On the number of equivalence classes of Boolean functions under a transformation group (Corresp.)

1980 ◽  
Vol 26 (5) ◽  
pp. 625-626 ◽  
Author(s):  
J. Denev ◽  
V. Tonchev
Keyword(s):  
Boolean Functions ◽  
Transformation Group ◽  
Equivalence Classes

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.

SAT-Based Generation of Optimum Circuits with Polymorphic Behavior Support

2019 ◽  
Vol 28 (supp01) ◽  
pp. 1940010
Author(s):  
Petr Fišer ◽  
Ivo Háleček ◽  
Jan Schmidt ◽  
Václav Šimek
Keyword(s):  
Boolean Functions ◽  
Equivalence Classes ◽  
Experimental Results ◽  
Behavior Support ◽  
Operating Environment ◽  
Polymorphic Behavior ◽  
Multi Level ◽  
Boolean Optimization ◽  
Hardware Structure

This paper presents a method for generating optimum multi-level implementations of Boolean functions based on Satisfiability (SAT) and Pseudo-Boolean Optimization (PBO) problems solving. The method is able to generate one or enumerate all optimum implementations, while the allowed target gate types and gates costs can be arbitrarily specified. Polymorphic circuits represent a newly emerging computation paradigm, where one hardware structure is capable of performing two or more different intended functions, depending on instantaneous conditions in the target operating environment. In this paper we propose the first method ever, generating provably size-optimal polymorphic circuits. Scalability and feasibility of the method are documented by providing experimental results for all NPN-equivalence classes of four-input functions implemented in AND–Inverter and AND–XOR–Inverter logics without polymorphic behavior support being used and for all pairs of NPN–equivalence classes of three-input functions for polymorphic circuits. Finally, several smaller benchmark circuits were synthesized optimally, both in standard and polymorphic logics.

Affine equivalence of monomial rotation symmetric Boolean functions: A Pólya’s theorem approach

2016 ◽  
Vol 10 (3-4) ◽  
Author(s):  
Thomas W. Cusick ◽  
K. V. Lakshmy ◽  
M. Sethumadhavan
Keyword(s):  
Boolean Functions ◽  
Affine Transformation ◽  
Equivalence Classes ◽  
Affine Equivalence ◽  
Symmetric Boolean Functions ◽  
Rotation Symmetric ◽  
Long Time ◽  
Input Variables ◽  
Large Groups ◽  
Rotation Symmetric Boolean Functions

AbstractTwo 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

scholarly journals On the number of equivalence classes of invertible Boolean functions under action of permutation of variables on domain and range

2016 ◽  
Vol 100 (114) ◽  
pp. 95-99 ◽  
Author(s):  
Marko Caric ◽  
Miodrag Zivkovic
Keyword(s):  
Boolean Functions ◽  
Equivalence Classes

Let Vn be the number of equivalence classes of invertible maps from {0,1}n to {0,1}n, under action of permutation of variables on domain and range. So far, the values Vn have been known for n ? 6. This paper describes the procedure by which the values of Vn are calculated for n ? 30.

Computing Affine Equivalence Classes of Boolean Functions by Group Isomorphism

2016 ◽  
Vol 65 (12) ◽  
pp. 3606-3616 ◽  
Author(s):  
Yan Zhang ◽  
Guowu Yang ◽  
William N.N. Hung ◽  
Juling Zhang
Keyword(s):  
Boolean Functions ◽  
Equivalence Classes ◽  
Affine Equivalence ◽  
Group Isomorphism
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