Maiorana-McFarland Functions Research Articles

Minimal linear codes from Maiorana-McFarland functions

Minimal linear codes have important applications in secret sharing schemes and secure multi-party computation, etc. In this paper, we study the minimality of a kind of linear codes over GF(p) from Maiorana-McFarland functions. We first obtain a new sufficient condition for this kind of linear codes to be minimal without analyzing the weights of its codewords, which is a generalization of some works given by Ding et al. in 2015. Using this condition, it is easy to verify that such minimal linear codes satisfy wminwmax≤p−1p for any prime p, where wmin and wmax denote the minimum and maximum nonzero weights in a code, respectively. Then, by selecting the subsets of GF(p)s, we present two new infinite families of minimal linear codes with wminwmax≤p−1p for any prime p. In addition, the weight distributions of the presented linear codes are determined in terms of Krawtchouk polynomials or partial spreads.

Secondary constructions of RSBFs with good cryptographic properties

Rotation symmetric Boolean functions have been used as components of different cryptosystems. How to construct rotation symmetric Boolean functions with good cryptographic properties is an interesting and significant problem. So far little attempt has been made to provide secondary constructions of rotation symmetric Boolean function. In this paper, based on indirect sum, Maiorana-McFarland functions and concatenation, we propose secondary constructions of rotation symmetric Boolean functions with resiliency, bentness and high algebraic immunity.

Modified planar functions and their components

Zhou (J. Combin. Des. 21(12), 563–584, 2013) introduced modified planar functions in order to describe (2n, 2n, 2n, 1) relative difference sets R as a graph of a function on the finite field \(\mathbb {F}_{2^{n}}\), and pointed out that projections of R are difference sets that can be described by negabent or bent4 functions, which are Boolean functions given in multivariate form. One of the objectives of this paper is to contribute to the understanding of these component functions of modified planar functions. We identify the versions of the Walsh transforms that apply to modified planar functions on \(\mathbb {F}_{2^{n}}\) and their components, obtain a description of modified planar functions by their components which is similar to that of the classical planar functions in odd characteristic as a vectorial bent function, and point out some further properties of these components. Finally we introduce vectorial bent4 functions (over finite fields), which correspond to relative difference sets in certain groups. We give conditions under which Maiorana-McFarland functions are such functions, hence give rise to relative difference sets in \(\mathbb {Z}_{2}^{n/2}\times \mathbb {Z}_{4}^{n/2}\).

Generalized bent functions -sufficient conditions and related constructions

The necessary and sufficient conditions for a class of functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$, where $q \geq 2$ is an even positive integer, have been recently identified for $q=4$ and $q=8$. In this article we give an alternative characterization of the generalized Walsh-Hadamard transform in terms of the Walsh spectra of the component Boolean functions of $f$, which then allows us to derive sufficient conditions that $f$ is generalized bent for any even $q$. The case when $q$ is not a power of two, which has not been addressed previously, is treated separately and a suitable representation in terms of the component functions is employed. Consequently, the derived results lead to generic construction methods of this class of functions. The main remaining task, which is not answered in this article, is whether the sufficient conditions are also necessary. There are some indications that this might be true which is also formally confirmed for generalized bent functions that belong to the class of generalized Maiorana-McFarland functions (GMMF), but still we were unable to completely specify (in terms of necessity) gbent conditions.

On the approximation of S‐boxes via Maiorana–McFarland functions

Substitution boxes (S-boxes) are the key components of conventional cryptographic systems. To quantify the confusion property of S-boxes, different non-linearity criteria are proposed such as usual non-linearity (NF ), unrestricted non-linearity (UN F ), generalised non-linearity (GN F ), higher order non-linearity (HN F ) and so on. Although these different criteria come from the idea of linear (or non-linear) approximation of S-boxes, the algebraic structures of Boolean functions that are used to approximate to S-boxes have not been considered yet. In this study, the concept of the extended non-linearity of S-boxes (denoted by EN F ) is introduced by measuring the distance of a given function to a subset of Maiorana–McFarland functions. This approximation appears to be appealing because of a particular structure of this class of functions, namely their representation as a concatenation of affine functions. The complexity of computing the rth order extended non-linearity for S-boxes over GF(2) n is less than O(( n r )2 n−r ), (r > 1). Moreover, a theoretical upper bound for the rth order extended non-linearity is proved, which is much lower than previous generalised non-linearity which might give a rise to more efficient attacks that combine a generalised correlation approach with guess and determine techniques. Furthermore, the relationship between the r-order extended non-linearity and the generalised non-linearity is derived.

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.

Polynomials With Linear Structure and Maiorana–McFarland Construction

In this paper, we study permutation polynomials over the finite fields that have linear structures. We present some results on such a permutation which transforms a hyperplane to another hyperplane. We fully characterize the bilinear polynomial with linear structure. The most important result of this paper is to show the relation between a Maiorana-McFarland function with an affine derivative and a polynomial with a linear structure. Moreover, we highlight this result in the context of resilient functions which are based on Maiorana-McFarland construction.

Construction of almost optimal resilient Boolean functions via concatenating Maiorana-McFarland functions

We describe a method to construct an n-variable (n even) almost optimal resilient Boolean function via concatenating 2d properly chosen (n−d)-variable Maiorana-McFarland functions. As an example, a 5-resilient 42-variable Boolean function with currently best known nonlinearity 241 −220 −217 is provided.

A resynchronization attack on stream ciphers filtered by Maiorana-McFarland functions

A resynchronization attack is proposed on stream ciphers filtered by Maiorana-McFarland (M-M) functions and equipped with a linear resynchronization mechanism. The proposed attack utilizes the linear weakness of the resynchronization mechanism, the partial linearity of M-M functions, and applies the linear consistency test method to recover the secret key. It is shown that an M-M function should not be implemented by itself but rather in combination with other nonlinear components in stream ciphers using linear mechanisms to prevent the proposed attack. It is also shown that the use of linear resynchronization mechanisms should be avoided despite their high efficiency in stream ciphers filtered by M-M functions.

Time-Memory-Data Trade-Off Attack on Stream Ciphers Based on Maiorana-McFarland Functions

In this paper, we present the time-memory-data (TMD) trade-off attack on stream ciphers filter function generators and filter cominers based on Maiorana-McFarland functions. This can be considered as a generalization of the time-memory-data trade-off attack of Mihaljevic and Imai on Toyocrypt. First, we substitute the filter function in Toyocrypt (which has the same size as the LFSR) with a general Maiorana-McFarland function. This allows us to apply the attack to a wider class of stream ciphers. Second, we highlight how the choice of different Maiorana-McFarland functions can affect the effectiveness of our attack. Third, we show that the attack can be modified to apply on filter functions which are smaller than the LFSR and on filter-combiner stream ciphers. This allows us to cryptanalyze other configurations commonly found in practice. Finally, filter functions with vector output are sometimes used in stream ciphers to improve the throughput. Therefore the case when the Maiorana-McFarland functions have vector output is investigated. We found that the extra speed comes at the price of additional weaknesses which make the attacks easier.

The ranks of Maiorana-McFarland bent functions

In this paper, the ranks of a special family of Maiorana-McFarland bent functions are discussed. The upper and lower bounds of the ranks are given and those bent functions whose ranks achieve these bounds are determined. As a consequence, the inequivalence of some bent functions are derived. Furthermore, the ranks of the functions of this family are calculated when t ⩽ 6.

On the confusion and diffusion properties of Maiorana–McFarland's and extended Maiorana–McFarland's functions

A practical problem in symmetric cryptography is finding constructions of Boolean functions leading to reasonably large sets of functions satisfying some desired cryptographic criteria. The main known construction, called Maiorana–McFarland, has been recently extended. Some other constructions exist, but lead to smaller classes of functions. Here, we study more in detail the nonlinearities and the resiliencies of the functions produced by all these constructions. Further we see how to obtain functions satisfying the propagation criterion (among which bent functions) with these methods, and we give a new construction of bent functions based on the extended Maiorana–McFarland's construction.