Partially Symmetric Functions Are Efficiently Isomorphism-Testable

formula omitted]-Regular languages are testable with a constant number of queries

Two Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

Two important classes of symmetric Boolean functions are the equal-weight Boolean functions and the elementary (or homogeneous) symmetric Boolean functions. In this paper we studied the equal-weight symmetric Boolean functions. First the Walsh spectra of the equal-weight symmetric Boolean functions are given. Second the sufficient and necessary condition on correlation-immunity of the equal-weight symmetric Boolean function is derived and other cryptology properties such as the nonlinearity, balance and propagation criterion are taken into account. In particular, the nonlinearity of the equal-weight symmetric Boolean functions with n (n ≥ 10) variables is determined by their Hamming weight. Considering these properties will be helpful in further investigations of symmetric Boolean functions.

The number and probability of canalizing functions

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.

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A recursive formula for weights of Boolean rotation symmetric functions

International audience

Boolean functions with a variety of secure cipher properties are the key factors to design cryptosystem with the ability to resist multiple cipher attacks and good safety performance. In this paper, using the derivative of the Boolean functions and the e-derivative defined by ourselves as the main research tools, we study algebraic immunity, correlation immunity and other cryptographic properties of the quadratic rotation symmetric Boolean functions. We determine the quadratic rotation symmetric Boolean functions which are H Boolean functions, and the range of weight distribution of the quadratic rotation symmetry H Boolean functions. Besides, we get the compatibility among propagation, balance, correlation immunity and algebraic immunity of the quadratic rotation symmetry H Boolean functions, and also focus on the relationship of balance, correlation immunity and dimension. Furthermore, we check the existence of the cubic rotation symmetry H Boolean functions, and obtain the relationship between existence and dimension of the cubic rotation symmetry H Boolean functions. Moreover, we obtain a more convenient method for solving annihilator. Such researches are important in cryptographic primitive designs, and have significance and role in the theory and application range of cryptosystems.

The multiplicative complexity of a Boolean function f is defined as the minimum number of binary conjunction (AND) gates required to construct a circuit representing f, when only exclusive-or, conjunction and negation gates may be used. This article explores in detail the multiplicative complexity of symmetric Boolean functions. New techniques that allow such exploration are introduced. They are powerful enough to give exact multiplicative complexities for several classes of symmetric functions. In particular, the multiplicative complexity of computing the Hamming weight of n bits is shown to be exactly n−Hℕ(n), where Hℕ(n) is the Hamming weight of the binary representation of n. We also show a close relationship between the complexity of symmetric functions and fractals derived from the parity of binomial coefficients.

One of the most fundamental questions in graph property testing is to characterize the combinatorial structure of properties that are testable with a constant number of queries. We work towards an answer to this question for the bounded-degree graph model introduced in [Goldreich, Ron, 2002], where the input graphs have maximum degree bounded by a constant $d$. In this model, it is known (among other results) that every \emph{hyperfinite} property is constant-query testable [Newman, Sohler, 2013], where, informally, a graph property is hyperfinite, if for every $\delta >0$ every graph in the property can be partitioned into small connected components by removing $\delta n$ edges. In this paper we show that hyperfiniteness plays a role in \emph{every} testable property, i.e. we show that every testable property is either finite (which trivially implies hyperfiniteness and testability) or contains an infinite hyperfinite subproperty. A simple consequence of our result is that no infinite graph property that only consists of expander graphs is constant-query testable. Based on the above findings, one could ask if every infinite testable non-hyperfinite property might contain an infinite family of expander (or near-expander) graphs. We show that this is not true. Motivated by our counter-example we develop a theorem that shows that we can partition the set of vertices of every bounded degree graph into a constant number of subsets and a separator set, such that the separator set is small and the distribution of $k$-disks on every subset of a partition class, is roughly the same as that of the partition class if the subset has small expansion.

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 address a sequential decision problem that arises in the computation of symmetric Boolean functions of distributed data. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and we wish to compute a given Boolean function of these measurements. We suppose that the measurements are independent and Bernoulli distributed. Thus, the problem of optimal computation becomes the problem of optimally ordering nodes' transmissions so as to minimize the total expected number of bits. We solve the ordering problem for the class of Boolean threshold functions. The optimal ordering is dynamic, i.e., it could potentially depend on the values of previously transmitted bits. Further, it depends only on the ordering of the marginal probabilites, but not on their exact values. This provides an elegant structure for the optimal strategy. For the case where each node has a block of measurements, the problem is significantly harder, and we conjecture the optimal strategy.

A technique for implementing totally symmetric Boolean functions using hierarchical modules is presented. First, a simple cellular module is designed for synthesizing unate symmetric functions. The structure is universal, admits a recursive design, and uses only 2-input AND-OR gates. A universal test set of size (n2/8 + 3n/4) for detecting all single stuck-at faults can be easily determined for an n-input module, where n = 2k, k � 3. General symmetric functions are then realized following a unate decomposition method. The synthesis procedure guarantees full robust path-delay fault testability in the circuit. Experimental results on several symmetric functions reveal that the hardware cost of the proposed design is low, and the number of paths in the circuit is reduced significantly compared to those of earlier designs. Results on circuit area and delay for a few benchmark examples are also reported.

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.