Symmetric Boolean Functions Research Articles

Symmetric Boolean functions depending on an odd number of variables with maximum algebraic immunity

To resist algebraic attacks, Boolean functions should possess high algebraic immunity. In 2003, Courtois and Meier showed that the algebraic immunity of an n-variable Boolean function is upper bounded by /spl lceil/n/2/spl rceil/. And then several papers studied how to find symmetric Boolean functions with maximum algebraic immunity. In this correspondence, we prove that for each odd n, there is exactly one trivially balanced n-variable symmetric Boolean function achieving the maximum algebraic immunity.

The influence of oppositely classified examples on the generalization complexity of Boolean functions

In this paper, we analyze Boolean functions using a recently proposed measure of their complexity. This complexity measure, motivated by the aim of relating the complexity of the functions with the generalization ability that can be obtained when the functions are implemented in feed-forward neural networks, is the sum of a number of components. We concentrate on the case in which we use the first two of these components. The first is related to the "average sensitivity" of the function and the second is, in a sense, a measure of the "randomness" or lack of structure of the function. In this paper, we investigate the importance of using the second term in the complexity measure, and we consider to what extent these two terms suffice as an indicator of how difficult it is to learn a Boolean function. We also explore the existence of very complex Boolean functions, considering, in particular, the symmetric Boolean functions.

Implementing Symmetric Functions with Hierarchical Modules for Stuck-At and Path-Delay Fault Testability

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.

ON ASYNCHRONOUS CELLULAR AUTOMATA

We study asynchronous cellular automata (ACA) induced by symmetric Boolean functions [1]. These systems can be considered as sequential dynamical systems (SDS) over words, a class of dynamical systems that consists of (a) a finite, labeled graph Y with vertex set {v1,…,vn} and where each vertex vi has a state xvi in a finite field K, (b) a sequence of functions (Fvi,Y)i, and (c) a word w = (w1,…,wk), where each wi is a vertex in Y. The function Fvi,Y updates the state of vertex vi as a function of the state of vi and its Y-neighbors and maps all other vertex states identically. The SDS is the composed map [Formula: see text]. In the particular case of ACA, the graph is the circle graph on n vertices (Y = Circ n), and all the maps Fvi are induced by a common Boolean function. Our main result is the identification of all w-independent ACA, that is, all ACA with periodic points that are independent of the word (update schedule) w. In general, for each w-independent SDS, there is a finite group whose structure contains information about for example SDS with specific phase space properties. We classify and enumerate the set of periodic points for all w-independent ACA, and we also compute their associated groups in the case of Y = Circ 4. Finally, we analyze invertible ACA and offer an interpretation of S35 as the group of an SDS over the three-dimensional cube with local functions induced by nor3 + nand3.

Symmetric polynomials over [formula omitted] and simultaneous communication protocols

We study the problem of representing symmetric Boolean functions as symmetric polynomials over Z m . We prove an equivalence between representations of Boolean functions by symmetric polynomials and simultaneous communication protocols. We show that computing a function f on 0–1 inputs with a polynomial of degree d modulo pq is equivalent to a two player simultaneous protocol for computing f where one player is given the first ⌈ log p d ⌉ digits of the weight in base p and the other is given the first ⌈ log q d ⌉ digits of the weight in base q . This equivalence allows us to show degree lower bounds by using techniques from communication complexity. For example, we show lower bounds of Ω ( n ) on symmetric polynomials weakly representing classes of Mod r and Threshold functions. Previously the best known lower bound for such representations of any function modulo pq was Ω ( n 1 2 ) [D.A. Barrington, R. Beigel, S. Rudich, Representing Boolean functions as polynomials modulo composite numbers, Comput. Complexity 4 (1994) 367–382]. The equivalence also allows us to use results from number theory to prove upper bounds for Threshold- k functions. We show that proving bounds on the degree of symmetric polynomials strongly representing the Threshold- k function is equivalent to counting the number of solutions to certain Diophantine equations. We use this to show an upper bound of O ( nk ) 1 2 + ɛ for Threshold- k assuming the abc-conjecture. We show the same bound unconditionally for k constant. Prior to this, non-trivial upper bounds were known only for the OR function [D.A. Barrington, R. Beigel, S. Rudich, Representing Boolean functions as polynomials modulo composite numbers, Comput. Complexity 4 (1994) 367–382]. We show an almost tight lower bound of Ω ( nk ) 1 2 , improving the previously known bound of Ω ( max ( k , n ) ) [S.-C. Tsai, Lower bounds on representing Boolean functions as polynomials in Z m , SIAM J. Discrete Math. 9 (1996) 55–62].

One-way communication complexity of symmetric Boolean functions

We study deterministic one-way communication complexity of functions with Hankel communication matrices. In this paper some structural properties of such matrices are established and applied to the one-way two-party communication complexity of symmetric Boolean functions. It is shown that the number of required communication bits does not depend on the communication direction, provided that neither direction needs maximum complexity. Moreover, in order to obtain an optimal protocol, it is in any case sufficient to consider only the communication direction from the party with the shorter input to the other party. These facts do not hold for arbitrary Boolean functions in general. Next, gaps between one-way and two-way communication complexity for symmetric Boolean functions are discussed. Finally, we give some generalizations to the case of multiple parties.

On the computational power of probabilistic and quantum branching program

In this paper, we show that one-qubit polynomial time computations are as powerful as NC 1 circuits. More generally, we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power. We show that any NC 1 language can be accepted exactly by a width-2 quantum branching program of polynomial length, in contrast to the classical case where width 5 is necessary unless NC 1 = ACC. This separates width-2 quantum programs from width-2 doubly stochastic programs as we show the latter cannot compute the middle bit of multiplication. Finally, we show that bounded-width quantum and stochastic programs can be simulated by classical programs of larger but bounded width, and thus are in NC 1 . For read-once quantum branching programs (QBPs), we give a symmetric Boolean function which is computable by a read-once QBP with O (log n ) width, but not by a deterministic read-once BP with o ( n ) width, or by a classical randomized read-once BP with o ( n ) width which is “stable” in the sense that its transitions depend on the value of the queried variable but do not vary from step to step. Finally, we present a general lower bound on the width of read-once QBPs, showing that our O (log n ) upper bound for this symmetric function is almost tight.

Symmetric Boolean Functions

We present an extensive study of symmetric Boolean functions, especially of their cryptographic properties. Our main result establishes the link between the periodicity of the simplified value vector of a symmetric Boolean function and its degree. Besides the reduction of the amount of memory required for representing a symmetric function, this property has some consequences from a cryptographic point of view. For instance, it leads to a new general bound on the order of resiliency of symmetric functions, which improves Siegenthaler's bound. The propagation characteristics of these functions are also addressed and the algebraic normal forms of all their derivatives are given. We finally detail the characteristics of the symmetric functions of degree at most 7, for any number of variables. Most notably, we determine all balanced symmetric functions of degree less than or equal to 7.

Sequential dynamical systems over words

This paper is motivated by the theory of sequential dynamical systems (SDS), developed as a basis for a mathematical theory of computer simulation. A sequential dynamical system is a collection of symmetric Boolean local update functions, with the update order determined by a permutation of the Boolean variables. In this paper, the notion of SDS is generalized to allow arbitrary functions over a general finite field, with the update schedule given by an arbitrary word on the variables. The paper contains generalizations of some of the known results about SDS with permutation update schedules. In particular, an upper bound on the number of different SDS over words of a given length is proved and open problems are discussed.

Exploiting causal independence in large Bayesian networks

The assessment of a probability distribution associated with a Bayesian network is a challenging task, even if its topology is sparse. Special probability distributions based on the notion of causal independence have therefore been proposed, as these allow defining a probability distribution in terms of Boolean combinations of local distributions. However, for very large networks even this approach becomes infeasible: in Bayesian networks which need to model a large number of interactions among causal mechanisms, such as in fields like genetics or immunology, it is necessary to further reduce the number of parameters that need to be assessed. In this paper, we propose using equivalence classes of binomial distributions as a means to define very large Bayesian networks. We analyse the behaviours obtained by using different symmetric Boolean functions with these probability distributions as a means to model joint interactions. Some surprisingly complicated behaviours are obtained in this fashion, and their intuitive basis is examined.

Polynomial Representations of Symmetric Partial Boolean Functions

For Boolean polynomials in $\mathbb{Z}_p$ of sufficiently low degree we derive a relation expressing their values on one level set in terms of their values on another level set. We use this relation to derive linear upper and lower bounds, tight to within constant factor, on the degrees of various approximate majority functions, namely, functions that take the value 0 on one level set, the value 1 on a different level set, and arbitrary 0-1 values on other Boolean inputs. We show sublinear upper bounds in the case of moduli that are not prime powers.

On the Degree, Nonlinearity, Algebraic Thickness, and Nonnormality of Boolean Functions, With Developments on Symmetric Functions

The two main criteria evaluating, from cryptographic viewpoint, the complexity of Boolean functions are the nonlinearity and the algebraic degree. Two other criteria can also be considered: the algebraic thickness and the nonnormality. Simple proofs are given that, asymptotically, almost all Boolean functions have high algebraic thicknesses and are deeply nonnormal, as well as they have high algebraic degrees and high nonlinearities. We also study in detail the relationship between nonnormality and nonlinearity. We derive simple proofs of known results on symmetric Boolean functions and we prove several new and more general results on a class containing all symmetric functions.

Novel Reconfigurable Logic Gates Using Spin Metal–Oxide–Semiconductor Field-Effect Transistors

We propose and numerically simulate novel reconfigurable logic gates employing spin metal-oxide-semiconductor field-effect transistors (spin MOSFETs). The output characteristics of the spin MOSFETs depend on the relative magnetization configuration of the ferromagnetic contacts for the source and drain, that is, high current-drive capability in parallel magnetization and low current-drive capability in antiparallel magnetization [S. Sugahara and M. Tanaka: Appl. Phys. Lett. 84 (2004) 2307]. A reconfigurable NAND/NOR logic gate can be realized by using a spin MOSFET as a driver or an active load of a complimentary MOS (CMOS) inverter with a neuron MOS input stage. Its logic function can be switched by changing the relative magnetization configuration of the ferromagnetic source and drain of the spin MOSFET. A reconfigurable logic gate for all symmetric Boolean functions can be configured using only five CMOS inverters including four spin MOSFETs. The operation of these reconfigurable logic gates was confirmed by numerical simulations using a simple device model for the spin MOSFETs.

On Some Special Classes of Sequential Dynamical Systems

Sequential Dynamical Systems (SDSs) are mathematical models for analyzing simulation systems. We investigate phase space properties of some special classes of SDSs obtained by restricting the local transition functions used at the nodes. We show that any SDS over the Boolean domain with symmetric Boolean local transition functions can be efficiently simulated by another SDS which uses only simple threshold and simple inverted threshold functions, where the same threshold value is used at each node and the underlying graph is d-regular for some integer d. We establish tight or nearly tight upper and lower bounds on the number of steps needed for SDSs over the Boolean domain with 1-, 2- or 3-threshold functions at each of the nodes to reach a fixed point. When the domain is a unitary semiring and each node computes a linear combination of its inputs, we present a polynomial time algorithm to determine whether such an SDS reaches a fixed point. We also show (through an explicit construction) that there are Boolean SDSs with the NOR function at each node such that their phase spaces contain directed cycles whose length is exponential in the number of nodes of the underlying graph of the SDS.

A constructive count of rotation symmetric functions

In this paper we present a constructive detection of minimal monomials in the algebraic normal form of rotation symmetric Boolean functions (immune to circular translation of indices). This helps in constructing rotation symmetric Boolean functions by respecting the rules we present here.

Correlation immunity and resiliency of symmetric Boolean functions

Correlation immunity of symmetric Boolean functions is studied in this paper. Lower bounds on the number of constructible correlation immune symmetric functions are given. Constructions for such new balanced functions are presented. These functions are also known as 1-resilient functions. In 1985, Chor et al. conjectured that the only 1-resilient symmetric functions are the exclusive-or of all n variables and its negation. This conjecture, however, was disproved by Gopalakrishnan, Hoffman and Stinson in 1993 by giving a class of infinite counterexamples, and they noted that it does not seem to extend any further in an obvious way. In this paper two more infinite classes of such examples are presented for n being even and being odd, respectively, and consequently one of the two open problems proposed by Gopalakrishnan et al., is addressed by constructing new symmetric resilient functions.

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

Abstract In 1999, Pieprzyk and Qu presented rotation symmetric (RotS) functions as components in the rounds of hashing algorithm. Later, in 2002, Cusick and Stǎnicǎ presented further advancement in this area. This class of Boolean functions are invariant under circular translation of indices. In this paper, using Burnside's lemma, we prove that the number of n-variable rotation symmetric Boolean functions is 2gn, where gn and φ is the Euler phi-function. Moreover, we find the number of short and long cycles of elements in Zn2 having fixed weight, under the RotS action. As a consequence we obtain the number of homogeneous RotS functions having algebraic degree w. Our results make the search space of RotS functions much reduced and we successfully analyzed important cryptographic properties of such functions by executing computer programs. We found that there are exactly 8, 48, and 15104, RotS bent functions on 4, 6, and 8 variables respectively. Experimental results up to 10 variables show that there is no homogeneous rotation symmetric bent function with degree > 2. Further, we studied the RotS functions on 5, 6, 7 variables for correlation immunity and propagation characteristics and found some functions with very good cryptographic properties which were not known earlier.

Balancedness and Correlation Immunity of Symmetric Boolean Functions

Abstract New subsets of symmetric balanced and symmetric correlation immune functions are identified. The method involves interesting relations on binomial coefficients and highlights the combinatorial richness of these classes. As a consequence of our constructive techniques, we improve upon the existing lower bounds on the cardinality of the above sets. We consider higher order correlation immune functions and show how to construct n -variable, 3rd order correlation immune function for each perfect square n ≥ 9.

Maximum nonlinearity of symmetric Boolean functions on odd number of variables

In this correspondence, we establish that for odd n, the maximum nonlinearity achievable by an n-variable symmetric Boolean function is 2/sup n-1/-2/sup (n-1)///sup 2/ and characterize the set of functions which achieve this value of nonlinearity. In particular, we show that for each odd n/spl ges/3, there are exactly four possible symmetric Boolean functions achieving the nonlinearity 2/sup n-1/-2/sup (n-1)/2/.

Rivest–Vuillemin conjecture is true for monotone boolean functions with twelve variables

A Boolean function f( x 1, x 2,…, x n ) is elusive if every decision tree computing f must examine all n variables in the worst case. It is a long-standing conjecture that every nontrivial monotone weakly symmetric Boolean function is elusive. In this paper, we prove this conjecture for Boolean functions with twelve variables.