Concatenations of the hidden weighted bit function and their cryptographic properties

Cryptographic properties of the hidden weighted bit function

Hybrid classes of balanced Boolean functions with good cryptographic properties

On the size of binary decision diagrams representing Boolean functions

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.

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.

Based on the hidden weighted bit function, we propose a family of cryptographically significant Boolean functions. We investigate its algebraic degree and use Schur polynomials to study its algebraic immunity. For a subclass of this family, we deduce a lower bound on its nonlinearity. Moreover, we give an infinite class of balanced functions with very good cryptographic properties: optimum algebraic degree, optimum algebraic immunity, high nonlinearity (higher than the Carlet-Feng function and the function proposed by [25]) and a good behavior against fast algebraic attacks. These functions seem to have the best cryptographic properties among all currently known functions.

In this note, we deduce a bound between fast algebraic immunity and higher order nonlinearity (it is the first time that a bound between these two cryptographic criteria is given), and find that a Boolean function should have high r-order nonlinearity to resist fast algebraic attacks. As a corollary, we find that no matter how much effort we make, the Tu-Deng functions cannot be repaired in a standard way to behave well against fast algebraic attacks. Therefore, we should give up repairing this class of Boolean functions and try to find other classes of functions with good cryptographic properties or to prove that the Carlet-Feng function behaves well.

In this paper, we study a class of Boolean functions with good cryptographic properties. We show that the functions of this class are 1-resilient and have optimal algebraic degree and good nonlinearity. Further, we prove that the functions of this class have at least sub-maximum algebraic immunity. We also check that, at least for small values of the number of variables, the functions of this class have very good nonlinearity, maximum algebraic immunity and almost perfect immunity to fast algebraic attacks.KeywordsBoolean functionscorrelation immunityresiliencynonlinearityalgebraic immunityfast algebraic attacks

Reachability analysis of large circuits using disjunctive partitioning and partial iterative squaring

A new method to construct Boolean functions with good cryptographic properties

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.

It is known that Boolean functions used in stream ciphers should have good cryptographic properties to resist fast algebraic attacks. In this paper, we study a new class of Boolean functions with good cryptographic properties: balancedness, optimum algebraic degree, optimum algebraic immunity and a high nonlinearity.

The use of Binary Decision Diagrams (BDDs) has proliferated in numerous fields. When a system criterion is formulated in form of a Boolean function, its BDD is constructed. Each node in the BDD is further mapped into another form to be exploited in the system analysis. However, the cost of the resultant mapping form is directly related to the BDD size which can be effectively reduced through applying proper variable reordering followed by applying reduction rules that preserve the fidelity of the BDD in correctly representing the input Boolean function. Although several algorithms have been proposed in the literature to find the optimal order of variables in the BDD, the scalability of such algorithms is a serious barrier when it comes to complex systems with exponential explosion in the possible number of orders in the search space. Furthermore, solely exploring the search space in BDD reordering is not sufficient since better permutations might be obtained with slight tuning of the candidate solutions. Thus, a sufficient degree of equilibrium between exploration and exploitation should be preserved during the evolution of the reordering algorithm. In this paper, we propose a BDD optimizer driven by either Genetic Algorithm (GA) or swarm engines. The proposed GA-based BDD reordering optimizer iteratively processes an essentially large population with a randomized mixing of low destructive crossover/mutation operators. The proposed swarm-based optimizer, on the other hand, maps a vector of real numbers into a permutation to further construct its companion BDD. The generation of the next vector is guided by recent parameter and parameter-less swarm algorithms that are armed with effective mechanisms to simultaneously conduct exploration and exploitation. Experimental results show that our proposed optimizer effectively reduces the resultant BDD size for input Boolean functions with almost linear computational complexity. Furthermore, it has been found that exploiting recent swarm optimizers with spiral movement in BDD reordering problem can outperform GA for large scale Boolean functions. Finally, as a real-world application, our proposed algorithm is applied to reversible logic synthesis to show the achieved reduction in the Quantum Cost (QC) associated with BDD-based synthesis.

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

This paper proposes a general method to construct 1-resilient Boolean functions by modifying the Tu-Deng and Tang-Carlet-Tang functions. Cryptographic properties such as algebraic degree, nonlinearity and algebraic immunity are also considered. A sufficient condition of the modified functions with optimal algebraic degree in terms of the Siegenthaler bound is proposed. The authors obtain a lower bound on the nonlinearity of the Tang-Carlet-Tang functions, which is slightly better than the known result. If the authors do not break the “continuity” of the support and zero sets, the functions constructed in this paper have suboptimal algebraic immunity. Finally, four specific classes of 1-resilient Boolean functions constructed from this construction and with the mentioned good cryptographic properties are proposed. Experimental results show that there are many 1-resilient Boolean functions have higher nonlinearities than known 1-resilient functions modified by Tu-Deng and Tang-Carlet-Tang functions.