Boolean Functions with Five Controllable Cryptographic Properties

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

Rotation symmetric Boolean functions (RSBFs) are used widely in symmetric cryptography. In this paper, a systematic construction of balanced odd-variable RSBFs satisfying strict avalanche criterion is proposed. Hence, a class of $(6k+3)$ -variable resilient RSBFs satisfying strict avalanche criterion is also presented for any $k\geq 2$ . Some of the obtained RSBFs have many other good cryptographic properties at the same time, that is, optimal algebraic degree, good global avalanche characteristics, high nonlinearity and nonexistence of nonzero linear structures. Moreover, we obtain some count results of RSBFs satisfying strict avalanche criterion. This is the first time that the autocorrelation properties of RSBFs are investigated systemically.

To further enhance the nonlinear and cryptographic properties of chaotic S-boxes, a novel S-box design is proposed based on the discrete chaotic map and improved artificial bee colony algorithm. Firstly, the S-box generated by chaotic sequence is initialized by the opposition-based optimization. Then, the dual transposition artificial bee colony algorithm is used to optimize the population, where the Gaussian swap is engaged to strengthen the optimum efficiency, and make the population acquire a better quality with each other. With the Gaussian mutation operator, the opposition-based optimization performs self-learning to obtain better population. Comparing with some recent S-boxes based on chaos. The S-boxes generated by the proposed scheme can achieve better performance in bijection, nonlinearity, strict avalanche criterion(SAC), bit independence criterion(BIC), I / O XOR distribution and linear approximation probability(LP). It provides a new and effective method for designing S-boxes with good cryptographic properties.

A new S-box construction method meeting strict avalanche criterion

Global avalanche characteristics and propagation criterion of balanced Boolean functions

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.

We develop an improved cost function to be used in simulated annealing followed by hill-climbing to find Boolean functions satisfying multiple desirable criteria such as high nonlinearity, low autocorrelation, balancedness, and high algebraic degree. Using this cost function that does not necessitate experimental search for parameter tuning, the annealing-based algorithm reaches the desired function profiles more rapidly. Some Boolean functions of eight and nine variables have been found, which are unattained in the computer search based literature, in terms of joint optimization of nonlinearity and autocorrelation. Global characteristics of eight-variable Boolean functions generated by algebraic construction or computer search are compared with respect to the sum-of-squared-errors in their squared spectra, which is also proportional to the sum-of-squared-errors in their autocorrelation function, the term ‘error’ denoting the deviation from bent function characteristics. Preliminary results consisting of cryptographically strong Boolean functions of nine, ten and eleven variables obtained using a three-stage optimization technique are also presented.

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.

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.

Since 1970, Boolean functions have been the focus of a lot of attention in cryptography. An important topic in symmetric ciphers concerns the cryptographic properties of Boolean functions and constructions of Boolean functions with good cryptographic properties, that is, good resistance to known attacks. An important progress in cryptanalysis areas made in 2003 was the introduction by Courtois and Meier of algebraic attacks and fast algebraic attacks which are very powerful analysis concepts and can be applied to almost all cryptographic algorithms. To study the resistance against algebraic attacks, the notion of algebraic immunity has been introduced. In this paper, we use a parameter introduced by Liu and al., called fast algebraic immunity, as a tool to measure the resistance of a cryptosystem (involving Boolean functions) to fast algebraic attacks. We prove an upper bound on the fast algebraic immunity. Using our upper bound, we establish the weakness of trace inverse functions against fast algebraic attacks confirming a recent result of Feng and Gong.

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

Ever since Baptista in 1998 introduced his cryptographic scheme utilizing the only in online version. ergodic property of chaotic maps which is able to produce different cipher values for the same plaintext within the same message, intense scrutiny has been given upon the design. The capability to do the above mentioned output is akin to the Vigenere cipher and thus has the capacity to render an attacker with infinitely many choices (theoretically speaking) or in cryptographic terms would render an attacker to have a set off possible ciphertexts that could all have the possibility to just be mapped to a unique plaintext. This makes it computationally infeasible for the attacker to re-construct the correct plaintext. The Baptista design has been attacked and repaired many times. Alvarez noticed the characteristic of the cryptosystem that generates a sequence which can be exploited by an attacker. The attack which is dubbed the one-time pad attack is akin to an attack upon a One-Time-Pad (OTP) cryptosystem that reuses its key. Since then, attempts were made to redefine the cryptosystem such that it would be resistant towards the attack. Most of the attempts failed due to either the repaired cryptosystem still generates an exploitable sequence or it is not invertible. In this work we pair the Baptista design with a concept taken from the Iterated Function Systems (IFS). Although we did not encompass the whole concept of iterating the IFS, it could be seen that this could be easily done with the same desirable results. Four main outcomes are discussed. Beginning with the discussion on the infeasibility of Alvarez’s one-time pad attack on the design, we then discuss the quantitative properties of the design in discussing its cryptographic properties namely the Maximum Deviation Factor (MDF), Correlation Coefficient Factor (CCF) and the Strict Avalanche Criterion (SAC). Each experimental result shows promising results for this new design.KeywordsCryptosystemStrict Avalanche Criterion (SAC)Iterated Function Systems (IFS)Chaotic MapsOne-time Pad (OTP)These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In 2008, Carlet and Feng studied a class of functions with good cryptographic properties. Based on that function, [18] proposed a family of cryptographically significant Boolean functions which contains the functions proposed by [28, 30]. However, their study is not in-depth. In this paper, we investigate the properties of those functions further, and find that they can be divided into some affine equivalent classes. The bent functions proposed by [18] are in fact in the same class with the function proposed by [30]. We then prove that those functions have optimum algebraic immunity if and only if a combinatorial conjecture is correct, which gives a new direction to prove the conjecture. Furthermore, we improve upon the lower bound on the nonlinearity, and our bound is higher than all other similar bounds. Finally, we extend the construction to a balanced function, and give an example of a 12-variable function which has the best cryptographic properties among all currently known functions.KeywordsBoolean functionAlgebraic immunityNonlinearity

It is an interesting and challenging topic in cryptography to construct Boolean (or vector Boolean) functions with various of good cryptographic properties. In this paper, by analyzing of a class of linearized equations, we prove that the differential uniformity of the functions constructed by exchanging two values of the Gold functions is 4, and the derived functions also have high algebraic degree and high nonlinearity.