Boolean functions constructed using digital sequences of linear recurrences

Error-Free and Best-Fit Extensions of Partially Defined Boolean Functions

The well known impossibility result of Cleve (STOC 1986) implies that in general it is impossible to securely compute a function with complete fairness without an honest majority. Since then, the accepted belief has been that nothing non-trivial can be computed with complete fairness in the two party setting. The surprising work of Gordon, Hazay, Katz and Lindell (STOC 2008) shows that this belief is false, and that there exist some non-trivial (deterministic, finite-domain) boolean functions that can be computed fairly. This raises the fundamental question of characterizing complete fairness in secure two-party computation.In this work we show that not only that some or few functions can be computed fairly, but rather an enormous amount of functions can be computed with complete fairness. In fact, almost all boolean functions with distinct domain sizes can be computed with complete fairness (for instance, more than 99.999% of the boolean functions with domain sizes 31 ×30). The class of functions that is shown to be possible includes also rather involved and highly non-trivial tasks, such as set-membership, evaluation of a private (Boolean) function and private matchmaking.In addition, we demonstrate that fairness is not restricted to the class of symmetric boolean functions where both parties get the same output, which is the only known feasibility result. Specifically, we show that fairness is also possible for asymmetric boolean functions where the output of the parties is not necessarily the same. Moreover, we consider the class of functions with non-binary output, and show that fairness is possible for any finite range.The constructions are based on the protocol of Gordon et. al, and the analysis uses tools from convex geometry.KeywordsComplete fairnesssecure two-party computationfoundationsmalicious adversaries

This paper investigates the optimum OBDD representation problem based on two classes of Boolean functions. The first class is defined by OBDDs, in which the number of non-terminal nodes is equal to the number of input variables. We refer to such OBDDs and their corresponding Boolean functions as thin OBDDs and thin Boolean functions. The second class is the thin factored Boolean functions, which is defined by factored forms with each variable appearing exactly once. Many interesting properties of these two classes of functions are presented. Based on which, a revised dynamic shortest cube first OBDD variable ordering algorithm is developed. This algorithm is shown to be optimum for thin factored Boolean functions.

In this paper, we study the problem of using statistical query (SQ) to learn a class of highly correlated boolean functions, namely, a class of functions where any pair agree on significantly more than 1/2 fraction of the inputs. We give an almost-tight bound on how well one can approximate all the functions without making any query, and then we show that beyond this bound, the number of statistical queries the algorithm has to make increases with the “extra” advantage the algorithm gains in learning the functions. Here the advantage is defined to be the probability the algorithm agrees with the target function minus the probability the algorithm doesn’t agree. An interesting consequence of our results is that the class of booleanized linear functions over a finite field (f(a(x) = 1 iff ø(a.x) = 1, where ø is an arbitrary boolean function that maps any elements in GFp to ±1) is not efficiently learnable. This result is useful since the hardness of learning booleanized linear functions over a finite field is related to the security of certain cryptosystems ([]). In particular, we prove that the class of linear threshold functions over a finite field (f(a,b(x) = 1 iff a. x ≥ b) cannot be learned efficiently using statistical query. This contrasts with Blum et. al.’s result [] that linear threshold functions over reals (perceptions) are learnable using the SQ model. Finally, we describe a PAC-learning algorithm that learns a class of linear threshold functions in time that is provably impossible for statistical query algorithms. With properly chosen parameters, this class of linear threshold functions become an example of PAC-learnable, but not SQlearnable functions that are not parity functions.

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.

Efficient hashing is a centerpiece of modern cryptography. The progress in computing technology enables us to use 64-bit machines with the promise of 128-bit machines in the near future. To exploit fully the technology for fast hashing, we need to be able to design cryptographically strong Boolean functions in many variables which can be evaluated faster using partial evaluations from the previous rounds. We introduce a new class of Boolean functions whose evaluation is especially efficient and we call them rotation symmetric. Basic cryptographic properties of rotation-symmetric functions are investigated in a broader context of symmetric functions. An algorithm for the design of rotation-symmetric functions is given and two classes of functions are examined. These classes are important from a practical point of view as their forms are short. We show that shortening of rotation-symmetric functions paradoxically leads to more expensive evaluation process.

Several underlying structural and functional factors that determine the fault behavior of a network are not yet well understood. In this paper, we show that there exists a large class of Boolean functions, called root functions, which can never appear as faulty response in an irredundant two-level circuit even when any arbitrary multiple stuck-at faults are injected. Conversely, we show that any other Boolean function can appear as a faulty response in an irredundant realization of some root function under certain stuck-at faults. We characterize this new class of functions and show that for n variables, their number is exactly equal to the number of independent dominating sets (Harary and Livingston, Appl. Math. Lett., 1993) in a Boolean n-cube. Similar properties are observed for multiple-valued logic functions as well. Finally, we discuss its application to logic design and point out some open problems.

We consider quantum versions of two well-studied models of learning Boolean functions: Angluin's model of exact learning from membership queries and Valiant's probably approximately correct (PAC) model of learning from random examples. For each of these two learning models we establish a polynomial relationship between the number of quantum or classical queries required for learning. These results contrast known results that show that testing black-box functions for various properties, as opposed to learning, can require exponentially more classical queries than quantum queries. We also show that, under a widely held computational hardness assumption (the intractability of factoring Blum integers), there is a class of Boolean functions which is polynomial-time learnable in the quantum version but not the classical version of each learning model. For the model of exact learning from membership queries, we establish a stronger separation by showing that if any one-way function exists, then there is a class of functions which is polynomial-time learnable in the quantum setting but not in the classical setting. Thus, while quantum and classical learning are equally powerful from an information theory perspective, the models are different when viewed from a computational complexity perspective.

In 2015, Cao and Hu (Cao, X., Hu, L.: Two boolean functions with five-valued walsh spectra and high nonlinearity. International Journal of Foundations of Computer Science, pp. 537–556 (2015)) introduced a certain class of Boolean functions, possessing low Walsh spectra, high nonlinearity, and high algebraic degree. For this class of Boolean functions, computation of higher-order nonlinearities (even second-order) is a tedious task. Therefore, in this article, we study the lower bound on the second-order nonlinearity of the above-mentioned class of Boolean functions for $$n=4.$$ Also, we deduce that the bound, thus obtained is the maximum possible bound. We also demonstrated that our lower bound is greater than the lower bound on the second-order nonlinearity of other classes of cubic Boolean functions.

The number and probability of canalizing functions

Valiant introduced a new computational model of concept learning by examples, gave the definition of learnability of classes of Boolean functions, and derived algorithms for learning specific classes of Boolean functions. Using his model as a base, the authors show that the class of Boolean functions expressed by monotone disjunctive normal form formulae with at most a fixed number of monomials and the class of Boolean threshold functions are polynomial time learnable when the examples are generated according to the uniform distribution.

Recent research shows that the class of rotation symmetric Boolean functions is potentially rich in functions of cryptographic significance. In this paper, two classes of rotation symmetric Boolean functions having optimal algebraic immunity on even number of variables are presented. We give a lower bound of the algebraic degree of the functions in the first class, and derive the algebraic degree of the second class of functions. Moreover, the algebraic degree of the second class of functions is high enough. It is shown that both classes of functions have much better nonlinearity than all the previously obtained rotation symmetric Boolean functions with optimal algebraic immunity, and have good behavior against fast algebraic attacks at least for small numbers of input variables.

We prove that polynomial size discrete synchronous Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial space-bounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly.

We consider the problem of determining whether a given function $f:{\{0,1\}}^n\to{\{0,1\}}$ belongs to a certain class of Boolean functions $\cal F$ or whether it is far from the class. More precisely, given query access to the function f and given a distance parameter $\epsilon$, we would like to decide whether $f \in \cal F$ or whether it differs from every $g\in \cal F$ on more than an $\epsilon$-fraction of the domain elements. The classes of functions we consider are singleton ("dictatorship") functions, monomials, and monotone disjunctive normal form functions with a bounded number of terms. In all cases we provide algorithms whose query complexity is independent of n (the number of function variables), and linear in $1/\epsilon$.

We prove that polynomial size discrete Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial space-bounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly, i.e., the class computed by polynomial time-bounded nonuniform Turing machines.