RETRACTED CHAPTER: On the Minimum the Sum-of-Squares Indicator of a Balanced Boolean Function

Boolean functions possessing multiple cryptographic criteria play an important role in the design of symmetric cryptosystems. The following criteria for cryptographic Boolean functions are often considered: high nonlinearity, balancedness, strict avalanche criterion, and global avalanche characteristics. The trade-off among these criteria is a difficult problem and has attracted many researchers. In this paper, two construction methods are provided to obtain balanced Boolean functions with high nonlinearity. Besides, the constructed functions satisfy strict avalanche criterion and have good global avalanche characteristics property. The algebraic immunity of the constructed functions is also considered.

Nonlinearity, local and global avalanche characteristics of balanced Boolean functions

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.

Substitution boxes (S-boxes) are often used as the most important nonlinear components in many symmetric encryption algorithms. The cryptographic properties of an S-box directly affect the security of the whole cipher system. Recently, generalized global avalanche characteristics (GGAC) were introduced to measure the correlation between two arbitrary Boolean functions. In this paper, to better evaluate the security of an S-box, we present two cross-correlation indicators for it. In addition, by studying the related properties of the cross-correlation between two balanced Boolean functions, we propose the lower bounds on the sum-of-squares indicator related to GGAC for two balanced functions and also for an S-box.

Global avalanche characteristics and propagation criterion of balanced Boolean functions

Boolean functions are used as nonlinear filter functions and combiner functions in several stream ciphers. The security of these stream ciphers largely depends upon cryptographic properties of Boolean functions. Finding a balanced Boolean function with optimal cryptographic properties is an open research problem in the cryptographic community. Since the number of n-variable Boolean functions is $$2^{2^n}$$ , it is not computationally feasible to search the entire space of such functions for cryptographically significant functions when $$n \ge 6$$ . In general, the construction of Boolean functions with optimal or near optimal cryptographically significant properties is formulated as combinatorial optimization problems. In this paper, we apply the Genetic algorithm with the integration of a local search procedure called hybrid GA (HGA) searching for the Boolean functions with high nonlinearity and low autocorrelation. The performance of the Genetic algorithm depends upon the tuning parameters such as crossover mechanism, mutation probability, selection criteria, and the choice of fitness/cost function. To achieve an optimal trade-off among two cryptographic properties, the choice of the cost function plays an important role for finding an optimal solution. So our main focus is to construct balanced Boolean functions using HGA with a new cost function and compare it with existing cost functions. Our experimental results shows that the HGA is more efficient than GA and the new cost function is more efficient than existing cost functions for reaching the optimal solutions.

We prove that for any decision tree calculating a Boolean function f : {-1,1}/sup n/ /spl rarr/ {-1, 1}, Var[f] /spl les/ /spl Sigma/ /sub i=1/ /sup n/ /spl delta//sup i/Inf/sub i/(f), i = 1 where /spl delta//sup i/ is the probability that the ith input variable is read and Inf/sub i/(f) is the influence of the ith variable on f. The variance, influence and probability are taken with respect to an arbitrary product measure on {-1, 1}/sup n/n. It follows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal of the largest influence of any input variable. Likewise, any balanced Boolean function with a decision tree of depth d has a variable with influence at least 1/d. The only previous nontrivial lower bound known was /spl Omega/(d2/sup -d/). Our inequality has many generalizations, allowing us to prove influence lower bounds for randomized decision trees, decision trees on arbitrary product probability spaces, and decision trees with nonBoolean outputs. As an application of our results we give a very easy proof that the randomized query complexity of nontrivial monotone graph properties is at least/spl Omega/(v/sup 4/3//p/sup 1/3/), where v is the number of vertices and p /spl les/ 1/2 is the critical threshold probability. This supersedes the milestone /spl Omega/(v/sup 4/3//p/sup 1/3/) bound of Hajnal (1991) and is sometimes superior to the best known lower bounds of Chakrabarti-Khot (2001) and Friedgut-Kahn-Wigderson (2002).

Given a positive integer n, let and . The generator matrix of the kth-order Reed–Muller code RM is an important tool in the study of Boolean functions' algebraic immunity. In this paper, choosing the last s column vectors in as a basis of the vector space , we study the values of the coefficients in the linear expressions of 's column vectors over this basis. As an application, we present a new construction of balanced even-variable Boolean functions on with optimal algebraic immunity by modifying the outputs of majority function. The nonlinearities of these constructed functions are also determined.

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.

In this paper, the analysis of cyclotomic coset has been done for construction of Boolean functions. Boolean function used in cryptographic systems should have good cryptographic properties against standard or fast algebraic attacks. The cyclotomic coset plays very important role in cryptography. By using cyclotomic coset leader and minimal polynomial, we construct more balanced Boolean function. In this paper, cryptographic properties; balancedness, algebraic-degree, algebraic-immunity and non-linearity are also defined in preliminaries.

Boolean functions play an important role in symmetric ciphers. One of important open problems on Boolean functions is determining the maximum possible resiliency order of n-variable Boolean functions with optimal algebraic immunity. In this letter, we search Boolean functions in the rotation symmetric class, and determine the maximum possible resiliency order of 9-variable Boolean functions with optimal algebraic immunity. Moreover, the maximum possible nonlinearity of 9-variable rotation symmetric Boolean functions with optimal algebraic immunity-resiliency trade-off is determined to be 224.

We prove that a balanced Boolean function on Sn whose Fourier transform is highly concentrated on the first two irreducible representations of Sn, is close in structure to a dictatorship, a function which is determined by the image or pre-image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on Sn generated by the transpositions. Our proof works in the case where the expectation of the function is bounded away from 0 and 1. In contrast, [6] deals with Boolean functions of expectation O(1/ n) whose Fourier transform is highly concentrated on the first two irreducible representations of Sn. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point-stabilizers. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 494–530, 2015

Boolean functions play a central role in security applications because they constitute one of the basic primitives for modern cryptographic services. In the last decades, research on Boolean functions has been boosted due to the importance of security in many diverse public systems relying on such technology. A main focus is to find Boolean functions with specific properties. An open problem in this context is to find a balanced Boolean function with an 8-bit input and nonlinearity 118. Theoretically, such a function has been shown to exist, but it has not been found yet. In this work we focus on specific classes of Boolean functions, and analyze the landscape of results obtained by integrating algebraic and evolutionary computation (EC) based approaches. Results indicate that combinations of these approaches give better results although not reaching 118 nonlinearity.

In this paper, we study the algebraic immunity of Boolean functions and consider in particular the problem of constructing Boolean functions with a good algebraic immunity. We first give heuristic arguments which seem to indicate that the algebraic immunity of a random Boolean function on n variables is at least lfloorn/2rfloor with a very high probability (while the upper bound is lceiln/2rceil, the "ceiling" of n/2). We give an upper bound, under a reasonable assumption, on the algebraic immunity of Boolean functions constructed through Maiorana-MacFarland construction. At last we give examples of balanced functions with optimal algebraic immunity and a good nonlinearity and of balanced functions with a good algebraic immunity, a good nonlinearity and a good correlation immunity, which can be used for cryptographic purposes

We improve the steepest descent algorithm and increase the double threshold parameter, which significantly improves the algorithm’s efficiency. And we design a new cost function so that in terms of search, various characteristics of Boolean functions can be taken into account simultaneously. Applying our algorithm, there are excellent results regarding the 9, 10, 11, and 12 variables. We find a Boolean function with a nonlinearity of 242 in 9 variables and the whole search space. Previously, this result only appeared in the rotational symmetry class. The best-achieved nonlinearity result for permutation (6, 5, 1, 4, 7, 2, 3, 0, 8) and (0, 7, 2, 5, 4, 1, 3, 6, 8) class is 238 and 239 introduced by Kavut in Information and Computation (2010). Still, applying our algorithm, we obtain a balanced Boolean function with a nonlinearity of 240 under the same permutation, indicating that our method is more general. Among the 11 variables, a Boolean function with a higher nonlinearity and a lower transparency level and the absolute value spectrum are maintained at a lower level. The algorithm performs well when considering all aspects of the property. There are similarly promising results in even-numbered variables.