Classes Of Balanced Boolean Functions Research Articles

SAO 1-Resilient Functions With Lower Absolute Indicator in Even Variables

In 2018, Tang and Maitra presented a class of balanced Boolean functions in $n$ variables with the absolute indicator $\Delta _{f} and the nonlinearity $NL(f)> 2^{n-1}-2^{n/2}$ , that is, $f$ is SAO (strictly almost optimal), for $n=2k\equiv 2\,\,({\mathrm {mod}\,\,}4)$ and $n\geq 46$ in [IEEE Ttans. Inf. Theory 64 (1) : 393-402, 2018]. However, there is no evidence to show that the absolute indicator of any 1-resilient function in $n$ variables can be strictly less than $2^{\lfloor ({n+1})/{2}\rfloor }$ , and the previously best known upper bound of which is $5\cdot 2^{n/2}-2^{n/4+2}+4$ . In this paper, we concentrate on two directions. Firstly, to complete Tang and Maitra’s work for $k$ being even, we present another class of balanced functions in $n$ variables with the absolute indicator $\Delta _{f} and the nonlinearity $NL(f)> 2^{n-1}-2^{n/2}$ for $n\equiv 0~({\mathrm {mod}~}4)$ and $n\geq 48$ . Secondly, we obtain two new classes of 1-resilient functions possessing very high nonlinearity and very low absolute indicator, from bent functions and plateaued functions, respectively. Moreover, one class of them achieves the currently known highest nonlinearity $2^{n-1}-2^{n/2-1}-2^{n/4}$ , and the absolute indicator of which is upper bounded by $2^{n/2}+2^{n/4+1}$ that is a new upper bound of the minimum of absolute indicator of 1-resilient functions, as it is clearly optimal than the previously best known upper bound $5\cdot 2^{n/2}-2^{n/4+2}+4$ .

Constructing Two Classes of Boolean Functions With Good Cryptographic Properties

Wu et al. proposed a generalized Tu-Deng conjecture over $\mathbb {F}_{2^{rm}}\times {\mathbb {F}_{2^{m}}}$ , and constructed Boolean functions with good properties. However the proof of the generalized conjecture is still open. Based on Wu’s work and assuming that the conjecture is true, we come up with a new class of balanced Boolean functions which has optimal algebraic degree, high nonlinearity and optimal algebraic immunity. The Boolean function also behaves well against fast algebraic attacks. Meanwhile we construct another class of Boolean functions by concatenation, which is 1-resilient and also has other good cryptographic properties.

A characterization of balanced Boolean functions with optimal algebraic immunity

In this paper, we characterize balanced Boolean functions with optimal algebraic immunity by Schur functions. By applying this characterization, three classes of balanced Boolean functions with optimal algebraic immunity are constructed. Some examples of the constructed functions with other desired cryptographic properties are also presented.

A combinatorial condition and Boolean functions with optimal algebraic immunity

This paper first proposes an infinite class of 2k-variable Boolean functions with high non- linearity and high algebraic degree. Then an infinite class of balanced Boolean functions are proposed by modifying the above Boolean functions. This class of balanced Boolean functions have optimal al- gebraic degree and high nonlinearity. Both classes have optimal algebraic immunity based on a general combinatorial conjecture.

CONSTRUCTING 2m-VARIABLE BOOLEAN FUNCTIONSWITH OPTIMAL ALGEBRAIC IMMUNITY BASED ON POLAR DECOMPOSITION OF $\mathbb{F}^\ast_{2^{2m}}$

Constructing 2m-variable Boolean functions with optimal algebraic immunity based on decomposition of additive group of the finite field [Formula: see text] seems to be a promising approach since Tu and Deng's work. In this paper, we consider the same problem in a new way. Based on polar decomposition of the multiplicative group of [Formula: see text], we propose a new construction of Boolean functions with optimal algebraic immunity. By a slight modification of it, we obtain a class of balanced Boolean functions achieving optimal algebraic immunity, which also have optimal algebraic degree and high nonlinearity. Computer investigations imply that this class of functions also behaves well against fast algebraic attacks.

Hybrid classes of balanced Boolean functions with good cryptographic properties

Cryptographically strong Boolean functions play an imperative role in the design of almost every modern symmetric cipher. In this context, the cryptographic properties of Boolean functions, such as non-linearity, algebraic degree, correlation immunity and propagation criteria, are critically considered in the process of designing these ciphers. More recently, with the emergence of algebraic and fast algebraic attacks, algebraic immunity has also been included as an integral property to be considered. As a result, several constructions of Boolean functions with high non-linearity, maximal algebraic degree and optimal algebraic immunity have been devised since then. This paper focuses on some of these constructions and presents two hybrid classes of Boolean functions. The functions constructed within these classes possess maximal algebraic degree for balanced functions, optimal algebraic immunity, high non-linearity and good resistance to algebraic and fast algebraic attacks. A hybrid class of 1-resilient functions has also been proposed that also possesses high algebraic degree, optimal algebraic immunity, high non-linearity and good resistance to algebraic and fast algebraic attacks.

The Judgment of Algebraic Immunity and Resilience of Balanced H Boolean Functions

Using the derivative of the Boolean function and the e-derivative defined by ourselves as main research tools, we study the relationship among e-derivative, algebraic immunity and resilience of balanced H Boolean functions.We get some theorems which connect algebraic immunity, annihilators, resilience, derivative and e-derivative of balanced H Boolean functions together. Besides, we also get the judgment of algebraic immunity and resilience for three classes of balanced Boolean functions by the e-derivative.

On the distribution of auto-correlation value of balanced Boolean functions

In this paper, we study the lower bound on the sum-of-square indicator of balanced Boolean functions obtained by Son, et al. in 1998, and give a sufficient and necessary condition under which balanced Boolean functions achieve this lower bound. We introduce a new general class of balanced Boolean functions in $n$ variables $(n\geq 4)$ with optimal auto-correlation distribution, and we study two sub-classes more explicitely. Finally, we study the sets of Boolean functions having a same auto-correlation distribution, and derive a lower bound on the number of elements in such set.

Several Classes of Even-Variable Balanced Boolean Functions with Optimal Algebraic Immunity

In this paper, we constructed six infinite classes of balanced Boolean functions. These six classes of Boolean functions achieved optimal algebraic degree, optimal algebraic immunity and high nonlinearity. Furthermore, we gave the proof of the lower bound of the nonlinearities of these balanced Boolean functions and proved the better lower bound of nonlinearity for Carlet-Feng's Boolean function.

On the security of the Feng–Liao–Yang Boolean functions with optimal algebraic immunity against fast algebraic attacks

In the past few years, algebraic attacks against stream ciphers with linear feedback function have been significantly improved. As a response to the new attacks, the notion of algebraic immunity of a Boolean function f was introduced, defined as the minimum degree of the annihilators of f and f + 1. An annihilator of f is a nonzero Boolean function g, such that fg = 0. There is an increasing interest in construction of Boolean functions that possess optimal algebraic immunity, combined with other characteristics, like balancedness, high nonlinearity, and high algebraic degree. In this paper, we investigate a recently proposed infinite class of balanced Boolean functions with optimal algebraic immunity, optimum algebraic degree and much better nonlinearity than all the previously introduced classes of Boolean functions with maximal algebraic immunity. More precisely, we study the resistance of the functions against one of the new algebraic attacks, namely the fast algebraic attacks (FAAs). Using the special characteristics of the family members, we introduce an efficient method for the evaluation of their behavior against these attacks. The new algorithm is based on the well studied Berlekamp---Massey algorithm.

Further properties of several classes of Boolean functions with optimum algebraic immunity

Based on a method proposed by the first author, several classes of balanced Boolean functions with optimum algebraic immunity are constructed, and they have nonlinearities significantly larger than the previously best known nonlinearity of functions with optimal algebraic immunity. By choosing suitable parameters, the constructed n-variable functions have nonlinearity \({2^{n-1}-{n-1\choose\frac{n}{2}-1}+2{n-2\choose\frac{n}{2}-2}\Big/(n-2)}\) for even \({n\geq 8\,{\rm and}\,2^{n-1}-{n-1\choose\frac{n-1}{2}}+\Delta(n)}\) for odd n, where Δ(n) is a function increasing rapidly with n. The algebraic degrees of some constructed functions are also discussed.

On the construction of cryptographically strong boolean functions with desirable trade-off

This paper proposes a practical algorithm for systematically generating strong Boolean functions (f:GF(2) n →GF(2)) with cryptographic meaning. This algorithm takes bent function as input and directly outputs the resulted Boolean function in terms of truth table sequence. This algorithm was used to develop two classes of balanced Boolean functions, one of which has very good cryptographic properties: nl(f)=2 2k−1 −2 k +2 k−2 (n=2k), with the sum-of-squares avalanche characteristic of f satisfying σf=2 4k +2 3k+2 +2 3k +2 3k−2 and the absolute avalanche characteristic of ∆f satisfying ∆f=2 k+1 . This is the best result up to now compared to existing ones. Instead of bent sequences, starting from random Boolean functions was also tested in the algorithm. Experimental results showed that starting from bent sequences is highly superior to starting from random Boolean functions.