Whereas the design and properties of single-output almost optimal five-valued spectra (AOFVS) Boolean functions on Fn2, whose Walsh spectra take the values in {0,±2⌊n/2⌋,±2⌊n/2⌋+1}, have been considered in several works, to the best of our knowledge the design of their vectorial counterpartF: Fn2→ Fm2has not been addressed so far. Based on a special kind of partitioning the vector space Fn2into disjoint linear codes (not all of them having the same dimension), for the first time we were able to specify vectorial AOFVS functionsF: Fn2→ Fn/2+12, for evenn. This has been achieved by identifying certain properties of the dual codes in the partition of Fn2. Moreover, for the first time, we could specify suitable quadruples (f1,...,f4) of AOFVS functions in a generic manner, whose concatenation f =f1||f2||f3||f4is bent. Due to a particular specification of the constituent functionsfi, we could also establish their exclusion from the completed Maiorana-McFarland (M#) class. Lastly, we introduce theD0class of AOFVS functions (similarly to the Carlet’sD0class of bent functions) and specify again suitable quadruples within this class, whose concatenation is provably bent and outside theM#class. Most notably, by doubly modifying functions in the generalized Maiorana-McFarland (GMM) class, we obtain AOFVS quadruples (f1,...,f4) for which we can fully specify the so-calledM-subspaces of eachfi. This allows us to determine their linearity index (the maximal dimension of any subspaceVfor whichDaDbfi= 0, for alla, b∈V), which is shown to be at most two, for any suchfi∈B2k+2andk≥ 3. Consequently, we deduce thatf=f1||f2||f3||f4∉M#, and additionally these bent functions have the lowest possible linearity index in certain cases, so thatDaDbf≢ 0 for any linearly independentaandb.

The problem of true random number generators (TRNGs) traces back to von Neumann’s 1951 work that aims to simulate an unbiased coin by using a biased coin with unknown probability. The core component in a TRNG is the corrector which is a post-processing function used to reduce or eliminate statistical weaknesses of physical random number generators. Note that an$(n,m,t)$-resilient function is an$(n,m,t)$-corrector. Hence, a natural question is how to construct an$(n,m,t)$-corrector which is not$(n,m,t)$-resilient? In this paper, a framework concerning the construction of nonlinear$(n,m,t)$-correctors with algebraic degree$m+1$is proposed based on an equidistant linear code. We show that the derived correctors are$(n,m,t-1)$-resilient, but not$(n,m,t)$-resilient. Given the importance of equidistant linear codes, we discuss how to get such a code with relatively flexible length, and how to get a pair of disjoint equidistant linear codes. In addition, the parameters comparison with linear correctors is given. It is shown that our method achieves the same correction order compared to the optimal linear method. As far as we know, the$(n,m,t)$-correctors we constructed also possess the best-known correction order compared with the known nonlinear$(n,m)$resilient functions. The algebraic degree and nonlinearity of the constructed correctors are also analyzed. Through a pair of disjoint equidistant linear codes, the nonlinearity of the nonlinear$(n,m,t)$-correctors can be improved. The results show that our$(n,m,t)$-correctors also possess the best algebraic degree and nonlinearity for fixed$(n,m)$.

Whereas the design and properties of bent and plateaued functions have been frequently addressed during the past few decades, there are only a few design methods of the so-called five-valued spectra Boolean functions whose Walsh spectra take the values in {0, ±2λ1, ±2λ2}. Moreover, these design methods mainly regard the specification of these functions in their algebraic normal form (ANF) domain. In this paper, we give a precise characterization of this class of functions in their spectral domain using the concept of a dual of plateaued functions. Both necessary and sufficient conditions on the Walsh support of these functions are given, which then connects their design (in the spectral domain) to a family of the so-called totally (non-overlap) disjoint spectra plateaued functions. We identify some suitable families of plateaued functions having this property, thus providing some generic methods in the spectral domain. Furthermore, we also provide an extensive analysis of their constructions in the ANF domain and provide several generic design methods. The importance of this class of functions is manifolded, where apart from being suitable for some cryptographic applications, we emphasize their property of being constituent functions in the so-called four-bent decomposition.

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This paper considers the construction of resilient Boolean functions on an odd number of variables with strictly almost optimal (SAO) nonlinearity. Through introducing the fragmentary Walsh transform, a construction technique called “High-Meets-Low” is proposed. The detailed design procedures of a 39-variable 3-resilient Boolean function with SAO nonlinearity 238- 219+ 216+ 214are given. It is shown that the nonlinearity of an n-variable t-resilient Boolean function can reach 2n-1-2(n-1)/2+5·2(n-11)/2or 2n-1-2(n-1)/2+2(n-7)/2, which are the largest known values for the corresponding n and t values. Finally, by constructing a 29-variable balanced Boolean function with SAO nonlinearity 228-214+ 210+ 29, we show an alternative method to realize the High-Meets-Low construction technique.

Permutation polynomials over finite fields constitute an active research area. Permutation trinomials attract researchers' interest due to their simple algebraic form and some additional extraordinary properties. In this paper, we present six new classes of permutation trinomials over $\mathbb{F}_{2^{n}}$ which have explicit forms by determining the solutions of some equations.

Resilient substitution boxes (S‐boxes) with high non‐linearity are important cryptographic primitives in the design of certain encryption algorithms. There are several trade‐offs between the most important cryptographic parameters and their simultaneous optimisation is regarded as a difficult task. In this study, the authors provide a construction technique to obtain resilient S‐boxes with so‐called strictly almost optimal non‐linearity for a larger number of output bits m than previously known. This is the first time that the non‐linearity bound 2 n −1 − 2 n /2 of resilient ( n , m ) S‐boxes, where n and m denote the number of the input and output bits, respectively, has been exceeded for m >⌊ n /4⌋. Thus, resilient S‐boxes with extremely high non‐linearity and a larger output space compared with other design methods have been obtained.

In this paper, we employ the so-called semi-bent functions to achieve significant improvements over currently known methods, regarding the number of orthogonal sequences per cell that can be assigned to a regular tessellation of hexagonal cells, typical for certain code-division multiple-access systems. Our initial design method generates a large family of orthogonal sets of sequences derived from vectorial semi-bent functions. A modification of the original approach is proposed to avoid a hard combinatorial problem of allocating several such orthogonal sets to a single cell of a regular hexagonal network, while preserving the orthogonality to adjacent cells. This modification increases the number of users per cell by starting from shorter codewords and then extending the length of these codewords to the desired length. The specification and assignment of these orthogonal sets to a regular tessellation of hexagonal cells have been solved, regardless of the parity and size of m (where 2mis the length of the codewords). In particular, when the re-use distance is D = 4, the number of users per cell is 2m-2for almost all m, which is twice as many as can be obtained by the best known methods.

Abstract Resilient Boolean functions with high nonlinearity and good algebraic properties play an important role in designing certain stream cipher schemes. In this paper, two construction methods are proposed to obtain such functions. It is shown that a class of resilient functions with high algebraic degree and currently best known nonlinearity can be constructed by using our technique. The algebraic immunity of the constructed functions is also analyzed. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a novel approach of finding disjoint linear codes is presented. The cardinality of a set of [u, m, t+1] disjoint linear codes largely exceeds all the previous best known methods used for the same purpose. Using such sets of disjoint linear codes, not necessarily of the same length, we have been able to provide a construction technique of t-resilient S-boxes F:F2n→2m( n even, ) with strictly almost optimal nonlinearity . This is the first time that the bound 2n-1-2n/2has been exceeded by multiple output resilient functions. Actually, the nonlinearity of our functions is in many cases equal to the best known nonlinearity of balanced Boolean functions. A large class of previously unknown cryptographic resilient S-boxes is obtained, and several improvements of the original approach are proposed. Some other relevant cryptographic properties are also briefly discussed. It is shown that these functions may reach Siegenthaler's bound n-t-1, and can be either of optimal algebraic immunity or of slightly suboptimal algebraic immunity, which was confirmed by simulations.

A new framework concerning the construction of small-order resilient Boolean functions whose nonlinearity is strictly greater than 2n-1- 2[n/2]is given. First, a generalized Maiorana-McFarland construction technique is described, which extends the current approaches by combining the usage of affine and nonlinear functions in a controllable manner. It is shown that for any given m, this technique can be used to construct a large class of n-variable (n both even and odd) m-resilient degree-optimized Boolean functions with currently best known nonlinearity. This class may also provide functions with excellent algebraic properties, measured through the resistance to (fast) algebraic attacks, if the number of n/2-variable affine subfunctions used in the construction is relatively low. Due to a potentially low hardware implementation cost, along with overall good cryptographic properties, this class of functions is an attractive candidate for the use in certain stream cipher schemes.

Substitution boxes (S-boxes) play a central role in the modern design of iterative block ciphers. While in substitution-permutation networks the S-boxes are bijective, thus ensuring the invertibility of the encryption algorithm, the property of being bijective is not mandatory for Feistel kind of networks. In this paper, two methods of constructing highly nonlinear balanced S-boxes (whose nonlinearity > 2n-1-2n/2is better than the nonlinearity of the commonly used inverse S-box) with good algebraic and differential properties are given. The first method employs two vectorial Boolean functions from the Maiorana-McFarland class that need to fulfill certain conditions. In particular, these conditions are shown to be satisfied by maximum length sequences. The second method is based on a suitable modification of a certain class of vectorial bent functions. The differential properties of these boxes, measured as a deviation from an optimal uniform distribution, also appear to be better than those of the inverse S-box. Both methods are susceptible to further optimizations of the relevant cryptographic parameters due to the underlying design ideas.

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any m, one can construct infinitely many re-variable (n even), m-resilient functions with nonlinearity > 2n-1- 2n-2. A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.