Classification of Boolean functions of n variables

1972 ◽  
Vol 15 (11) ◽  
pp. 1352-1360
Author(s):  
I. V. Kotel'nikov
Keyword(s):  
Boolean Functions

On a new classification of Boolean functions

2019 ◽  
Vol 10 (2) ◽  
pp. 159-168
Author(s):  
Sergei N Fedorov
Keyword(s):  
Boolean Functions ◽  
New Classification

Рассматривается недавно предложенный подход к исследованию булевых функций, в основе которого лежит понятие класса $\Delta$-эквивалентности: множества булевых функций с одной и той же функцией автокорреляции. Такая классификация представляется полезной, поскольку многие криптографические характеристики булевых функций, принадлежащих одному и тому же классу $\Delta$-эквивалентности, одинаковы.

scholarly journals Boolean differential equations: A common model for classes, lattices, and arbitrary sets of Boolean functions

2015 ◽  
Vol 28 (1) ◽  
pp. 51-76 ◽  
Author(s):  
Bernd Steinbach ◽  
Christian Posthoff
Keyword(s):  
Differential Equations ◽  
Boolean Functions ◽  
Differential Calculus ◽  
Short Introduction ◽  
The Core ◽  
Boolean Equation ◽  
General Difference ◽  
Derivatives Of

The Boolean Differential Calculus (BDC) significantly extends the Boolean Algebra because not only Boolean values 0 and 1, but also changes of Boolean values or Boolean functions can be described. A Boolean Differential Equation (BDe) is a Boolean equation that includes derivative operations of the Boolean Differential Calculus. This paper aims at the classification of BDEs, the characterization of the respective solutions, algorithms to calculate the solution of a BDe, and selected applications. We will show that not only classes and arbitrary sets of Boolean functions but also lattices of Boolean functions can be expressed by Boolean Differential equations. In order to reach this aim, we give a short introduction into the BDC, emphasize the general difference between the solutions of a Boolean equation and a BDE, explain the core algorithms to solve a BDe that is restricted to all vectorial derivatives of f (x) and optionally contains Boolean variables. We explain formulas for transforming other derivative operations to vectorial derivatives in order to solve more general BDEs. New fields of applications for BDEs are simple and generalized lattices of Boolean functions. We describe the construction, simplification and solution. The basic operations of XBOOLE are sufficient to solve BDEs. We demonstrate how a XBooLe-problem program (PRP) of the freely available XBooLe-Monitor quickly solves some BDes.

scholarly journals Classification of Boolean Functions Where Affine Functions Are Uniformly Distributed

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ranjeet Kumar Rout ◽  
Pabitra Pal Choudhury ◽  
Sudhakar Sahoo
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Cartesian Product ◽  
Affine Function ◽  
Recursive Procedure ◽  
Symmetry Properties ◽  
Affine Functions

The present paper on classification of -variable Boolean functions highlights the process of classification in a coherent way such that each class contains a single affine Boolean function. Two unique and different methods have been devised for this classification. The first one is a recursive procedure that uses the Cartesian product of sets starting from the set of one variable Boolean functions. In the second method, the classification is done by changing some predefined bit positions with respect to the affine function belonging to that class. The bit positions which are changing also provide us information concerning the size and symmetry properties of the classes/subclasses in such a way that the members of classes/subclasses satisfy certain similar properties.

Classification of cubic (n-4)-resilient Boolean functions

2006 ◽  
Vol 52 (4) ◽  
pp. 1670-1676 ◽  
Author(s):  
An Braeken ◽  
Y. Borissov ◽  
S. Nikova ◽  
B. Preneel
Keyword(s):  
Boolean Functions

Topological Equivalence Classification of Balanced Linearly Separable Boolean Functions on n-Dimensional Hypercube

2021 ◽  
Vol 31 (02) ◽  
pp. 2150031
Author(s):  
Qinbin He ◽  
Fangyue Chen ◽  
Wei Jin
Keyword(s):  
Spatial Structure ◽  
Boolean Functions ◽  
Conformal Transformation ◽  
Classification Algorithm ◽  
Equivalence Classes ◽  
Topological Equivalence ◽  
Novel Algorithm

The concept of conformal transformation is proposed through the study of the spatial structure of [Formula: see text]-dimensional hypercubes. Based on conformal transformation, a novel algorithm, called topological equivalence classification algorithm, is proposed for classifying balanced linearly separable Boolean functions. By the proposed algorithm, the topological equivalence classes of all balanced linearly separable Boolean functions and the number of Boolean functions in each of the topological equivalence classes are obtained. In addition, the properties of conformal transformation also show an application prospect for decomposing nonlinearly separable Boolean functions.

Classification of correlation-immune and minimal correlation-immune Boolean functions of 4 and 5 variables

2015 ◽  
Vol 25 (4) ◽  
Author(s):  
Evgeniy. K. Alekseev ◽  
Ekaterina K. Karelina
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Equivalence Classes ◽  
Constant Function ◽  
Immune Functions

AbstractA classification of correlation-immune and minimal corelation-immune Boolean function of 4 and 5 variables with respect to the Jevons group is given. Representatives of the equivalence classes of correlationimmune functions of 4 and 5 variables are decomposed into minimal correlation-immune functions. Characteristics of various decompositions of the constant function 1 into minimal correlation-immune functions are presented.

CLASSIFYING WITH A TWO NEURON CNN

2012 ◽  
Vol 22 (02) ◽  
pp. 1250035
Author(s):  
MIREIA VINYOLES-SERRA ◽  
XAVIER VILASÍS-CARDONA
Keyword(s):  
Steady State ◽  
Turing Machine ◽  
Boolean Functions ◽  
Functional Relation ◽  
Input Output ◽  
Classification Problems ◽  
Universal Turing Machine ◽  
Binary Input ◽  
Universal Cell

We analyze the two neuron CNN for the particular parameter range where the system converges to constant outputs. The functional relation between the external inputs and the steady state values of the neuron states is found and proves to be useful to solve classification problems. In fact, an exhaustive classification of the binary input–output relations that can be achieved by a two neuron CNN is established. From this relation, we propose an algorithm relating the CNN parameters and each one of the different classification problems. As an illustration, we attempt to implement the header action of a universal Turing machine and Boolean functions. Our results are compared to the CNN universal cell.

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