Nonlinear Boolean Functions Research Articles

On the nonlinearity of Boolean functions with restricted input

Very recently, Carlet, Meaux and Rotella have studied the main cryptographic features of Boolean functions when, for a given number n of variables, the input to these functions is restricted to some subset E of $\mathbb {F}_{2}^{n}$. Their study includes the particular case when E equals the set of vectors of fixed Hamming weight, which is important in the robustness of the Boolean function involved in the FLIP stream cipher. In this paper we focus on the nonlinearity of Boolean functions with restricted input and present new results related to the analysis of this nonlinearity improving the upper bound given by Carlet et al.

Reliable computation from contextual correlations

An operational approach to the study of computation based on correlations considers black-boxes with one-bit inputs and outputs, controlled by a limited classical computer capable only of performing sums modulo-2. In this setting, it was shown that non-contextual correlations do not provide any extra computational power, while contextual correlations were found to be necessary for the deterministic evaluation of non-linear Boolean functions. Here we investigate the requirements for reliable computation in this setting, that is, the evaluation of any Boolean function with success probability bounded away from 1/2. We show that bipartite CHSH quantum correlations suffice for reliable computation. We also prove that an arbitrarily small violation of a multipartite GHZ non-contextuality inequality also suffices for reliable computation.

Results on highly nonlinear Boolean functions with provably good immunity to fast algebraic attacks

In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSR-based stream ciphers. Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been researched in a large number of papers. However, there are few results with respect to Boolean functions with provably good immunity against fast algebraic attacks. In previous literatures, only Carlet–Feng function was proven to have good immunity to fast algebraic attacks.In this paper, we first study a large family of highly nonlinear Boolean functions in terms of the immunity to fast algebraic attacks, which includes the functions of Tu–Deng, the functions of Tang et al. and the functions of Jin et al. Based on a sufficient and necessary condition for measuring the immunity of Boolean functions against fast algebraic attacks using bivariate polynomial representation, we propose an efficient method for estimating the immunity of the functions of such family. Then we prove that a family of 2k-variable Boolean functions, including the function recently constructed by Tang et al., are almost perfect algebraic immune for any integer k ≥ 3. More exactly, they achieve optimal algebraic immunity and almost perfect immunity to fast algebraic attacks. The functions of such family are balanced and have optimal algebraic degree. Besides, we prove a lower bound on their nonlinearity based on the work of Tang et al. which is better than that of Carlet–Feng function. It is also checked for 3 ≤ k ≤ 9 that the exact nonlinearity of such functions is very good, which is slightly smaller than that of Carlet–Feng function, and some functions of this family even have a slightly larger nonlinearity than Tang’s et al. function. To sum up, among the known functions with provably good immunity against fast algebraic attacks, the functions of this family make a trade-off between the exact value and the lower bound of nonlinearity.

Efficient probabilistic algorithm for estimating the algebraic properties of Boolean functions for large n

Although several methods for estimating the resistance of a random Boolean function against (fast) algebraic attacks were proposed, these methods are usually infeasible in practice for relatively large number of input variables n (for instance n ≥ 30) due to increased computational complexity. An efficient estimation of the resistance of Boolean functions, with relatively large number of inputs n, against (fast) algebraic attacks appears to be a rather difficult task. In this paper, the concept of partial linear relations decomposition is introduced, which decomposes any given nonlinear Boolean function into many linear (affine) subfunctions by using the disjoint sets of input variables. Based on this result, a general probabilistic decomposition algorithm for nonlinear Boolean functions is presented which gives a new framework for estimating the resistance of Boolean function against (fast) algebraic attacks. It is shown that our new probabilistic method gives very tight estimates (lower and upper bound) and it only requires about O(n22n) operations for a random Boolean function with n variables, thus having much less time complexity than previously known algorithms.

Construction of Highly Nonlinear 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Provably High Fast Algebraic Immunity

In 2013, Tang, Carlet, and Tang [IEEE TIT 59(1): 653–664, 2013] presented two classes of Boolean functions. The functions in the first class are unbalanced and the functions in the second one are balanced. Both of those two classes of functions have high nonlinearity, high algebraic degree, optimal algebraic immunity, and high fast algebraic immunity. However, they are not 1-resilient which represents a drawback for their use as filter functions in stream ciphers. In this paper, we first propose a large family of 1-resilient Boolean functions having high lower bound on nonlinearity, optimal algebraic immunity, and optimal algebraic degree, that is, meeting the Siegenthaler bound. Most notably, we can mathematically prove that every function in $n$ variables belonging to this family has fast algebraic immunity no less than $n-6$ , which is the first time that an infinite family of 1-resilient functions with provably high fast algebraic immunity has been invented. Furthermore, we exhibit a subclass of the family which has higher lower bound on nonlinearity than all the known 1-resilient functions with (potentially) optimal algebraic immunity and potentially high fast algebraic immunity.

An algorithm and architecture for non-recursive pseudorandom sequence generation using sequence folding technique

In this paper, a new algorithm along with its first order hardware architecture is proposed to build the pseudorandom non-recursive binary sequence using sequence folding technique to protect the confidential information and data integrity in cryptography applications. Unlike the existing techniques to build the pseudorandom recursive sequence. The novelty of this proposed algorithm along with first order architecture is to generate non-recursive nibble of thirty one (O) from the input seed of four () bits. In the sequence folding technique, an arbitrary linear recursive binary sequence is folded to a minimum state by an empirical approach to build the non-linear Boolean function and realized using algebraic normal form. In this proposed first order architecture, the constructed non-linear Boolean functions are utilized to engender the pseudorandom binary non-recursive sequence. The proposed first order architecture is implemented using Verilog HDL and further prototyped on commercially available FPGA device ‘Spartan 3E xc3s500e-4fg320’. The realization of proposed architecture in this FPGA device accomplishes an improved data throughput, and efficiency (i.e. throughput/area) is compared to existing techniques. The generated binary sequence from the experiment is further analyzed briefly for sequence size, number of stages, correlation and verified for randomness by using NIST statistical test suites.

On Relevance of Nonlinearity, Derivative, and E-Derivative for Boolean Functions

In this paper, we discuss the impact of weight structure of the derivative and e-derivative on the nonlinearity of Boolean functions, and the relationship among the nonlinearity, derivative, and e-derivative of Boolean functions. And get six kinds of relations among nonlinearity, derivative, and e-derivative of Boolean functions. The methods to solve the nonlinearity are derived based on those relationships, thus solving the problem that it’s difficult to calculate the nonlinearity only in accordance with the definition.

Bent functions linear on elements of some classical spreads and presemifields spreads

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They have attracted a lot of research for four decades because of their own sake as interesting combinatorial objects, and also because of their relations to coding theory, sequences and their applications in cryptography and other domains such as design theory. In this paper we investigate explicit constructions of bent functions which are linear on elements of spreads. After presenting an overview on this topic, we study bent functions which are linear on elements of presemifield spreads and give explicit descriptions of such functions for known commutative presemifields. A direct connection between bent functions which are linear on elements of the Desarguesian spread and oval polynomials over finite fields was proved by Carlet and the second author. Very recently, further nice extensions have been made by Carlet in another context. We introduce oval polynomials for semifields which are dual to symplectic semifields. In particular, it is shown that from a linear oval polynomial for a semifield one can get an oval polynomial for transposed semifield.

A comparison of Carlet's second-order nonlinearity bounds

ABSTRACTCarlet provides two bounds on the second-order nonlinearity of Boolean functions. We construct a family of Boolean functions where the first bound (the presumed weaker bound) is tight and the second bound is strictly worse than the first bound. We show that the difference between the two bounds can be made arbitrarily large.

The Nonlinearity of Sum and Product for Boolean Functions

In this paper, we study the relationship between the nonlinearity of Boolean function and the nonlinearity of the sum and product of Boolean function, while derivative and e-derivative are used to study the problem further. We obtain that the sum of two functions’ nonlinearity is not less than the nonlinearity of the sum of two functions. The relationship between the nonlinearity of function and the nonlinearity of the sum and product of two functions are also obtained. Furthermore, we also get the relationship between the nonlinearity of the product of functions, and the derivative and e-derivative of function. Moreover, we also deduced some important applications on the basis of the above work.

Linear Codes From Some 2-Designs

A classical method of constructing a linear code over $\gf(q)$ with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over $\gf(q)$ of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of $2$-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of $2$-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented.

Higher Order Nonlinearity of Niho Functions

The r-th order nonlinearity of Boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers. It is also very useful in coding theory, since it is related to the covering radii of Reed-Muller codes. In this paper we investigate the lower bound of the higher-order nonlinearity of Niho Boolean functions fx = trλxd over F2n, where $\lambda \isin F_2_n^*$, $d = 2^{n-1/2} + 2^{n-1/4}-1$ when n = 1 mod 4 or $d = 2^{n-1/2} + 2^{3n-1/4}-1$ when n = 3 mod 4.

Secondary constructions of highly nonlinear Boolean functions and disjoint spectra plateaued functions

In this paper, we modify a generalized indirect sum construction to construct functions with high nonlinearity. By utilizing the modified construction, highly nonlinear functions in (n+m) variables can be obtained from known bent functions in n variables and highly nonlinear functions in m variables. It is possible to obtain new (n+15)-variable functions with nonlinearity 2n+15-1-2(n+15-1)/2+20×2n/2 and new 12-variable 2-resilient functions with nonlinearity 2000 and algebraic degree 8, which achieve optimal algebraic immunity. Moreover, the modified construction can also be used as an iterative construction of a quadruple of disjoint spectra plateaued functions. In addition, we present sufficient conditions for a quadruple of disjoint spectra plateaued functions to have no nonzero linear structure.

Research on Design and Construction Method of Bent Functions

Bent function is a class of the highest nonlinear Boolean functions. In this paper three methods of design and construction are discussed with examples, which are algebra method, the character function in projective geometry and random researching method. Also, the Bent function of class is implemented with C language. At last, the concatenate construction from m = 2n-k Bent functions of k variables to a Bent function of n variables is given and verified with Walsh spectra.

On Third-Order Nonlinearity of Biquadratic Monomial Boolean Functions

The rth-order nonlinearity of Boolean function plays a central role against several known attacks on stream and block ciphers. Because of the fact that its maximum equals the covering radius of the rth-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the rth-order nonlinearity of a Boolean function is very complicated problem, especially when r>1. This paper is concerned with the computation of the lower bounds for third-order nonlinearities of two classes of Boolean functions of the form Tr1nλxd for all x∈𝔽2n, λ∈𝔽2n*, where a d=2i+2j+2k+1, where i, j, and k are integers such that i>j>k≥1 and n>2i, and b d=23ℓ+22ℓ+2ℓ+1, where ℓ is a positive integer such that gcdℓ,𝓃=1 and n>6.

On second-order nonlinearity and maximum algebraic immunity of some bent functions in $\mathcal{PS}^{+}$

The rth-order nonlinearity and algebraic immunity of Boolean function play a central role against several known attacks on stream and block ciphers. Since its maximum equals the covering radius of the rth-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the rth-order nonlinearity of a Boolean function is very complected/challenging problem, especially when r>1. In this article, we identify a subclass of \({\mathcal{D}}_{0}\) type bent functions constructed by modifying well known Dillon functions having sharper bound on their second-order nonlinearity. We further, identify a subclass of bent functions in \({\mathcal {PS}}^{+}\) class with maximum possible algebraic immunity. The result is proved by using the well known conjecture proposed by Tu and Deng (Des. Codes Cryptogr. 60(1):1–14, 2011). To obtain rth-order nonlinearity (r>2), that is, whole nonlinearity profile of the constructed bent functions is still an open problem.

A Survey on Nonlinear Boolean Functions with Optimal Algebraic Immunity Suitable for Stream Ciphers

This paper is devoted to Boolean functions for cryptography; more precisely those which can be used for filtering linear feedback shift registers, in the pseudo-random generators of stream ciphers, for allowing resistance to the main cryptanalyses (algebraic, fast algebraic, Berlekamp–Massey, Ronjom–Helleseth and fast correlation attacks).

On a method of derivation of lower bounds for the nonlinearity of Boolean functions

The calculation of the exact value of the rth order nonlinearity of a Boolean function (the power of the distance between the function and the set of functions is at most r) or the derivation of a lower bound for it is a complicated problem (especially for r > 1). Lower bounds for nonlinearities of different orders in terms of the value of algebraic immunity were obtained in a number of papers. These estimates turn out to be sufficiently strong if the value of algebraic immunity is maximum or close to maximum. In the present paper, we prove a statement that allows us to obtain fairly strong lower bounds for nonlinearities of different orders and for many functions with low algebraic immunity.

Highly Nonlinear Boolean Functions With Optimal Algebraic Immunity and Good Behavior Against Fast Algebraic Attacks

Inspired by the previous work of Tu and Deng, we propose two infinite classes of Boolean functions of 2k variables where k ≥ 2. The first class contains unbalanced functions having high algebraic degree and nonlinearity. The functions in the second one are balanced and have maximal algebraic degree and high nonlinearity (as shown by a lower bound that we prove; as a byproduct we also prove a better lower bound on the nonlinearity of the Carlet-Feng function). Thanks to a combinatorial fact, first conjectured by the authors and later proved by Cohen and Flori, we are able to show that they both possess optimal algebraic immunity. It is also checked that, at least for numbers of variables n ≤ 16, functions in both classes have a good behavior against fast algebraic attacks. Compared with the known Boolean functions resisting algebraic attacks and fast algebraic attacks, both of them possess the highest lower bounds on nonlinearity. These bounds are however not enough for ensuring a sufficient nonlinearity for allowing resistance to fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for bounded numbers of variables (n ≤ 38). Moreover, these values are very good. The infinite class of functions we propose in Construction 2 presents, among all currently known constructions, the best provable tradeoff between all the important cryptographic criteria.

On the Construction of Cryptographically Significant Boolean Functions Using Objects in Projective Geometry Spaces

Recently, several construction methods of highly nonlinear Boolean functions with relatively good algebraic properties were proposed. These approaches manage in optimizing most of the relevant cryptographic criteria, but not all of them at the same time. Usually, either the nonlinearity bounds are rather loose (though the actual nonlinearity is relatively high) or the functions do not provide a good resistance to fast algebraic cryptanalysis. In this paper, we develop a theoretical framework for using objects in suitable projective geometry spaces for construction of highly nonlinear Boolean functions. This allows us to establish tight bounds on the nonlinearity using simple counting arguments, thus avoiding rather complicated estimates of certain trace sums. Our method generates a class of almost fully optimized functions, that is the functions apart from very high nonlinearity also have the maximum algebraic degree and optimal algebraic immunity. Compared to the classes of functions proposed by Carlet and Feng, Wang , and Zeng , our functions achieve a slightly better nonlinearity which is traded-off against a little worse resistance against fast algebraic attacks. On the other hand, compared to the functions by Tang and Tu and Deng, our nonlinearity is somewhat lower, but the algebraic properties are slightly better.