Strict Avalanche Criterion Research Articles

4‐bit Boolean functions in generation and cryptanalysis of secure 4‐bit crypto S‐boxes

AbstractIn modern ciphers of commercial computer cryptography, 4‐bit crypto substitution boxes or 4‐bit crypto S‐boxes have been of utmost importance since the late 1960s. Since then, the 4‐bit Boolean functions (BFs) have been proved to be the best tool to generate the said 4‐bit crypto S‐boxes. In this article, the crypto‐related properties of the 4‐bit BFs, such as the algebraic normal form (ANF) of the 4‐bit BFs, the balancedness, the linearity, the nonlinearity, the affinity and nonaffinity of the 4‐bit BFs, and the strict avalanche criterion (SAC) of 4‐bit BFs, are studied in detail. An exhaustive study of 4‐bit BFs with some new observations and algorithms on SAC of 4‐bit BFs is also reported in this article. In the end of 1990s, the Galois field polynomials over Galois field GF(28) are in use to generate the 8‐bit crypto S‐box of the Advance Encryption Standard. A detailed study on generation of the 4‐bit crypto S‐boxes with such Galois field polynomials over the binary as well as nonbinary extended Galois fields is also given in this article. The generated 4‐bit crypto S‐boxes are analyzed with four cryptanalysis techniques and the well‐defined SAC algorithms of 4‐bit crypto S‐boxes to search for the best possible 4‐bit crypto S‐boxes. Some existing 4‐bit crypto S‐boxes such as the thirty‐two 4‐bit crypto S‐boxes of the data encryption standard (DES) and the four 4‐bit crypto S‐boxes of the two variants of the Lucifer are analyzed to report the weakness of such S‐boxes. A comparative study of the ancient as well as the modern 4‐bit crypto S‐boxes with the generated 4‐bit crypto S‐boxes proves the said generated 4‐bit crypto S‐boxes to be the best possible one.

Enhancement strict avalanche criterion value in robust S-boxes construction using selected irreducible polynomial and affine matrixes

One indicator that a robust S-box is an ideal strict avalanche criterion value of 0.5. One method for applying the ideal strict avalanche criterion value is the selection of the precise irreducible polynomial and affine matrix. In this paper, we will discuss the robust S-boxes construction with the selected irreducible polynomial r(x) = x8 + x5 + x3 + x2 + 1 combined with the three selected affine matrixes i.e. t1, t2, and t3. Hence, the combination of selected irreducible polynomial and affine matrixes results in S-box1, S-box2, and S-box3. The test of strict avalanche criterion shows that S-box2 and S-box3 produce the same value of 0.49951. Nonlinearity and bit independence criterion tests are also conducted to complete the S-box testing.This result shows S-box2 and S-box3 are the best S-boxes compared to S-box1 and S- boxes from previous studies.

On the Construction of Balanced Boolean Functions with Strict Avalanche Criterion and Optimal Algebraic Immunity

On the Construction of Balanced Boolean Functions with Strict Avalanche Criterion and Optimal Algebraic Immunity

An approach for constructing the S-box using the CML system

We propose an approach for constructing S-box by using the CML chaotic system. The chaotic sequences produced by the CML chaotic system are used to permute and diffuse the elements of the S-box randomly. The obtained S-box has a strong randomness property due to the high dimensional feature of the CML chaotic system. We analyze bijective property, nonlinearity, strict avalanche criterion, outputs bit independence criterion, input/output XOR distribution and linear approximation probability of the constructed S-box. The experimental results show that the proposed approach for constructing S-box is effective.

Construction of new substitution boxes using linear fractional transformation and enhanced chaos

Substitution boxes are used in different security techniques and cryptosystems to ensure the secure communication of data. To enhance the randomness and perplexity of data, chaos theory has utmost importance in encryption schemes and multimedia security. In this paper, Substitution boxes are developed by using linear fractional transformation and combination of chaotic systems with the increased chaotic range as compared to their seed maps. The Substitution boxes are assessed by using various analyses which include nonlinearity, strict Avalanche criterion, bit independence criterion, linear and differential approximation probabilities. Majority logic criterion is also performed to evaluate its application in various encryption systems.

S-Box on Subgroup of Galois Field

In substitution–permutation network as a cryptosystem, substitution boxes play the role of the only nonlinear part. It would be easy for adversaries to compromise the security of the system without them. 8-bit S-boxes are the most used cryptographic components. So far, cryptographers were constructing 8-bit S-boxes used in cryptographic primitives by exhaustive search of permutations of order 256. However, now for cryptographic techniques with 8-bit S-boxes as confusion layers, researchers are trying to reduce the size of S-box by working with a small unit of data. The aim is to make the techniques compact, fast and elegant. The novelty of this research is the construction of S-box on the elements of the multiplicative subgroup of the Galois field instead of the entire Galois field. The sturdiness of the proposed S-box against algebraic attacks was hashed out by employing the renowned analyses, including balance, nonlinearity, strict avalanche criterion, and approximation probabilities. Furthermore, the statistical strength of the S-box was tested by the majority logic criterion. The fallouts show that the S-box is appropriate for applications for secure data communications. The S-box was also used for watermarking of grayscale images with good outcomes.

Steps Towards Redesigning Cryptosystems by a Non-associative Algebra of IP-Loops

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.

Quantum cryptographic property testing of multi-output Boolean functions

Compared with Boolean functions, multi-output Boolean functions (a.k.a. vectorial Boolean functions) are commonly used in classical cryptography. More generally, many cryptographic primitives can be treated as multi-output Boolean functions. Hence, the research on property testing of multi-output Boolean functions is meaningful for the design and cryptanalysis of symmetric cryptography. This paper mainly focuses on the cryptographic property testing of multi-output Boolean functions in the quantum world. Firstly, the generalized Deutsch–Jozsa algorithm is proposed to distinguish balanced multi-output Boolean functions from constant ones with a single query. This algorithm has a wider scope of applications with arbitrary ancillary inputs. The first generalized Bernstein–Vazirani algorithm suitable for multi-output Boolean functions is presented to recover the linear coefficients of linear functions. Then, combined with the generalized Deutsch–Jozsa algorithm, the quantum algorithm for estimating Walsh coefficients of multi-output Boolean functions is proposed with the same idea of quantum approximate counting algorithm, accompanied with an algorithm for computing the Walsh coefficient at a specified point based on quantum exact counting algorithm. Finally, with the usage of algorithms mentioned above, the cryptographic property testing of multi-output Boolean functions is studied. In order to describe the distances from having the certain properties, Euclidean distance and Manhattan distance are introduced as complements of Hamming distance. According to the definition, the first balance testing of multi-output Boolean functions is presented by testing the uniformity of images. The second algorithm exploits the relationship between balance and the Walsh coefficients at the point 0 which could be easily extended to k-order resiliency testing. We also briefly analyze the query complexities of strict avalanche criterion testing and k-order propagation criteria testing. The linearity testing algorithm for multi-output Boolean functions based on the generalized Bernstein–Vazirani algorithm can be adapted to Boolean functions achieving a further speedup. The non-junta testing algorithm is proposed with lower query complexity.

Construction of S-Box Based on Chaotic Map and Algebraic Structures

The Advanced Encryption Standard (AES) is widely used in different kinds of security applications. The substitution box (S-box) is the main component of many modern symmetric encryption ciphers that provides confusion between the secret key and ciphertext. The S-box component that is used in AES is fixed. If we construct this component dynamically, the encryption strength of AES would be greater than before. In this manuscript, we used chaotic logistic map, Mobius transformation and symmetric group S 256 to construct S-box for AES. The idea behind the proposed work is to make supplementary safe S-box. The presented S-box is analyzed for the following analyses: linear approximation probability (LP), nonlinearity (NL), differential approximation probability (DP), strict avalanche criterion (SAC), and bit independence criterion (BIC). The analyses show that the proposed technique is useful in generating high resistance S-box to known attacks.

Construction of Cryptographic S-Boxes Based on Mobius Transformation and Chaotic Tent-Sine System

Over the last few decades, different mediums of secure communication use chaos which is demonstrated by some nonlinear dynamical systems. Chaos shows unpredictable behavior and this characteristic is quite helpful in different encryption techniques and for multimedia security. In this work, the chaotic behavior of the improved Tent-Sine map is conferred and ultimately a new method to construct substitution-boxes is proposed. This new method explores the features of chaos through TSS map and algebraic Mobius transformation to generate strong S-boxes. The S-boxes are assessed using standard tests suit which includes nonlinearity, strict avalanche criterion, bit independence criterion, linear approximation probability and differential uniformity. Moreover, the proposed S-boxes show excellent statistical properties under majority logic criterions such as correlation, homogeneity, energy, entropy, contrast. The statistical encryption results are demonstrate the better performance of the proposed S-boxes when compared with some of state of the art S-boxes including AES, Gray, APA S8 AES, Skipjack and validate the suitability of anticipated method.

A Novel Design of Cryptographic SP-Network Based on Gold Sequences and Chaotic Logistic Tent System

Substitution permutation network (SP-network) is a chain of linked mathematical primitives used in block cipher algorithms. The proposed novel design of cryptographic SP-network consists of three cryptographic primitives: substitution box, permutation box, and random key sequences, including one key whitening operation. A new design is being proposed for each cryptographic primitive. The cryptographic strength of the proposed SP-network is evaluated by employing various standard tests; strict avalanche criterion, differential approximation probability, bit independent criterion, linear approximation probability, nonlinearity test, unified averaged changed intensity, histogram analysis, and coefficient correlation tests. The outcomes of the investigations validate that the designed cryptosystem is stable for secure communication and attains better cryptographic strength as compared with other state-of-the-art techniques.

A Novel Cryptographic Substitution Box Design Using Gaussian Distribution

In this paper, a novel method is being proposed to construct a substitution box or Boolean function for block ciphers using Gaussian distribution and linear fractional transform. The substitution box is constructed by employing a linear fractional transform based on Box–Muller transform, polarization decision, and central limit algorithm. The cryptographic strength of the proposed S-boxes is evaluated with standardized tests such as linear approximation probability, unified averaged changed intensity, bit independent criterion, histogram analysis, nonlinearity score, strict avalanche criterion, and differential approximation probability. The results show that the proposed substitution box achieves better cryptographic strength as compared with the state-of-the-art techniques.

Design of AL-13 Block Cipher Algorithm Based On Extended Feistel Network

In this paper, we design a new block cipher algorithm called AL-13. The structure of AL-13 is simple. We use Extended Feistel Network Type II with input-output 128 bits and key input 256 bits. We also use Dragon’s key schedule, binary addition, modulus addition, and XOR to give diffusion, and two functions F and N for confusion. We conducted Strict Avalanche Criterion (SAC) tests to determine whether AL-13 satisfy cryptographic measurement parameters.

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

Substitution boxes with resilient cryptographic possessions are normally utilized in block ciphers to give the substantial property of nonlinearity. They are important to resist standard attacks such as linear and differential cryptanalysis. A cryptographically robust S-box must be sound with respect to cryptographic properties like nonlinearity, bit independent criteria, strict avalanche criteria, linear and differential approximation probability. In this paper, we have developed an innovative construction scheme of nonlinear component of block cipher based on the action of projective linear groups on the projective line, and the permutation triangle groups. This nonlinear component, namely S-box, is responsible for making the relation between plaintext and ciphertext intractable which is one of the most important requirements of any modern block ciphers. By widening the scope of the proposed S-boxes, we have applied these lightweight nonlinear components in watermarking scheme.

Sine-Cosine Optimization-Based Bijective Substitution-Boxes Construction Using Enhanced Dynamics of Chaotic Map

This paper proposes a novel method of constructing strong substitution-boxes (S-boxes) of order n (4 ≤ n ≤ 8) based on a recent optimization algorithm known as sine-cosine algorithm (SCA). The paper also proposes a new 1D chaotic map, which owns enhanced dynamics compared to conventional chaotic map, for generating initial population of S-boxes and facilitating the optimization mechanism of SCA. The proposed method applies the SCA with enhanced chaotic map to explore and exploit the search space for obtaining optimized S-boxes on the basis of maximization of nonlinearity as fitness function. The S-box construction involves three phases such as initialization of population, optimization, and adjustment. The simulation and performance analyses are done using standard measures of nonlinearity, strict avalanche criterion, bits independence criterion, differential uniformity, linear approximation probability, and autocorrelation function. The obtained experimental results are compared with some immediate optimization-based and other S-boxes to show the strength of proposed method for constructing bijective S-boxes of salient cryptographic features.

Design of Highly Nonlinear Substitution Boxes Based on I-Ching Operators.

This paper is to design substitution boxes (S-Boxes) using innovative I-Ching operators (ICOs) that have evolved from ancient Chinese I-Ching philosophy. These three operators-intrication, turnover, and mutual- inherited from I-Ching are specifically designed to generate S-Boxes in cryptography. In order to analyze these three operators, identity, compositionality, and periodicity measures are developed. All three operators are only applied to change the output positions of Boolean functions. Therefore, the bijection property of S-Box is satisfied automatically. It means that our approach can avoid singular values, which is very important to generate S-Boxes. Based on the periodicity property of the ICOs, a new network is constructed, thus to be applied in the algorithm for designing S-Boxes. To examine the efficiency of our proposed approach, some commonly used criteria are adopted, such as nonlinearity, strict avalanche criterion, differential approximation probability, and linear approximation probability. The comparison results show that S-Boxes designed by applying ICOs have a higher security and better performance compared with other schemes. Furthermore, the proposed approach can also be used to other practice problems in a similar way.

Construction of highly nonlinear S-boxes for degree 8 primitive irreducible polynomials over ℤ2

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 novel efficient substitution-box design based on firefly algorithm and discrete chaotic map

Substitution boxes are essential nonlinear components responsible to impart strong confusion and security in most of modern symmetric ciphers. Constructing efficient S-boxes has been a prominent topic of interest for security experts. With an aim to construct cryptographically efficient S-box, a novel scheme based on firefly (FA) optimization and chaotic map is proposed in this paper. The anticipated approach generates initial S-box using chaotic map. The meta-heuristic FA is applied to find notable configuration of S-box that satisfies the criterions by guided search for near-optimal features by minimizing fitness function. The performance of proposed approach is assessed through well-established criterions such as bijectivity, nonlinearity, strict avalanche criteria, bit independence criteria, differential uniformity, and linear approximation probability. The obtained experimental results are compared with some recently investigated S-boxes to demonstrate that the proposed scheme has better proficiency of constructing efficient S-boxes.

The replacement of irreducible polynomial and affine mapping for the construction of a strong S-box

Substitution box (S-box) is a critical part of the data encryption and decryption procedures. The primary function of the S-box in advanced encryption standard algorithm is to randomize the 8-bit input into 8-bit output. This paper presents a novel approach to S-box construction based on the replacement of irreducible polynomial and affine mapping. The strength of the created S-box is assessed by applying several standard tests, e.g., balance, bijective, nonlinearity, strict avalanche criterion, and bit independence criterion-nonlinearity. The strength of the S-box outperforms those of available S-boxes.

Enhancing Security of Double Random Phase Encoding Based on Random S-Box

In this paper, we propose a novel asymmetric cryptosystem for double random phase encoding (DRPE) using random S-Box. While utilising S-Box separately is not reliable and DRPE does not support non-linearity, so, our system unites the effectiveness of S-Box with an asymmetric system of DRPE (through Fourier transform). The uniqueness of proposed cryptosystem lies on employing high sensitivity dynamic S-Box for our DRPE system. The randomness and scalability achieved due to applied technique is an additional feature of the proposed solution. The firmness of random S-Box is investigated in terms of performance parameters such as non-linearity, strict avalanche criterion, bit independence criterion, linear and differential approximation probabilities etc. S-Boxes convey nonlinearity to cryptosystems which is a significant parameter and very essential for DRPE. The strength of proposed cryptosystem has been analysed using various parameters such as MSE, PSNR, correlation coefficient analysis, noise analysis, SVD analysis, etc. Experimental results are conferred in detail to exhibit proposed cryptosystem is highly secure.