Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary linear codes from vectorial Boolean functions and determine their parameters, by further studying a generic construction developed by Ding \emph{et al.} recently. First, by employing perfect nonlinear functions and almost bent functions, we obtain several classes of six-weight linear codes which contains the all-one codeword. Second, we investigate a subcode of any linear code mentioned above and consider its parameters. When the vectorial Boolean function is a perfect nonlinear function or a Gold function in odd dimension, we can completely determine the weight distribution of this subcode. Besides, our linear codes have larger dimensions than the ones by Ding et al.'s generic construction.
Recently, much progress has been made to construct minimal linear codes due to their preference in secret sharing schemes and secure two-party computation. In this paper, we put forward a new method to construct minimal linear codes by using vectorial Boolean functions. Firstly, we give a necessary and sufficient condition for a generic class of linear codes from vectorial Boolean functions to be minimal. Based on that, we derive some new three-weight minimal linear codes and determine their weight distributions. Secondly, we obtain a necessary and sufficient condition for another generic clas…
Vectorial Boolean functions are crucial building blocks in symmetric ciphers. Different known attacks on block ciphers have resulted in diverse cryptographic criteria of vectorial Boolean functions,such as differential distribution table and nonlinearity. Very recently, Bar-On et al. introduced at Eurocrypt'19 a new tool, called the Differential-Linear Connectivity Table (DLCT).This paper is a follow-up work, which presents further theoretical characterization of the DLCT of vectorial Boolean functions and also investigates this new criterion of functions with certain forms. In this paper we i…
In this paper we study the relationship between vectorial (Boolean) functions and cyclic codes in the context of algebraic attacks. We first derive a direct link between the annihilators of a vectorial function (in univariate form) and certain $2^{n}$-ary cyclic codes (which we prove that they are LCD codes) extending results due to Rønjom and Helleseth. The knowledge of the minimum distance of those codes gives rise to a lower bound on the algebraic immunity of the associated vectorial function. Furthermore, we solve an open question raised by Mesnager and Cohen. We also present some properti…
Vectorial Boolean functions are crucial building-blocks in symmetric ciphers. Different known attacks on block ciphers have resulted in diverse cryptographic criteria for vectorial Boolean functions, such as differential uniformity and nonlinearity. Very recently, Bar-On et al. introduced at Eurocrypt'19 a new tool, called the differential-linear connectivity table (DLCT), which allows for taking into account the dependency between the two subciphers $E_0$ and $E_1$ involved in differential-linear attacks. This new notion leads to significant improvements of differential-linear attacks on seve…
Propagation criterion of degree $l$ and order $k$ ($PC(l)$ of order $k$) and resiliency of vectorial Boolean functions are important for cryptographic purpose (see [1, 2, 3,6, 7,8,10,11,16]. Kurosawa, Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying $PC(l)$ of order $k$ from binary linear or nonlinear codes in. In this paper, algebraic-geometric codes over $GF(2^m)$ are used to modify Carlet and Kurosawa-Satoh's construction for giving vectorial resilient Boolean functions satisfying $PC(l)$ of order $k$. The new construction is compared with previously known results…
There are two construction methods of designs from $(n,m)$-bent functions, known as translation and addition designs. In this paper we analyze, which equivalence relation for Boolean bent functions, i.e. $(n,1)$-bent functions, and vectorial bent functions, i.e. $(n,m)$-bent functions with $2\le m\le n/2$, is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial $(n,m)$-bent functions and isomorphism of addition designs are the same concepts for a…
The notion of o-polynomial comes from finite projective geometry. In 2011 and later, it has been shown that those objects play an important role in symmetric cryptography and coding theory to design bent Boolean functions, bent vectorial Boolean functions, semi-bent functions and to construct good linear codes. In this note, we characterize o-polynomials by the Walsh transform of the associated vectorial functions.
In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of vectorial/Boolean bent functions via the so-called $(P_U)$ property was introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the construction of $p$-ary weakly regular bent functions. The objective of this paper is to further generalize these constructions, following the ideas in \cite{Bapic, Zheng}, for secondary constructions of vectorial $p$-ary weakly regular bent and plateaued functions. We also present some infinite families of such functions via the $p$-ary Maiorana-McFarland class. Addition…
Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing binary minimal codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high …
In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$. Explicitly, we describe gbent functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$, which can be seen as a gbent version of Dillon's $PS_{ap}$ class. For the first time, we also introduce the concept of a vectorial gbent function from $\F_2^n$ to $\Z_q^m$, and determine the maximal value which $m$ can attain for the case $q=2^t$. Finally we point to a relation between vectorial gbent functions and relative difference sets.
Let $n$ be an even positive integer, and $m<n$ be one of its positive divisors. In this paper, inspired by a nice work of Tang et al. on constructing large classes of bent functions from known bent functions [27, IEEE TIT, 63(10): 6149-6157, 2017], we consider the construction of vectorial bent and vectorial plateaued $(n,m)$-functions of the form $H(x)=G(x)+g(x)$, where $G(x)$ is a vectorial bent $(n,m)$-function, and $g(x)$ is a Boolean function over $\mathbb{F}_{2^{n}}$. We find an efficient generic method to construct vectorial bent and vectorial plateaued functions of this form by establi…
The study of non-linearity (linearity) of Boolean function was initiated by Rothaus in 1976. The classical non-linearity of a Boolean function is the minimum Hamming distance of its truth table to that of affine functions. In this note we introduce new "multidimensional" non-linearity parameters $(N_f,H_f)$ for conventional and vectorial Boolean functions $f$ with $m$ coordinates in $n$ variables. The classical non-linearity may be treated as a 1-dimensional parameter in the new definition. $r$-dimensional parameters for $r\geq 2$ are relevant to possible multidimensional extensions of the Fas…
Linear codes with a few weights are an important class of codes in coding theory and have attracted a lot of attention. In this paper, we present several constructions of $q$-ary linear codes with two or three weights from vectorial dual-bent functions, where $q$ is a power of an odd prime $p$. The weight distributions of the constructed $q$-ary linear codes are completely determined. We illustrate that some known constructions in the literature can be obtained by our constructions. In some special cases, our constructed linear codes can meet the Griesmer bound. Furthermore, based on the const…
In this paper, we show that there is no vectorial Boolean function of degree 4e, with e satisfaying certain conditions, which is APN over infinitely many extensions of its field of definition. It is a new step in the proof of the conjecture of Aubry, McGuire and Rodier
In this paper, we investigate permutation rotation-symmetric (shift-invariant) vectorial Boolean functions on $n$ bits that are liftings from Boolean functions on $k$ bits, for $k\leq n$. These functions generalize the well-known map used in the current Keccak hash function, which is generated via the Boolean function on $3$ variables, $x_1+(x_2+1)x_3$. We provide some general constructions, and also study the affine equivalence between rotation-symmetric S-boxes and describe the corresponding relationship between the Boolean function they are associated with.
Boolean functions have very nice applications in cryptography and coding theory, which have led to a lot of research focusing on their applications. The objective of this paper is to construct binary linear codes with few weights from the defining set, which is defined by some special Boolean functions and some additional restrictions. First, we provide two general constructions of binary linear codes with three or four weights from Boolean functions with at most three Walsh transform values and determine the parameters of their dual codes. Then many classes of binary linear codes with explici…
We construct trace codes over $\Z_4$ by using Boolean functions and skew sets, respectively. Their Lee weight distribution is studied by using a Galois ring version of the Walsh-Hadamard transform and exponential sums. We obtain a new family of optimal two-weight codes over $\Z_4.$
In this paper, we present a construction of linear codes over $\f_{2^t}$ from Boolean functions, which is a generalization of Ding's method \cite[Theorem 9]{Ding15}. Based on this construction, we give two classes of linear codes $\tilde{\C}_{f}$ and $\C_f$ (see Theorem \ref{thm-maincode1} and Theorem \ref{thm-maincodenew}) over $\f_{2^t}$ from a Boolean function $f:\f_{q}\rightarrow \f_2$, where $q=2^n$ and $\f_{2^t}$ is some subfield of $\f_{q}$. The complete weight enumerator of $\tilde{\C}_{f}$ can be easily determined from the Walsh spectrum of $f$, while the weight distribution of the co…
On quadratic binomial vectorial functions with maximal bent components
Anhui Agricultural University（安徽农业大学）；University of Science and Technology of China（中国科学技术大学）；Hefei National Laboratory（合肥国家实验室）；Hefei University of Technology（合肥工业大学）；State Key Laboratory of Cyberspace Security Defense, Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所网络空间安全防御国家重点实验室）
In 2008, Cusick {\it et al.} conjectured that certain elementary symmetric Boolean functions of the form $σ_{2^{t+1}l-1, 2^t}$ are the only nonlinear balanced ones, where $t$, $l$ are any positive integers, and $σ_{n,d}=\bigoplus_{1\le i_1<...<i_d\le n}x_{i_1}x_{i_2}...x_{i_d}$ for positive integers $n$, $1\le d\le n$. In this note, by analyzing the weight of $σ_{n, 2^t}$ and $σ_{n, d}$, we prove that ${\rm wt}(σ_{n, d})<2^{n-1}$ holds in most cases, and so does the conjecture. According to the remainder of modulo 4, we also consider the weight of $σ_{n, d}$ from two aspects: $n\equiv 3({\rm m…
An algorithm for computing the nonlinearity of a Boolean function from its algebraic normal form (ANF) is proposed. By generalizing the expression of the weight of a Boolean function in terms of its ANF coefficients, a formulation of the distances to linear functions is obtained. The special structure of these distances can be exploited to reduce the task of nonlinearity computation to solving an associated binary integer programming problem. The proposed algorithm can be used in cases where applying the Fast Walsh transform is infeasible, typically when the number of input variables exceeds 4…
Nonlinearity of rotation symmetric Boolean functions is an important topic on cryptography algorithm. Let $e\ge 1$ be any given integer. In this paper, we investigate the following question: Is the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ equal to its weight? We introduce some new simple sub-functions and develop new technique to get several recursive formulas. Then we use these recursive formulas to show that the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ i…
In this paper we obtain the average sensitivity of the laced Boolean functions. This confirms a conjecture of Shparlinski. We also compute the weights of the laced Boolean functions and show that they are almost balanced.
In this work, we employ the concept of {\em composite representation} of Boolean functions, which represents an arbitrary Boolean function as a composition of one Boolean function and one vectorial function, for the purpose of specifying new secondary constructions of bent/plateaued functions. This representation gives a better understanding of the existing secondary constructions and it also allows us to provide a general construction framework of these objects. This framework essentially gives rise to an {\em infinite number} of possibilities to specify such secondary construction methods (w…
Let $f_n(x_1, x_2, \ldots, x_n)$ denote the algebraic normal form (polynomial form) of a rotation symmetric Boolean function of degree $d$ in $n \geq d$ variables and let $wt(f_n)$ denote the Hamming weight of this function. Let $(1, a_2, \ldots, a_d)_n$ denote the function $f_n$ of degree $d$ in $n$ variables generated by the monomial $x_1x_{a_2} \cdots x_{a_d}.$ Such a function $f_n$ is called {\em monomial rotation symmetric} (MRS). It was proved in a $2012$ paper that for any MRS $f_n$ with $d=3,$ the sequence of weights $\{w_k = wt(f_k):~k = 3, 4, \ldots\}$ satisfies a homogeneous linear …
In this paper we find a complete characterization of plateaued Boolean functions in terms of the associated Cayley graphs. Precisely, we show that a Boolean function $f$ is $s$-plateaued (of weight $=2^{(n+s-2)/2}$) if and only if the associated Cayley graph is a complete bipartite graph between the support of $f$ and its complement (hence the graph is strongly regular of parameters $e=0,d=2^{(n+s-2)/2}$). Moreover, a Boolean function $f$ is $s$-plateaued (of weight $\neq 2^{(n+s-2)/2}$) if and only if the associated Cayley graph is strongly $3$-walk-regular (and also strongly $\ell$-walk-regu…
Boolean functions can be used to construct binary linear codes in many ways, and vice versa. The objective of this short article is to point out a connection between the weight distributions of all projective binary linear codes and the Walsh spectra of all Boolean functions. New research problems are also proposed.
Correlation-immune (CI) multi-output Boolean functions have the property of keeping the same output distribution when some input variables are fixed. Recently, a new application of CI functions has appeared in the system of resisting side-channel attacks (SCA). In this paper, three new methods are proposed to characterize the $t$ th-order CI multi-output Boolean functions ($n$-input and $m$-output). The first characterization is to regard the multi-output Boolean functions as the corresponding generalized Boolean functions. It is shown that a generalized Boolean functions $f_g$ is a $t$ th-ord…
We identify the weights $wt(f_n)$ of a family $\{f_n\}$ of rotation symmetric Boolean functions with the cardinalities of the sets of $n$-periodic points of a finite-type shift, recovering the second author's result that said weights satisfy a linear recurrence. Similarly, the weights of idempotent functions $f_n$ defined on finite fields can be recovered as the cardinalities of curves over those fields and hence satisfy a linear recurrence as a consequence of the rationality of curves' zeta functions. Weil's Riemann hypothesis for curves then provides additional information about $wt(f_n)$. W…
In this paper we define the (edge-weighted) Cayley graph associated to a generalized Boolean function, introduce a notion of strong regularity and give several of its properties. We show some connections between this concept and generalized bent functions (gbent), that is, functions with flat Walsh-Hadamard spectrum. In particular, we find a complete characterization of quartic gbent functions in terms of the strong regularity of their associated Cayley graph.
Let $f_n(x_0, x_1, \ldots, x_{n-1})$ denote the algebraic normal form (polynomial form) of a rotation symmetric (RS) Boolean function of degree $d$ in $n \geq d$ variables and let $wt(f_n)$ denote the Hamming weight of this function. Let $(0, a_1, \ldots, a_{d-1})_n$ denote the function $f_n$ of degree $d$ in $n$ variables generated by the monomial $x_0x_{a_1} \cdots x_{a_{d-1}}.$ Such a function $f_n$ is called monomial rotation symmetric (MRS). It was proved in a $2012$ paper that for any MRS $f_n$ with $d=3,$ the sequence of weights $\{w_k = wt(f_k):~k = 3, 4, \ldots\}$ satisfies a homogene…
This paper investigates the impact of puncturing on finite-length polar codes in which a puncturing pattern $\pv^{N}=(p_0,...,p_N)$ is applied to a length-$N$ polar code.. We first introduce two virtual channels to stochastically model the punctured (untransmitted) bits, which are respectively called {\em useless channel model} (UCM) and {\em deterministic channel model} (DCM). Under each model, we derive boolean functions in variables $p_0,...,p_{N-1}$ that can indicate which polarized channels should carry frozen bits. Based on this, we present an efficient method to jointly optimize a punct…
In this paper, we study a class of linear codes defined by characteristic functions of certain subsets of a finite field. We derive a sufficient and necessary condition for such a code to be a minimal linear code by a character-theoretical approach. We obtain new three-weight or four-weight minimal linear codes that do not satisfy the Ashikhmin-Barg condition by using partial difference sets. We show that our construction yields minimal linear codes that do not arise from cutting vectorial blocking sets, and also discuss their applications in secret sharing schemes.
