Differential Uniformity Research Articles

Constructing infinite families of low differential uniformity (n, m)-functions with $$m>n/2$$ m > n / 2

Little theoretical work has been done on (n, m)-functions when $$\frac{n}{2}<m<n$$ , even though these functions can be used in Feistel ciphers, and actually play an important role in several block ciphers. Nyberg has shown that the differential uniformity of such functions is bounded below by $$2^{n-m}+2$$ if n is odd or if $$m>\frac{n}{2}$$ . In this paper, we first characterize the differential uniformity of those (n, m)-functions of the form $$F(x,z)=\phi (z)I(x)$$ , where I(x) is the (m, m)-inverse function and $$\phi (z)$$ is an $$(n-m,m)$$ -function. Using this characterization, we construct an infinite family of differentially $$\Delta $$ -uniform $$(2m-1,m)$$ -functions with $$m\ge 3$$ achieving Nyberg’s bound with equality, which also have high nonlinearity and not too low algebraic degree. We then discuss an infinite family of differentially 4-uniform $$(m+1,m)$$ -functions in this form, which leads to many differentially 4-uniform permutations. We also present a method to construct infinite families of $$(m+k,m)$$ -functions with low differential uniformity and construct an infinite family of $$(2m-2,m)$$ -functions with $$\Delta \le 2^{m-1}-2^{m-6}+2$$ for any $$m\ge 8$$ . The constructed functions in this paper may provide more choices for the design of Feistel ciphers.

On the Boomerang Uniformity of Cryptographic Sboxes

The boomerang attack is a cryptanalysis technique against block ciphers which combines two differentials for the upper part and the lower part of the cipher. The dependency between these two differentials then highly affects the complexity of the attack and all its variants. Recently, Cid et al. introduced at Eurocrypt’18 a new tool, called the Boomerang Connectivity Table (BCT) that permits to simplify this complexity analysis, by storing and unifying the different switching probabilities of the cipher’s Sbox in one table. In this seminal paper a brief analysis of the properties of these tables is provided and some open questions are raised. It is being asked in particular whether Sboxes with optimal BCTs exist for even dimensions, where optimal means that the maximal value in the BCT equals the lowest known differential uniformity. When the dimension is even and differs from 6, such optimal Sboxes correspond to permutations such that the maximal value in their DDT and in their BCT equals 4 (unless APN permutations for such dimensions exist). We provide in this work a more in-depth analysis of boomerang connectivity tables, by studying more closely differentially 4-uniform Sboxes. We first completely characterize the BCT of all differentially 4-uniform permutations of 4 bits and then study these objects for some cryptographically relevant families of Sboxes, as the inverse function and quadratic permutations. These two families provide us with the first examples of differentially 4-uniform Sboxes optimal against boomerang attacks for an even number of variables, answering the above open question.

Characterizations of the Differential Uniformity of Vectorial Functions by the Walsh Transform

For every positive integers $n$ and $m$ and every even positive integer $\delta $ , we derive inequalities satisfied by the Walsh transforms of all vectorial $(n,m)$ -functions and prove that the case of equality characterizes differential $\delta $ -uniformity. This provides a generalization to all differentially $\delta $ -uniform functions of the characterization of Almost Perfect Nonlinear (APN) functions due to Chabaud and Vaudenay, by means of the fourth moment of the Walsh transform. Such generalization has been missing since the introduction of the notion of differential uniformity by Nyberg in 1994 and since Chabaud-Vaudenay’s result in the same year. Moreover, for each even $\delta \geq 2$ , we find several (in fact, an infinity of) such characterizations. In particular, when $\delta =2$ and $\delta =4$ , we have that, for any $(n,n)$ -function [resp. any $(n,n-1)$ -function)], the arithmetic mean of $W_{F}^{2}(u_{1},v_{1})W_{F}^{2}(u_{2},v_{2})W_{F}^{2}(u_{1}+u_{2},v_{1}+v_{2})$ when $u_{1},u_{2}$ range independently over ${\mathbb F}_{2}^{n}$ and $v_{1},v_{2}$ are nonzero and distinct and range independently over ${\mathbb F}_{2}^{m}$ is at least $2^{3n}$ , and that $F$ is APN (resp. is differentially 4-uniform) if and only if this arithmetic mean equals $2^{3n}$ (which is the value we would get with a bent function if such function could exist). These inequalities give more knowledge on the Walsh spectrum of $(n,m)$ -functions. We deduce in particular a property of the Walsh support of highly nonlinear functions. We also consider the completely open question of knowing if the nonlinearity of APN functions is necessarily non-weak (as it is the case for known APN functions); we prove new lower bounds which cover all power APN functions (and hence a large part of known APN functions), which explain why their nonlinearities are not bad, and we discuss the question of the nonlinearity of APN quadratic functions (since almost all other known APN functions are quadratic).

An Algorithm for Improving Algebraic Degree of S-Box Coordinate Boolean Functions Based on Affine Equivalence Transformation

The Substitution box (\(S\)-box) plays an important role in a block cipher as it is the only nonlinear part of the cipher in most cases. \(S\)-box \(S\) can be considered as a vectorial Boolean function consisting of \(m\) individual Boolean functions \(f_1,f_2,\dots,f_m\), where \(f_i: GF (2^n)\to GF (2)\) and \( f_i(x)=y_i\) for \(i=1,2,\dots,m\). These functions are called coordinate Boolean functions of the \(S\)-box \(S\). To avoid various attacks on the ciphers and for efficient software implementation, the coordinate Boolean functions of \(S\)-boxes are required to satisfy a lot of properties, for instance being a permutation defined on the fields with even degrees, with a high nonlinearity, a low differential uniformity and a high algebraic degree, etc. However, it seems very difficult to find an \(S\)-box with the coordinate Boolean functions to satisfy all the criteria. The \(S\)-box with low algebraic degree of the coordinate Boolean functions is vulnerable to many attacks such as linear and differential cryptanalysis, for instance higher-order differential attacks, algebraic attacks or cube attacks. In this paper we propose an algorithm for improving algebraic degree of the $S$-box coordinate Boolean functions while not affecting its other important properties. The algorithm is based on affine equivalence transformation of the \(S\)-boxes.

A New Hyperchaotic System-Based Design for Efficient Bijective Substitution-Boxes.

In this paper, we present a novel method to construct cryptographically strong bijective substitution-boxes based on the complicated dynamics of a new hyperchaotic system. The new hyperchaotic system was found to have good characteristics when compared with other systems utilized for S-box construction. The performance assessment of the proposed S-box method was carried out based on criteria, such as high nonlinearity, a good avalanche effect, bit-independent criteria, and low differential uniformity. The proposed method was also analyzed for the batch-generation of 8 × 8 S-boxes. The analyses found that through a proposed purely chaos-based method, an 8 × 8 S-box with a maximum average high nonlinearity of 108.5, or S-boxes with differential uniformity as low as 8, can be retrieved. Moreover, small-sized S-boxes with high nonlinearity and low differential uniformity are also obtainable. A performance comparison of the anticipated method with recent S-box proposals proved its dominance and effectiveness for a strong bijective S-box construction.

Componentwise APNness, Walsh uniformity of APN functions, and cyclic-additive difference sets

In [Characterizations of the differential uniformity of vectorial functions by the Walsh transform, IEEE Transactions on Information Theory 2017], the author has characterized differentially δ-uniform functions by equalities satisfied by their Walsh transforms. This generalizes the characterization of APN functions by the fourth moment of the Walsh transform. We study two notions which are related: (1) the componentwise APNness (CAPNness) of (n,n)-functions, which is a stronger version of APNness, related to the characterization by the fourth moment, in which the arithmetic mean of WF4(u,v) when u ranges over F2n and v is fixed nonzero in F2n equals 22n+1 (2) the componentwise Walsh uniformity (CWU) of (n,m)-functions (m=n, resp. m=n−1), which is a stronger version of APNness (resp. of differential 4-uniformity) related to one of the new characterizations, in which the arithmetic mean of WF2(u1,v1)WF2(u2,v2)WF2(u1+u2,v1+v2) when u1,u2 range independently over F2n and v1,v2 are fixed nonzero and distinct in F2m, equals 23n. Concerning the first notion, it is known from Berger, Canteaut, Charpin and Laigle-Chapuy that any plateaued function is CAPN if and only if it is AB and that APN power permutations are CAPN. We prove that CAPN functions can exist only if n is odd; this solves an eleven year old open problem by these authors. Concerning the second notion, we show that any crooked function (and in particular any quadratic APN function) is CWU, but we observe also that other APN functions like Kasami functions and the inverse of one of the Gold APN permutations are CWU for n≤11. We show that the CWUness of APN power permutations is equivalent to a property of ΔF={F(x)+F(x+1)+1;x∈F2n}. This new property, that we call cyclic-additive difference set property, is more complex than the cyclic difference set property (proved in the case of Kasami APN functions by Dillon and Dobbertin). We prove it in the case of the inverse of Gold function. In the case of Kasami functions, we observe that the cyclic-additive property is also true for n≤10 even and we leave the proof of the CWUness and the cyclic-additive property as open problems.

Results on symmetric S-boxes constructed by concatenation of RSSBs

In this paper, we first present an efficient exhaustive search algorithm to enumerate 6 × 6 bijective S-boxes with the best-known nonlinearity 24 in a class of S-boxes that are symmetric under the permutation τ(x) = (x0, x2, x3, x4, x5, x1), where x = (x0, \(x_{1}, \ldots , x_{5}) \in \mathbb {F}_{2}^{6}\). Since any S-box \(S: \mathbb {F}_{2}^{6}\rightarrow \mathbb {F}_{2}^{6}\) in this class has the property that S(τ(x)) = τ(S(x)) for every x, it can be considered as a construction obtained by the concatenation of 5 × 5 rotation-symmetric S-boxes (RSSBs). The size of the search space, i.e., the number of S-boxes belonging to the class, is 261.28. By performing our algorithm, we find that there exist 237.56 S-boxes with nonlinearity 24 and among them the number of those that are differentially 4-uniform is 233.99, which indicates that the concatenation method provides a rich class in terms of high nonlinearity and low differential uniformity. We then classify the S-boxes achieving the best possible trade-off between nonlinearity and differential uniformity in the class with respect to absolute indicator, algebraic degree, and transparency order. Secondly, we extend our construction method to the case of 8 × 8 bijective S-boxes and perform a steepest-descent-like iterative search algorithm in the respective class (of size 2243.74), which yields differentially 6-uniform permutations with high nonlinearity and algebraic degree.

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.

Cellular automata based S-boxes

Cellular Automata (CA) represent an interesting approach to design Substitution Boxes (S-boxes) having good cryptographic properties and low implementation costs. From the cryptographic perspective, up to now there have been only ad-hoc studies about specific kinds of CA, the best known example being the χ nonlinear transformation used in Keccak. In this paper, we undertake a systematic investigation of the cryptographic properties of S-boxes defined by CA, proving some upper bounds on their nonlinearity and differential uniformity. Next, we extend some previous published results about the construction of CAbased S-boxes by means of a heuristic technique, namely Genetic Programming (GP). In particular, we propose a “reverse engineering” method based on De Bruijn graphs to determine whether a specific S-box is expressible through a single CA rule. Then, we use GP to assess if some CA-based S-box with optimal cryptographic properties can be described by a smaller CA. The results show that GP is able to find much smaller CA rules defining the same reference S-boxes up to the size 7 × 7, suggesting that our method could be used to find more efficient representations of CA-based S-boxes for hardware implementations. Finally, we classify up to affine equivalence all 3 × 3 and 4 × 4 CA-based S-boxes.

ABC Optimization Based Construction of Strong Substitution-Boxes

Most of the block ciphers contain substitution-boxes as their cornerstone primitive to add requisite nonlinearity. S-boxes are responsible to ensure strong confusion and resistance to linear and differential assaults. The security of such ciphers relies on the strength of S-boxes deployed during substitution stage. It is challenging to construct cryptographically strong S-boxes that satisfy several properties such as high nonlinearity, good avalanche effect, bit-independent criteria, low differential uniformity and linear probability, etc. In this paper, we proposed to construct an S-box based on artificial bee colony optimization and chaotic map. The algorithm is designed to optimize an initial S-box to satisfy many properties. The simulation outcomes and comparison with recent proposals show that the proposed ABC optimization based algorithm performs reasonably well and construct S-box that exhibits higher security strength.

Parallel Fast Walsh Transform Algorithm and Its Implementation with CUDA on GPUs

Abstract Some of the most important cryptographic characteristics of the Boolean and vector Boolean functions (nonlinearity, autocorrelation, differential uniformity) are connected with the Walsh spectrum. In this paper, we present several algorithms for computing the Walsh spectrum implemented in CUDA for parallel execution on GPU. They are based on the most popular sequential algorithm. The algorithms differ in the complexity of implementations, resources used, optimization strategies and techniques. In the end, we give some experimental results.

Involutory differentially 4-uniform permutations from known constructions

Substitution boxes (S-boxes) are important components of block ciphers that can cause confusion in cryptosystems. The functions used as S-boxes should have low differential uniformity, high nonlinearity and high algebraic degree. When $$k>3$$ , due to the lack of knowledge about the existence of almost perfect nonlinear permutations over $$\mathbb {F}_{2^{2k}}$$ , which can offer optimal resistance to the differential cryptanalysis, S-boxes are often constructed from differentially 4-uniform permutations. To date, many infinite families of such functions have been constructed. In addition, the lower hardware implementation cost of S-boxes is an important criterion in the design of block ciphers. If the S-box is an involution, which means that the permutation is its own compositional inverse, then the implementation cost for its inverse can be saved. The same hardware circuit can thus be used for both encryption and decryption, which is an advantage in hardware implementation. In this paper, we investigate all of the differentially 4-uniform permutations that are known in the literature and determine whether they can be involutory. We find that some involutory differentially 4-uniform permutations with high nonlinearity and algebraic degree can be given from these known constructions. We also give some partial results and computer experiments to consider the problem of whether a permutation can be affine equivalent to an involution or it will become an involution upon adding an affine function. Some new families of differentially 4-uniform involutions constructed by composing the inverse function and cycles with length 3 are also given. This family of constructions has a high nonlinearity and a maximum algebraic degree.

Random Search Based Efficient Chaotic Substitution Box Design for Image Encryption

Cryptographic Substitution-boxes are source of nonlinearity in modern block encryption systems. The robustness and confusion imparted through these systems heavily rely on the strength of their S-boxes. This brings new challenges to design cryptographically potent S-boxes to develop strong encryption systems. In this paper, an effective method to design efficient 8×8 S-box is proposed. The design methodology incorporates piece-wise linear chaotic map based random search. The S-box obtained by the proposed methodology is tested against standard statistical tests like bijective property, strict avalanche criteria, nonlinearity, differential uniformity, bits independent criteria, and linear approximation probability, revealing its outstanding performance. The proposed S-box is compared with some recent chaos-based 8×8 S-boxes. Moreover, the proposed S-box is applied to encrypt plain-image with proposed S-box transformation to unveil and highlight its inherent great encryption strength. The results confirm that the design is consistent and suitable for building strong encryption systems for secure communication.

Current implementation of advance encryption standard (AES) S-Box

Although the attack on cryptosystem is still not severe, the development of the scheme is stillongoing especially for the design of S-Box. Two main approach has beenused, which areheuristic method and algebraic method. Algebraic method as in current AES implementationhas been proven to be the most secure S-Box design to date. This review paper willconcentrate on two kinds of method of constructing AES S-Box, which are algebraic approachand heuristic approach. The objective is to review a method of constructing S-Box, which arecomparable or close to the original construction of AES S-Box especially for the heuristicapproach. Finally, all the listed S-Boxes from these two methods will be compared in terms oftheir security performance which is nonlinearity and differential uniformity of the S-Box. Thefinding may offer the potential approach to develop a new S-Box that is better than theoriginal one.Keywords: block cipher; AES; S-Box

Super Optimal S-boxes Based on Pure Non-linear 3-quasigroups

In this paper, we present an approach to construct cryptographically strong 44-bit S-boxes with pure non-linear 3-quasigroups. PRESENT is a light weight block cipher included in ISO and IEC. The S-box used in PRESENT is optimal in linearity, differential uniformity, branch number and algebraic degree. Our methodology is based on 3-quasigroup operations and it enables someone to get 44-bit S-boxes optimal in all the above four properties that PRESENT S-box has.

Acceptance testing and performance evaluation of SPECT Gamma Camera

Objective The aim of this study is to perform the acceptance tests of Siemens Symbia E dual head gamma camera newly installed at the nuclear medicine department of NORI, Pakistan according to the specifications given by the manufacturer. Material and methods Acceptance tests performed include the measurement of system differential and integral uniformity, energy resolution, image resolution and linearity, center of rotation and total SPECT performance. Tc-99m, Co-57 and I-131 radioistopes of different amount are used according to respective calibration measurements. Different types of collimators are used for extrinsic measurement studies and Jaszczak phantom is used for the evaluation of total SPECT performance of the system. Results The results obtained for differential and integral uniformity values over central and useful field of views are 1.86%, 2.35% and 3.97%, 4.38% respectively for detector 1. Differential and integral uniformity values obtained for detector 2 over central and useful field of views are 2.01%, 2.57% and 4.06%, 4.71% respectively. The COR test performed for both detectors at 76°, 90°, 180° configuration and obtained results compared with the recommended limits. Conclusions The results obtained by these experimental studies showed that uniformity and integral values are within acceptable limits. Energy resolution, peaking, dead time and center of rotation for each detector configuration are within the recommended limits. It is also observed in the present study that intrinsic resolution is better than extrinsic resolution and system is showing good linearity behavior and sustainable sensitivity values.

Influence of Scintillation Camera Uniformity on Artifact Generation: A Simulation Study

Non-uniformity of a scintillation camera can result in artifacts on planar, projection, and single-photon emission computed tomography (SPECT) images. The purpose of this study was to evaluate the effect of field uniformity on artifact generation. Using a simulation phantom, we investigated the relationship between non-uniformity of the image and artifacts on planar, projection, and SPECT images. All the non-uniformity images were generated by decreasing the photomultiplier tube sensitivity ranging from 0% to 10%. Quantitative analysis was performed using integral and differential uniformity. We also visually assessed artifact magnitude. Integral and differential uniformity increased with decreasing the photomultiplier tube sensitivity and tended to be higher in SPECT images compared with planar and projection images. For visual assessment, mean scores in SPECT images were higher than in planar and projection images for artifact detection. Our results indicated that decreasing field uniformity is expected to produce artifacts in planar and SPECT images. Also, SPECT images require very high-field uniformity.

Non-Monomial Permutations with Differential Uniformity Six

In this paper, a family of non-monomial permutations over the finite field $${F_{{2^n}}}$$ with differential uniformity at most 6 is proposed, where n is a positive integer. The algebraic degree of these functions is also determined.

A method to calculate differential uniformity for permutations

We introduce a search algorithm to find permutation S-boxes with low differential uniformity, high nonlinearity and high algebraic degree, which play important roles in block ciphers. Inspired by the results of our search algorithm, we propose a method to calculate differential uniformity for permutations. We establish a sufficient condition for differentially 4-uniform permutations based on our method and construct some example classes of differentially 4-uniform permutations.

More permutation polynomials with differential uniformity six

More permutation polynomials with differential uniformity six