Maiorana-McFarland Class Research Articles

Quadratic permutations, complete mappings and mutually orthogonal latin squares

Abstract We investigate the permutation behavior of a special class of Dembowski-Ostrom polynomials over a finite field of characteristic 2 of the form P(X) = L 1(X)(L 2(X)+L 1(X)L 3(X)) where L 1, L 2, L 3 are linearized polynomials. To our knowledge, the given class has not been studied previously in the literature. We identify several new types of permutation polynomials of this class. While most of the newly identified polynomials are linearly equivalent to permutation monomials, we show that there exist subclasses that are not affine equivalent to monomials, and we describe their forms. One of the newly identified classes contains a subclass of complete mappings. We use these complete mappings to define new sets of mutually orthogonal Latin squares, as well as new vectorial bent functions from the Maiorana-McFarland class. Moreover, the quasigroup polynomials obtained in the process are different and inequivalent to the previously known ones.

Constructing Bent Functions Outside the Maiorana–McFarland Class Using a General Form of Rothaus

In the mid 1960s, Rothaus proposed the so-called “most general form” of constructing new bent functions by using three (initial) bent functions whose sum is again bent. In this paper, we utilize a special case of Rothaus construction when two of these three bent functions differ by a suitably chosen characteristic function of an $n/2$ -dimensional subspace. This simplification allows us to treat the induced bent conditions more easily, also implying the possibility to specify the initial functions in the partial spread class and most notably to identify several instances of the so-called non-normal bent functions. Affine inequivalent bent functions within this class are then identified using a suitable selection of initial bent functions within the partial spread class (stemming from the complete Desarguesian spread). It is also shown that when the initial bent functions belong to the class $\mathcal {D}$ , then, under certain conditions, the constructed functions provably do not belong to the completed Maiorana–McFarland class. We conjecture that our method potentially generates an infinite class of non-normal bent functions (all tested ten-variable functions are non-normal but unfortunately they are weakly normal) though there are no efficient computational tools for confirming this.

The graph of minimal distances of bent functions and its properties

A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph (V, E) where V is the set of all bent functions in 2k variables and $$(f, g) \in E$$ if the Hamming distance between f and g is equal to $$2^k$$ . It is shown that the maximum degree of the graph is equal to $$2^k (2^1 + 1) (2^2 + 1) \cdots (2^k + 1)$$ and all its vertices of maximum degree are quadratic bent functions. It is obtained that the degree of a vertex from Maiorana—McFarland class is not less than $$2^{2k + 1} - 2^k$$ . It is proven that the graph is connected for $$2k = 2, 4, 6$$ , disconnected for $$2k \ge 10$$ and its subgraph induced by all functions EA-equivalent to Maiorana—McFarland bent functions is connected.

On communication complexity of bent functions from Maiorana–McFarland class

In this article we study two party Communication Complexity of Boolean bent functions from Maiorana–McFarland class. In particular, we describe connections between Maiorana–McFarland construction of bent functions and operations on matrix form of Boolean functions and show that bent functions of 2n variables from Maiorana–McFarland class have deterministic communication complexity equal n + 1. Finally, we show that not all bent functions have high communication complexity lower bound by giving the example of such function.

An Analysis of the 𝒞 Class of Bent Functions

Two (so-called C;D) classes of permutation-based bent Boolean functions were introduced by Carlet two decades ago, but without specifying some explicit construction methods for their construction (apart from the subclass D0). In this article, we look in more detail at theC class, and derive some existence and nonexistence results concerning the bent functions in the C class for many of the known classes of permutations over F2n. Most importantly, the existence results induce generic methods of constructing bent functions in classC which possibly do not belong to the completed Maiorana-McFarland class. The question whether the specic permutations and related subspaces we identify in this article indeed give bent functions outside the completed Maiorana-McFarland class remains open.

Efficient implementation of generalized Maiorana–McFarland class of cryptographic functions

Recently, a class of cryptographic Boolean functions called generalized Maiorana–McFarland (GMM) functions was proposed in Zhang and Pasalic (IEEE Trans Inf Theory 60(10):6681–6695, 2014). In particular, it was demonstrated that certain subclasses within the GMM class satisfy all the relevant cryptographic criteria including a good resistance to (fast) algebraic cryptanalysis. However, the issue of efficient hardware implementation, which is essentially of crucial importance when such a function is used as a filtering function in certain stream cipher encryption schemes, has not been addressed in Zhang and Pasalic (2014). In this article, we analyze the complexity of hardware implementation of these subclasses and provide some exact estimates in terms of the number of elementary circuits needed. It turns out that these classes of cryptographically strong functions are also characterized with a very low hardware implementation cost, making these functions attractive candidates for the use in certain stream cipher schemes.

Infinite classes of vectorial plateaued functions, permutations and complete permutations

We use the well-known Maiorana–McFarland class to construct several important combinatorial structures. In the first place, we easily identify infinite classes of vectorial plateaued functions {F}:F2n→F2n such that all non-zero linear combinations of its component functions are also plateaued. More importantly, by setting certain restrictions on the component functions, the same approach also yields many infinite classes of permutations for any n≥6. Finally, we deduce some infinite classes of complete permutations, as a subclass of these permutations. Most notably, all these classes are of variable and controllable degree, the property being intrinsic to the construction method. The construction method is highly tweakable giving rise to many variations that again provide us with infinite classes of these structures.

Constructions of Bent—Negabent Functions and Their Relation to the Completed Maiorana—McFarland Class

The problem of constructing bent-negabent functions that do not belong to the completed Maiorana-McFarland class emerges implicitly through a series of construction methods proposed recently. These approaches manage to optimize the algebraic degree of bent-negabent functions, but all of the constructed bent-negabent functions belong to the completed Maiorana-McFarland class. In this paper, we use the indirect sum construction (proposed by Carlet in 2004) for constructing the bent-negabent functions that are not provably contained in this class, which is the first significant attempt in this direction. To achieve this, we first provide a class of bent functions with certain desirable properties that does not belong to this class and demonstrate the existence of the class members. Then, embedding these functions in the framework of the indirect sum construction, we are able to specify sufficient conditions for bent-negabent functions not being contained in the completed Maiorana-McFarland class.

On algebraic properties of S-boxes designed by means of disjoint linear codes

In a recent paper [W. Zhang and E. Pasalic, Constructions of resilient S-Boxes with strictly almost optimal nonlinearity through disjoint linear codes, IEEE Trans Inf Theory 60, no. 3 (2014), pp. 1638–1651], by using disjoint linear codes, Zhang and Pasalic presented a method for constructing t-resilient S-boxes ( even, with strictly almost optimal (currently best) nonlinearity exceeding the value . It was also shown that the algebraic degree and algebraic immunity of these resilient S-boxes are very good, but the resistance of these resilient S-boxes against fast algebraic attacks has not been treated in [W. Zhang and E. Pasalic,Constructions of resilient S-Boxes with strictly almost optimal nonlinearity through disjoint linear codes, IEEE Trans. Inf. Theory 60, no. 3 (2014), pp. 1638–1651]. In this work, we extend the method originally proposed in [E. Pasalic,Maiorana-McFarland class: Degree optimization and algebraic properties, IEEE Trans. Inf. Theory 52, no. 10 (2006), pp. 4581–4595] and used in deriving the upper bound on algebraic immunity of the Maiorana–McFarland class, for establishing the existence of low degree multiplier for the class of S-boxes that uses disjoint linear codes in the design. It is demonstrated that this class of functions has a substantial weakness against fast algebraic cryptanalysis. An alternative approach, based on the use of the associated dual codes is also developed.

Highly Nonlinear Balanced S-Boxes With Good Differential Properties

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 > 2 n-1 -2 n/2 is 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.

On cross-correlation spectrum of generalized bent functions in generalized Maiorana-McFarland class

In this paper, we obtain the cross-correlation spectrum of generalized b ent Boolean functions in a subclass of Maiorana- McFarland class (GMMF). An affine transformation which preserve the cross-correlation spectrum of two generalized Boolean functions, in absolute value is also presented. A construction of generaliz ed bent Boolean functions in (n+ 2) variables from four generalized Boolean functions in n variables is presented. It is demonstrated that the direct sum of two gene ralized bent Boolean functions is also generalized bent. Finally, we identify a class of affine fun ctions, in the generalized set up, each of its function is generalized bent.

Generalized Maiorana–McFarland class and normality of p-ary bent functions

A class of bent functions which contains bent functions with various properties like regular, weakly regular and not weakly regular bent functions in even and in odd dimension, is analyzed. It is shown that this class includes the Maiorana–McFarland class as a special case. Known classes and examples of bent functions in odd characteristic are examined for their relation to this class. In the second part, normality for bent functions in odd characteristic is analyzed. It turns out that differently to Boolean bent functions, many – also quadratic – bent functions in odd characteristic and even dimension are not normal. It is shown that regular Coulter–Matthews bent functions are normal.

Further Results on Niho Bent Functions

This paper consists of two main contributions. First, the Niho bent function consisting of 2r exponents (discovered by Leander and Kholosha) is studied. The dual of the function is found and it is shown that this new bent function is not of the Niho type. Second, all known univariate representations of Niho bent functions are analyzed for their relation to the completed Maiorana-McFarland class M. In particular, it is proven that two families do not belong to the completed class M. The latter result gives a positive answer to an open problem whether the class H of bent functions introduced by Dillon in his thesis of 1974 differs from the completed class M.

A note on the algebraic immunity of the Maiorana–McFarland class of bent functions

In Gupta et al. (2011) [5], the authors proved that the algebraic immunity of a subclass of Maiorana–McFarland functions is at most ⌈n4⌉+2 and claimed that this bound is tight. The main theorem of the upper bound is correct. However, their proof is incomplete and the bound is not tight. We will prove a more general theorem of a much larger subclass of Maiorana–McFarland functions and find that its algebraic immunity cannot achieve the optimum value. However, we find an 8-variable Maiorana–McFarland function which is not in that larger subclass of Maiorana–McFarland functions achieving the optimum algebraic immunity (this is the first time that a nontrivial Maiorana–McFarland function with the optimum algebraic immunity is given). Hence, this shows that there exist the Maiorana–McFarland functions achieving the optimum algebraic immunity.

Enumeration of the bent functions of least deviation from a quadratic bent function

We study a construction of the bent functions of least deviation from a quadratic bent function, describe all these bent functions of 2k variables, and show that the quantity of them is 2k(21 + 1) ... (2k + 1). We find some lower bound on the number of the bent functions of least deviation from a bent function of the Maiorana-McFarland class.

On Dillonʼs class H of bent functions, Niho bent functions and o-polynomials

One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family H. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well-known Maiorana–McFarland class. We first notice that H can be extended to a slightly larger class that we denote by H . We observe that the bent functions constructed via Niho power functions, for which four examples are known due to Dobbertin et al. and to Leander and Kholosha, are the univariate form of the functions of class H . Their restrictions to the vector spaces ω F 2 n / 2 , ω ∈ F 2 n ⋆ , are linear. We also characterize the bent functions whose restrictions to the ω F 2 n / 2 ʼs are affine. We answer the open question raised by Dobbertin et al. (2006) in [11] on whether the duals of the Niho bent functions introduced in the paper are affinely equivalent to them, by explicitly calculating the dual of one of these functions. We observe that this Niho function also belongs to the Maiorana–McFarland class, which brings us back to the problem of knowing whether H (or H ) is a subclass of the Maiorana–McFarland completed class. We then show that the condition for a function in bivariate form to belong to class H is equivalent to the fact that a polynomial directly related to its definition is an o-polynomial (also called oval polynomial, a notion from finite geometry). Thanks to the existence in the literature of 8 classes of nonlinear o-polynomials, we deduce a large number of new cases of bent functions in H , which are potentially affinely inequivalent to known bent functions (in particular, to Maiorana–McFarlandʼs functions).

Upper bound for algebraic immunity on a subclass of Maiorana McFarland class of bent functions

Studying algebraic immunity of Boolean functions is recently a very important research topic in cryptography. It is recently proved by Courtois and Meier that for any Boolean function of n-variable the maximum algebraic immunity is ⌈ n 2 ⌉ . We found a large subclass of Maiorana McFarland bent functions on n-variable with a proven low level of algebraic immunity ⩽ ⌈ n 4 ⌉ + 2 . To the best of our knowledge we provide for the first time a new upper bound for algebraic immunity for a nontrivial class of Boolean functions. We also discuss that this result has some fascinating implications.

On Guess and Determine Cryptanalysis of LFSR-Based Stream Ciphers

In this paper, the complexity of applying a guess and determine attack to so-called Linear Feedback Shift register (LFSR)-based stream ciphers is analyzed. This family of stream ciphers uses a single or several LFSR and a filtering function F : GF(2) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> rarr GF(2) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> to generate the blocks of m ges 1 keystream bits at the time. In difference to a classical guess and determine attack, a method based on guessing certain bits in order to determine the remaining secret key/state bits, our approach efficiently takes advantage of the reduced preimage space for relatively large m and at the same time employing the design structure of the cipher. Several variations of the algorithm are derived to circumvent the sensitivity of attack to the input data, n, m and the key length. In certain cases, our attack outperforms classical algebraic attacks; these being considered as one of the most efficient cryptanalyst tools for this type of ciphers. A superior performance of our attack over algebraic attacks is demonstrated in case the filtering function belongs to the extended Maiorana-McFarland class.

Cubic Monomial Bent Functions: A Subclass of $\mathcal{M}$

Based on a computer search, Anne Canteaut conjectured that the exponent $2^{2r}+2^r+1$ in ${\bf F}_{2^{6r}}$ and the exponent $(2^{r}+1)^2$ in ${\bf F}_{2^{4r}}$ yield bent monomial functions. These conjectures are proved in [A. Canteaut, P. Charpin, and G. Kyureghyan, A new class of monomial bent functions, in Proceedings of the 2006 IEEE International Symposium on Information Theory, (ISIT 06 Seattle), IEEE Press, Piscataway, NJ, 2006, pp. 903-906] and [N. G. Leander, IEEE Trans. Inform. Theory, 52 (2006), pp. 738-743]. Both exponents are of binary weight $3$ and define functions from the Maiorana-McFarland class $\mathcal{M}$ of bent functions to the subfield. In this paper we show that these are the only such exponents. Our proof is based on the classification of the permutation binomials $X^{2^k+2}+ \nu X$ of finite fields of even characteristics. We also extend the result of Leander, determining all bent monomial functions with the exponent $(2^{r}+1)^2$.

Maiorana–McFarland Class: Degree Optimization and Algebraic Properties

In this paper, we consider a subclass of the Maiorana-McFarland class used in the design of resilient nonlinear Boolean functions. We show that these functions allow a simple modification so that resilient Boolean functions of maximum algebraic degree may be generated instead of suboptimized degree in the original class. Preserving a high-nonlinearity value immanent to the original construction method, together with the degree optimization gives in many cases functions with cryptographic properties superior to all previously known construction methods. This approach is then used to increase the algebraic degree of functions in the extended Maiorana-McFarland (MM) class (nonlinear resilient functions F:GF(2)n |rarrGF(2)m derived from linear codes). We also show that in the Boolean case, the same subclass seems not to have an optimized algebraic immunity, hence not providing a maximum resistance against algebraic attacks. A theoretical analysis of the algebraic properties of extended Maiorana-McFarland class indicates that this class of functions should be avoided as a filtering function in nonlinear combining generators