability. The randomization using dispersion property is introduced within the design loop to achieve low differential uniformity possibly. Two steps are involved in generating the proposed S-box. In the first step, a piecewise linear chaotic map (PWLCM) is utilized to generate initial S-box positions. Generally, the dispersion property is a post-processing technique that measures maximum nonlinearity in a given random sequence. However, in the second step, the concept is carefully reverse engineered, and the dispersion property is used within the design loop for systematic dispersal of input substituting sequence. The proposed controlled randomization changes the probability distribution statistics of S-box’s differentials. The proposed methodology systematically substitutes the S-box positions that cause output differences to recur for a given input difference. The proposed S-box is analyzed using well-established and well-known statistical cryptographic criteria of nonlinearity, strict avalanche criteria (SAC), bit independence criteria (BIC), differential probability, and linear probability. Further, the S-box’s boomerang connectivity table (BCT) is generated to analyze its strength against boomerang attack. Boomerang is a relatively new attacking framework for cryptosystem. The proposed S-box is compared with the state-of-the-art latest related publications. Results show that the proposed S-box achieves an upper bound of cryptographic properties, especially differential probability. This work hypothesizes that highly dispersive hamming distances at output difference, generated a systematic S-box. The mixing property of chaos generated trajectories utilized for decimal mapping. To test the randomness of generated chaotic trajectories, a cryptographically secure pseudo-random sequence was generated using a chaotic map that was tested using the National Institute of Standards and Technology (NIST) NIST-800-22 test suit.

Statistical attacks on block ciphers make use of a property of the cipher so that an event occurs with different probabilities depending on whether or not the correct key is used. For instance, differential cryptanalysis [3] and truncated differential cryptanalysis [5] consider characteristics or differentials which show that a particular output difference should be obtained with a relatively high probability when a particular input difference is used. Hence, when the correct key is used, the predicted differences occur more frequently.

The protection of confidential information is a global issue, and block encryption algorithms are the most reliable option for securing data. The famous information theorist, Claude Shannon, has given two desirable characteristics that should exist in a strong cipher which are substitution and permutation in their fundamental research on “Communication Theory of Secrecy Systems.” block ciphers strictly follow the substitution and permutation principle in an iterative manner to generate a ciphertext. The actual strength of the block ciphers against several attacks is entirely based on its substitution characteristic, which is gained by using the substitution box (S-box). In the current literature, algebraic structure-based and chaos-based techniques are highly used for the construction of S-boxes because both these techniques have favourable features for S-box construction but also various attacks of these techniques have been identified including SAT solver, linear and differential attacks, Gröbner-based attacks, XSL attacks, interpolation attacks, XL-based attacks, finite precision effect, chaotic systems degradation, predictability, weak randomness, chaotic discontinuity, and limited control parameters. The main objective of this research is to design a novel technique for the dynamic generation of S-boxes that are safe against the cryptanalysis techniques of algebraic structure-based and chaos-based approaches. True randomness has been universally recognized as the ideal method for cipher primitives design because true random numbers are unpredictable, irreversible, and unreproducible. The biggest challenge we faced during this research was how can we generate the true random numbers and how can true random numbers utilized for strengthening the S-box construction technique. The basic concept of the proposed technique is the extraction of true random bits from underwater acoustic waves and to design a novel technique for the dynamic generation of S-boxes using the chain of knight's tour. Rather than algebraic structure- and chaos-based techniques, our proposed technique depends on inevitable high-quality randomness which exists in underwater acoustics waves. The proposed method satisfies all standard evaluation tests of S-boxes construction and true random numbers generation. Two million bits have been analyzed using the NIST randomness test suite, and the results show that underwater sound waves are an impeccable entropy source for true randomness. Additionally, our dynamically generated S-boxes have better or equal strength, over the latest published S-boxes (2020 to 2021). According to our knowledge first time, this type of research has been conducted, in which natural randomness of underwater acoustic waves has been used for the construction of block cipher's substitution box.

In this paper, we analyze the security of two recently proposed ultra-lightweight block ciphers, SLIM and LCB. SLIM is designed based on the Feistel paradigm, operating on 32-bit blocks and has an 80-bit key. The designers claim that SLIM is immune to differential cryptanalysis after they were only able to find a trail of up to 7 rounds by using a heuristic method. LCB is another ultra-lightweight block cipher with a 32-bit block and instead uses a 64-bit secret key. It was designed based on a hybrid of Feistel and substitution-permutation network structures. Although no concrete security analyses were performed, the designers claim that 10 rounds of the cipher is secure enough against various attacks including differential cryptanalysis. We verify these claims by proposing differential attacks on both ciphers. For SLIM, we first report optimal (i.e., having the best differential probability) trails for up to 32 rounds found using an SMT solver. We then propose practical key recovery attacks on up to 14 rounds that recover the final round key with time complexity \(2^{32}\). Next, a close inspection of LCB’s design revealed a lack of nonlinearity, whereby its S-box could be modelled as a permutation. As such, differential trails that hold with probability 1 can be trivially derived for any number of rounds of the cipher. A trivial distinguishing attack can be performed with just one known-ciphertext. We fix this flaw and go on to show that LCB is actually more secure (against differential cryptanalysis) than SLIM given the same number of rounds. To the best of our knowledge, these are the first third-party cryptanalysis attacks against both ciphers.KeywordsSLIMLCBDifferential cryptanalysisLightweight block cipherSMT

To further enhance the nonlinear and cryptographic properties of chaotic S-boxes, a novel S-box design is proposed based on the discrete chaotic map and improved artificial bee colony algorithm. Firstly, the S-box generated by chaotic sequence is initialized by the opposition-based optimization. Then, the dual transposition artificial bee colony algorithm is used to optimize the population, where the Gaussian swap is engaged to strengthen the optimum efficiency, and make the population acquire a better quality with each other. With the Gaussian mutation operator, the opposition-based optimization performs self-learning to obtain better population. Comparing with some recent S-boxes based on chaos. The S-boxes generated by the proposed scheme can achieve better performance in bijection, nonlinearity, strict avalanche criterion(SAC), bit independence criterion(BIC), I / O XOR distribution and linear approximation probability(LP). It provides a new and effective method for designing S-boxes with good cryptographic properties.