New Constructions of Resilient and Correlation Immune Boolean Functions Achieving Upper Bound on Nonlinearity

In this paper, the discrete Fourier transform (DFT) of Boolean functions over the complex field is introduced and the locations of zero-valued Fourier spectrum are studied. Then a Fourier spectral characterization of correlation immune and resilient Boolean functions is investigated. It is shown that a Boolean function $f$ is $m$ th-order correlation immune if and only if the Fourier spectrum of $f$ under any permutation of variables (or the equivalence class of $f$ defined by Golomb in 1959) vanishes at a specified location. This is an analog of using Walsh-Hadamard spectra to characterize correlation immunity of the Boolean functions. In particular, if $f$ is a symmetric function, $f$ is correlation immune if and only if its Fourier spectrum vanishes at a specified location. Similarly, zero-valued Fourier spectrum can also be used to characterize resilient functions. The application of the Fourier spectral analysis on studying the peak-to-mean envelope power ratio of the sequences is also addressed.

International audience

The correlation immunity of Boolean functions is a property related to cryptography, to error correcting codes, to orthogonal arrays (in combinatorics), and in a slightly looser way to sequences. Correlation-immune Boolean functions (in short, CI functions) have the property of keeping the same output distribution when some input variables are fixed. They have been widely used as combiners in stream ciphers to allow resistance to the Siegenthaler correlation attack. Very recently, a new use of CI functions has appeared in the framework of side channel attacks (SCA). To reduce the cost overhead of counter-measures to SCA, CI functions need to have low Hamming weights. This actually poses new challenges since the known constructions which are based on properties of the Walsh-Hadamard transform, do not allow to build unbalanced CI functions. In this paper, we propose constructions of low-weight d th-order CI functions based on the Fourier-Hadamard transform, while the known constructions of resilient functions are based on the Walsh-Hadamard transform. These two transforms are closely related but the resulting constructions are very different. We first prove a simple but powerful result, which makes that one only need to consider the case where d is odd in further research. Then, we investigate how constructing low Hamming weight CI functions through the Fourier-Hadamard transform (which behaves well with respect to the multiplication of Boolean functions). We use the characterization of CI functions by the Fourier-Hadamard transform and introduce a related general construction of CI functions by multiplication. By using the Kronecker product of vectors, we obtain more constructions of low-weight d-CI Boolean functions. Furthermore, we present a method to construct low-weight d-CI Boolean functions by making additional restrictions on the supports built from the Kronecker product.

Chapter 4 - Correlation Immune and Resilient Boolean Functions

We first give a survey of the known secondary constructions of Boolean functions, permitting to obtain resilient functions achieving the best possible trade-offs between resiliency order, algebraic degree and nonlinearity (that is, achieving Siegenthaler’s bound and Sarkar et al.’s bound). We introduce then, and we study, a general secondary construction of Boolean functions. This construction includes as particular cases the known secondary constructions previously recalled. We apply this construction to design more numerous functions achieving optimum trade-offs between the three characteristics (and additionally having no linear structure). We conclude the paper by indicating generalizations of our construction to Boolean and vectorial functions, and by relating it to a known secondary construction of bent functions.

We observe a property on Boolean functions which explains how work some secondary constructions recently obtained for Boolean bent functions. It leads to a generalization and to a unification of these constructions. It also permits to design highly nonlinear resilient functions from known ones. This construction does not increase the number of variables, contrary to the known general secondary constructions, and it permits to improve some cryptographic characters of the functions (e.g. their algebraic immunity) while keeping good the other characteristics

We extend the notions of correlation-immune functions and resilient functions to functions over any finite alphabet. A previous result due to Gopalakrishnan and Stinson is generalized as we give an orthogonal array characterization, a Fourier transform and a matrix characterization for correlation-immune and resilient functions over any finite alphabet endowed with the structure of an Abelian group. We then point out the existence of a tradeoff between the degree of the algebraic normal form and the correlation-immunity order of any function defined on a finite field and we construct some infinite families of t-resilient functions with optimal nonlinearity which are particularly well-suited for combining linear feedback shift registers. We also point out the link between correlation-immune functions and some cryptographic objects as perfect local randomizers and multipermutations.

We explore three applications of geometric sequences in constructing cryptographic Boolean functions. First, we construct 1-resilient functions of n Boolean variables with nonlinearity 2n-1-2(n-1)/2, n odd. The Hadamard transform of these functions is 3-valued, which limits the efficiency of certain stream cipher attacks. From the case for n odd, we construct highly nonlinear 1-resilient functions which disprove a conjecture of Pasalic and Johansson for n even. Our constructions do not have a potential weakness shared by resilient functions which are formed from concatenation of linear functions. Second, we give a new construction for balanced Boolean functions with high nonlinearity, exceeding 2n-1-2(n-1)/2, which is not based on the direct sum construction. Moreover, these functions have high algebraic degree and large linear span. Third, we construct balanced vectorial Boolean functions with nonlinearity 2n-1-2(n-1)/2 and low maximum correlation. They can be used as nonlinear combiners for stream cipher systems with high throughput.

Siegenthaler proved that an n input 1 output, m-resilient (balanced mth order correlation immune) Boolean function with algebraic degree d satisfies the inequality: m + d ≤ n − 1. We provide a new construction method using a small set of recursive operations for a large class of highly nonlinear, resilient Boolean functions optimizing Siegenthaler’s inequality m + d = n − 1. Comparisons to previous constructions show that better nonlinearity can be obtained by our method. In particular, we show that as n increases, for almost all m, the nonlinearity obtained by our method is better than that provided by Seberry et al in Eurocrypt’93. For small values of n, the functions constructed by our method is better than or at least comparable to those constructed using the methods provided in papers by Filiol et al and Millan et al in Eurocrypt’98. Our technique can be used to construct functions on large number of input variables with simple hardware implementation.KeywordsStream CipherBoolean FunctionAlgebraic DegreeCorrelation ImmunityNonlinearityBalancedness

Algebraic immunity is an important cryptographic property for Boolean functions against algebraic attacks. Constructions of Boolean functions with the maximum algebraic immunity (MAI Boolean functions) by using univariate polynomial representation of Boolean functions over finite fields have received more and more attention. In this paper, how to obtain more MAI Boolean functions from a known MAI Boolean function under univariate polynomial representation is further investigated. The sufficient condition of Boolean functions having the maximum algebraic immunity obtained by changing a known MAI Boolean function under univariate polynomial representation is given. With this condition, more balanced MAI Boolean functions under univariate polynomial representation can be obtained. The algebraic degree and the nonlinearity of these Boolean functions are analyzed.

Boolean functions exhibiting strong cryptographic characteristics are essential crypto primitive to be included in the design of secure cryptosystems. It is not only common to incorporate Boolean functions in the design of symmetric block and stream ciphers but they also have a key place in the design of cryptographic hash functions. Strong Boolean functions make the system secure and resistant against cryptanalytic attacks. A wide range of approaches have been adopted in the discovery of Boolean functions that excel in terms of several cryptographic characteristics. In this paper, we present a new scheme based on Genetic Algorithm to generate Boolean functions which satisfy balancedness, correlation immunity, algebraic degree and nonlinearity characteristics. Proposed scheme generates strong Boolean functions with desired values of these characteristics.

We go deep into the internal structure of the Boolean functions values, and study the relationship of algebraic immunity and algebraic degree of Boolean functions with the Hamming weight with the diffusion included. Then we get some theorems which relevance together algebraic immunity, annihilators and algebraic degree of H Boolean functions by the e-derivative which is a part of the H Boolean function. Besides, we also get the results that algebraic immunity and algebraic degree of Boolean functions can be linked together by the e-derivative of H Boolean functions and so on.

The cryptographic primitives such as Boolean function and S-Box are used as a building block to design stream cipher and block cipher, respectively. However, it is important to evaluate the cryptographic strength to measure the security of such crytposystem resistance against important cryptanalytic attacks. We develop a new tool called Boolean function and S-Box analysis tool (BSAT) to evaluate the cryptographic strength of the Boolean function and S-Box. As the size of S-Box increases, it takes a lot of computational effort to evaluate all the required cryptographic properties. In this paper, we develop a tool to evaluate important cryptographic properties of both Boolean function and S-Box to minimize the computational time. The computational time of BSAT is lesser than the SET tool in terms of calculating the algebraic normal form and algebraic degree. Our BSAT takes only 0.404 s to evaluate the AES properties faster than the SET, which takes 650 millisecond (0.65 s).

A Boolean function is called correlation immune if every input is independent of the output when the inputs are chosen from a uniform distribution. We show how an asymptotic formula of Denisov, which approximately counts the n-variable correlation immune functions, can be improved so as to be accurate even for fairly small n. We also provide error estimates for our approximations, which Denisov did not do, and show a connection to the behavior of random walks. Such information is useful to designers of machine learning algorithms and stream ciphers.

Boolean functions with maximum algebraic immunity have been considered as one class of cryptographically significant functions. It is known that Boolean functions on odd variables have maximum algebraic immunity if and only if a correlative matrix has column full rank, and Boolean functions on even variables have maximum algebraic immunity if and only if two correlative matrices have column full rank. Recently, a smaller matrix was used in the odd case. We find that one or two smaller matrices can be used in the even case and consequently present several sufficient and necessary conditions for Boolean functions with maximum algebraic immunity. This result advances the ability to identify whether Boolean functions on even variables achieve maximum algebraic immunity. We also present a construction algorithm for n-variable Boolean functions with maximum algebraic immunity, specially with the Hamming weights of \( \sum {_{i = 0}^{\left\lceil {\frac{n} {2}} \right\rceil - 1} } \left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right) \). It is easily realized for not too large n and helps construct balanced Boolean functions with maximum algebraic immunity on even variables. Furthermore, we present a sufficient and necessary condition for balanced Boolean functions to achieve maximum algebraic immunity and optimum algebraic degree, and modify the construction algorithm to construct Boolean functions on odd variables with maximum algebraic immunity, optimum algebraic degree and high nonlinearity.