Clonoids of Boolean functions with a monotone or discriminator source clone

Analysis and synthesis of textures through the inference of Boolean functions

Boolean functions and their generalization Vectorial Boolean functions or Substitution Boxes (S-Boxes) have attracted much attention in the domain of modern block ciphers that use only these elements to provide the necessary confusion against the cryptanalysis attacks. Thus, a significant number of research has been done to construct cryptographically strong Boolean functions and S-Boxes. Among these researches, several heuristics were applied and therefore the hill climbing heuristic was largely investigated. In this paper, we propose a new variant of Hill Climbing heuristic called Parallel Steepest Ascent Hill Climbing to construct Boolean functions and \(n \times m\) S-Boxes through the progressive construction and incorporation of their m coordinate Boolean functions. The obtained results demonstrate that this new variant provides solutions with high cryptographic properties.

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

Aim. The purpose of the article is to study the cryptographic strength characteristics of Boolean functions used in the blender algorithm. This algorithm is designed for information security tasks as part of a pseudorandom sequence generator. Methods. The methods of cryptographic Boolean functions theory, the methods of analysis and comparison were used in the work. Results. The general features of the blender Boolean functions are formulated. To assess their cryptographic strength, a computational study of compliance with criteria such as balancedness, algebraic degree, the presence of fictitious variables, the absence of linear structures, correlation immunity, high nonlinearity, avalanche criterion, etc. was performed. A comparison is made with similar results for Boolean functions that are part of nonlinear s-blocks of known cryptographic algorithms. Conclusions. The ability to adjust the blender dimension to obtain blocks of the desired size, forming the output sequence, allows increasing the values of some characteristics of Boolean functions cryptographic strength. It is detected that some cryptographic strength indicators of the Boolean blender functions (balancedness, nonlinearity and global avalanche characteristic) have asymptotic regularities of behavior. The results of the study makes it possible to consider the level of cryptographic strength of the blender Boolean functions acceptable for use in cryptographic protection systems.

Boolean functions have a prominent role in many real-world applications, which makes them a very active research domain. Throughout the years, various heuristic techniques proved to be an attractive choice for the construction of Boolean functions with different properties. One of the most important properties is nonlinearity, and in particular maximally nonlinear Boolean functions are also called bent functions. In this paper, instead of considering Boolean functions, we experiment with quaternary functions. The corresponding problem is much more difficult and presents an interesting benchmark as well as realworld applications. The results we obtain show that evolutionary metaheuristics, especially genetic programming, succeed in finding quaternary functions with the desired properties. The obtained results in the quaternary domain can also be translated into the binary domain, in which case this approach compares favorably with the state-of-the-art in Boolean optimization. Our techniques are able to find quaternary bent functions for up to 8 inputs, which corresponds to obtaining Boolean bent functions of 16 inputs.

The immunity of Boolean functions against fast algebraic attacks FAA's has been considered as an important cryptographic property for Boolean functions used in stream ciphers. An n-variable Boolean power function f can be represented as a monomial trace function over finite field $$\mathbb {F}_{2^n}$$, $$fx=Tr_1^n\lambda x^k$$, where $$\lambda \in \mathbb {F}_{2^n}$$ and k is the coset leader of cyclotomic coset $$C_k$$ modulo $$2^n-1$$. To determine the immunity of Boolean power functions, one may need the arithmetic in $$\mathbb {F}_{2^n}$$, which is not computationally efficient compared with the operations over $$\mathbb {F}_2$$. In this paper, we show that the linear affine invariance of the immunity of Boolean functions against FAA's can be exploited to observe the immunity of Boolean power functions against FAA's, i.e., the immunity of $$fx=Tr_1^n\lambda x^k$$ against FAA's is the same as that of $$rx=Tr_1^nx^k$$ if fx can be obtained from rx through a linear transformation. In particular, if $$\gcd k,2^n-1=1$$ then the immunity against FAA's of fx and that of rx are always the same. The immunity of Boolean power functions that satisfy this condition can be computed more efficiently.

An original approach to analytic canonical minimization of switching (Boolean) functions, which is based on the representation of Boolean functions (BF) as a function of one variable – the number of the set í of values of its arguments, is proposed. For this, using the Antje function, the dependence (in the form of formulas) of the values of each of n variables on the number í is established. To construct a minimal disjunctive or conjunctive normal form of the BF are repelled from the corresponding perfect forms. In this connection the need to describe the sequence of numbers of sets, on which the variables take the values 1 and 0, arises. The common terms of such sequences can be obtained using the formula for the common term of a numerical sequence in a representation through the finite differences of its members. Further the connection between the sum of the values of atoms (term) ó(í) and the numbers of their values is established. With the help of the values of ó(í) , the question of gluing the constituents of the unit or zero is solved. Threedimensional and four-dimensional cubes whose vertices are labeled with sigmas are depicted for a function of three and four variables, respectively. Since the necessary condition for gluing the constituents – the modulus of the difference between sigmas is equal to one – is not sufficient, the gluing criterion is used: the constituents of a unit or zero are glued if and only if the Gamming distance between two values of ó(í) is equal to one. The general order for gluing the constituent of a unit or zero on the basis of the described means is obtained. A matrix of size m× n , where m is the number of simple implicant or implicent, n is the number of sets, on which the BF unit or zero values takes, is used to build deadlock forms; they are not represented by letters, but by sets of ones and zeros that correspond to them. The proposed approach to minimizing BF is called the í -minimization method. An example of minimizing of BF of five variables, given by the numbers of the sets í of values of its arguments, on which it takes values equal to one, that is – with the property f = 1, is given. The possibility of using the í -minimization of BF in bases other than the basis (∧, ∨, ￣ ), and in the future – an application to the minimization of BF of integer mathematical programming is discussed.

Multi- and multi-polar capacities

A circuit C compresses a function f : {0, 1}n → {0, 1}m if given an input x ∈ {0, 1}n the circuit C can shrink x to a shorter e-bit string x' such that later, a computationally-unbounded solver D will be able to compute f(x) based on x'. In this paper we study the existence of functions which are incompressible by circuits of some fixed polynomial size s = nc. Motivated by cryptographic applications, we focus on average-case (e, e) incompressibility, which guarantees that on a random input x ∈ {0, 1}n, for every size s circuit C : {0, 1}n → {0, 1}e and any unbounded solver D, the success probability Prx[D(C(x)) = f(x)] is upper-bounded by 2-m + e. While this notion of incompressibility appeared in several works (e.g., Dubrov and Ishai [12]), so far no explicit constructions of efficiently computable incompressible functions were known. In this work we present the following results:1. Assuming that E is hard for exponential size nondeterministic circuits, we construct a polynomial time computable boolean function f : {0, 1}n → {0, 1} which is incompressible by size nc circuits with communication e = (1 - o(1)) · n and error e = n-c. Our technique generalizes to the case of PRGs against nonboolean circuits, improving and simplifying the previous construction of Shaltiel and Artemenko [5].2. We show that it is possible to achieve negligible error parameter e = n-ω(1) for nonboolean functions. Specifically, assuming that E is hard for exponential size Σ3-circuits, we construct a nonboolean function f : {0, 1}n → {0, 1}m which is incompressible by size nc circuits with e = Ω(n) and extremely small e = n-c · 2-m. Our construction combines the techniques of Trevisan and Vadhan [47] with a new notion of relative error deterministic extractor which may be of independent interest.3. We show that the task of constructing an incompressible boolean function f : {0, 1}n → {0, 1} with negligible error parameter e cannot be achieved by proof techniques. Namely, nondeterministic reductions (or even Σi reductions) cannot get e = n-ω(1) for boolean incompressible functions. Our results also apply to constructions of standard Nisan-Wigderson type PRGs and (standard) boolean functions that are hard on average, explaining, in retrospective, the limitations of existing constructions. Our impossibility result builds on an approach of Shaltiel and Viola [40].

We consider collocated wireless sensor networks, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and the goal of the network is to compute a given Boolean function of these measurements. We first consider the worst case setting and study optimal block computation strategies for computing symmetric Boolean functions. We study three classes of functions: threshold functions, delta functions and interval functions. We provide optimal strategies for the first two classes, and a scaling law order-optimal strategy with optimal preconstant for interval functions. We extend the results to the case of integer measurements and certain integer-valued functions. Next, we address the problem of minimizing the expected total number of bits that are transmitted when node measurements are random and drawn from independent Bernoulli distributions. In the case of computing a single instance of a Boolean threshold function, the problem reduces to one of determining the optimal order in which the nodes should transmit. We show that the optimal order of transmissions depends in an extremely simple way on the values of previously transmitted bits and the ordering of the marginal probabilities of the Boolean variables according to the k-th least likely rule: At any transmission, the node that transmits is the one that has the k-th least likely value of its Boolean variable, where k reduces by one whenever a node transmits a one. Initially the value of k is (n +1 - Threshold). Interestingly, the order of transmissions does not depend on the exact values of the probabilities of the Boolean variables. In the case of identically distributed measurements, we further show that the average-case complexity of block computation of a Boolean threshold function is O(θ), where θ is the threshold. We further show how to generalize to a pulse model of communication. One can also consider the related problem of approximate computation given a fixed number of bits. For the special case of the parity function, we show that the greedy strategy is optimal.

Ordered binary decision diagrams (OBDDs) are graph-based data structures for representing Boolean functions. They have found widespread use in computer-aided design and in formal verification of digital circuits. Minimal trellises are graphical representations of error-correcting codes that play a prominent role in coding theory. This paper establishes a close connection between these two graphical models, as follows. Let /spl Cscr/ be a binary code of length n, and let f/sub c/(x/sub 1/, ..., x/sub n/) be the Boolean function that takes the value 0 at x/sub 1/, ..., x/sub n/ if and only if (x/sub 1/, ..., x/sub n/)/spl isin//spl Cscr/. Given this natural one-to-one correspondence between Boolean functions and binary codes, we prove that the minimal proper trellis for a code /spl Cscr/ with minimum distance d>1 is isomorphic to the single-terminal OBDD for its Boolean indicator function f/sub c/(x/sub 1/, ..., x/sub n/). Prior to this result, the extensive research during the past decade on binary decision diagrams (in computer engineering) and on minimal trellises (in coding theory) has been carried out independently. As outlined in this work, the realization that binary decision diagrams and minimal trellises are essentially the same data structure opens up a range of promising possibilities for transfer of ideas between these disciplines.

This paper focuses on generalized threshold gates (GTGs) that implement Boolean logic functions using elements with negative differential resistance (NDR). GTGs are capable of implementing Boolean functions, however, no effective synthesis algorithms have been proposed so far. We present that GTGs can be effectively implemented using unate functions. Our synthesis algorithm ensures that the circuit implementing n variable Boolean function consists of at most n+2 NDR elements and can be further optimized by reducing the number of switching elements.

The variable-entered Karnaugh map (VEKM) is shown to be the natural map for representing and manipulating general ‘big’ Boolean functions that are not restricted to the switching or two-valued case. The VEKM is utilized herein in producing a compact general solution of a system of Boolean equations. It serves as a powerful manual tool for function inversion, implementation of the solution procedure, handling don't-care conditions and minimization of the final expressions. The rules of using the VEKM are semi-algebraic and collective in nature, and hence are much easier to state, remember and implement than are the tabular and per-cell rules of classical maps. As a result, the maps used are significantly smaller than those required by classical methods. As an offshoot, the paper contributes some pictorial insight into the representation of ‘big’ Boolean algebras and functions. It also predicts the correct number of particular solutions of a Boolean equation, and produces an exhaustive list of particular solutions. Details of the method are carefully explained and further demonstrated via an illustrative example.

An analysis of the class of gene regulatory functions implied by a biochemical model

We introduce and examine some properties of a new complexity measure for Boolean functions. Unlike classical approaches, which are largely concerned with resource requirements, the measure examined here aims at quantifying the potential for lazy evaluation in a function. This measure is motivated by issues arising in the implementation of demand-driven logic simulators. The range of values that can be taken by the measure is precisely identified and a lower bound on the complexity of 'almost all' Boolean functions derived. In addition asymptotically exact values are derived for the class of all Boolean symmetric functions. >