scholarly journals Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$

10.37236/9469 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Jozefien D'haeseleer ◽  
Ferdinand Ihringer ◽  
Jonathan Mannaert ◽  
Leo Storme
Keyword(s):  
Projective Geometry ◽  
Boolean Functions ◽  
Association Scheme ◽  
Affine Geometry ◽  
Algebraic Combinatorics ◽  
Character Table ◽  
Analysis Of Boolean Functions ◽  
The Relationship

We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\mathrm{AG}(n,q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\mathrm{PG}(n,q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\mathrm{AG}(n,q)$ correspond to certain equitable bipartitions of the association scheme of $k$-spaces in $\mathrm{AG}(n,q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\mathrm{AG}(n,q)$. We define Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ by intersection properties with $k$-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ and $\mathrm{PG}(n,q)$. As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler $k$-sets. This paper focuses on $\mathrm{AG}(n,q)$ for $n > 3$, while the case for Cameron-Liebler line classes in $\mathrm{AG}(3,q)$ was already treated separately.

On Algebraic Immunity of Weight Symmetric H Boolean Functions

2014 ◽  
Vol 643 ◽  
pp. 124-129
Author(s):  
Jing Lian Huang ◽  
Zhuo Wang ◽  
Juan Li
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Theoretical Basis ◽  
Research Method ◽  
Algebraic Immunity ◽  
Correlation Immunity ◽  
Cryptographic Property ◽  
Cryptographic Primitive ◽  
New Research ◽  
The Relationship

Using the derivative of Boolean functions and the e-derivative defined by ourselves as research tools, we discuss the relationship among a variety of cryptographic properties of the weight symmetric H Boolean functions in the range of the weight with the existence of H Boolean functions. We also study algebraic immunity and correlation immunity of the weight symmetric H Boolean functions and the balanced H Boolean functions. We obtain that the weight symmetric H Boolean function should have the same algebraic immunity, correlation immunity, propagation degree and nonlinearity. Besides, we determine that there exist several kinds of H Boolean functions with resilient, algebraic immunity and optimal algebraic immunity. The above results not only provide a theoretical basis for reducing nearly half of workload when studying the cryptographic properties of H Boolean function, but also provide a new research method for the study of secure cryptographic property of Boolean functions. Such researches are important in cryptographic primitive designs.

scholarly journals Analysis of Boolean functions based on interaction graphs and their influence in system biology

2019 ◽  
Vol 32 (12) ◽  
pp. 7803-7821
Author(s):  
Ranjeet Kumar Rout ◽  
Santi P. Maity ◽  
Pabitra Pal Choudhury ◽  
Jayanta Kumar Das ◽  
Sk. Sarif Hassan ◽  
...  
Keyword(s):  
Boolean Functions ◽  
System Biology ◽  
Interaction Graphs ◽  
Analysis Of Boolean Functions

Linear codes from vectorial Boolean functions in the context of algebraic attacks

2020 ◽  
pp. 2150032
Author(s):  
M. Boumezbeur ◽  
S. Mesnager ◽  
K. Guenda
Keyword(s):  
Boolean Functions ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Weight Enumerator ◽  
Direct Link ◽  
Algebraic Attacks ◽  
Vectorial Boolean Functions ◽  
Lcd Codes ◽  
Vectorial Function ◽  
The Relationship

In this paper, we study the relationship between vectorial (Boolean) functions and cyclic codes in the context of algebraic attacks. We first derive a direct link between the annihilators of a vectorial function (in univariate form) and certain [Formula: see text]-ary cyclic codes (which we show that they are LCD codes). We also present some properties of those cyclic codes as well as their weight enumerator. In addition, we generalize the so-called algebraic complement and study its properties.

scholarly journals FourierSAT: A Fourier Expansion-Based Algebraic Framework for Solving Hybrid Boolean Constraints

2020 ◽  
Vol 34 (02) ◽  
pp. 1552-1560
Author(s):  
Anastasios Kyrillidis ◽  
Anshumali Shrivastava ◽  
Moshe Vardi ◽  
Zhiwei Zhang
Keyword(s):  
Boolean Functions ◽  
Continuous Optimization ◽  
Conjunctive Normal Form ◽  
Cardinality Constraints ◽  
Sat Solvers ◽  
Boolean Satisfiability Problem ◽  
Algebraic Framework ◽  
Clause Learning ◽  
Sat Solver ◽  
Analysis Of Boolean Functions

The Boolean SATisfiability problem (SAT) is of central importance in computer science. Although SAT is known to be NP-complete, progress on the engineering side—especially that of Conflict-Driven Clause Learning (CDCL) and Local Search SAT solvers—has been remarkable. Yet, while SAT solvers, aimed at solving industrial-scale benchmarks in Conjunctive Normal Form (CNF), have become quite mature, SAT solvers that are effective on other types of constraints (e.g., cardinality constraints and XORs) are less well-studied; a general approach to handling non-CNF constraints is still lacking. In addition, previous work indicated that for specific classes of benchmarks, the running time of extant SAT solvers depends heavily on properties of the formula and details of encoding, instead of the scale of the benchmarks, which adds uncertainty to expectations of running time.To address the issues above, we design FourierSAT, an incomplete SAT solver based on Fourier analysis of Boolean functions, a technique to represent Boolean functions by multilinear polynomials. By such a reduction to continuous optimization, we propose an algebraic framework for solving systems consisting of different types of constraints. The idea is to leverage gradient information to guide the search process in the direction of local improvements. Empirical results demonstrate that FourierSAT is more robust than other solvers on certain classes of benchmarks.

Effects of the Internal Structure of H Boolean Functions on Algebraic Immunity and Algebraic Degree

2013 ◽  
Vol 774-776 ◽  
pp. 1721-1724
Author(s):  
Jing Lian Huang ◽  
Xiu Juan Yuan ◽  
Jian Hua Wang
Keyword(s):  
Boolean Function ◽  
Internal Structure ◽  
Boolean Functions ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
Hamming Weight ◽  
Relationship Of ◽  
The Relationship

We go deep into the internal structure of the Boolean functions values, and study the relationship of algebraic immunity and algebraic degree of Boolean functions with the Hamming weight with the diffusion included. Then we get some theorems which relevance together algebraic immunity, annihilators and algebraic degree of H Boolean functions by the e-derivative which is a part of the H Boolean function. Besides, we also get the results that algebraic immunity and algebraic degree of Boolean functions can be linked together by the e-derivative of H Boolean functions and so on.

A NOTE ON THE POLYNOMIAL REPRESENTATION OF BOOLEAN FUNCTIONS OVER GF(2)

1999 ◽  
Vol 10 (04) ◽  
pp. 535-542
Author(s):  
RICHARD BEIGEL ◽  
ANNA BERNASCONI
Keyword(s):  
Boolean Function ◽  
Open Problem ◽  
Boolean Functions ◽  
Polynomial Representation ◽  
Maximal Sensitivity ◽  
Input Strings ◽  
The Relationship ◽  
Block Sensitivity

We investigate the representation of Boolean functions as polynomials over the field GF(2), and prove an interesting characteriztion theorem: the degree of a Boolean function over GF(2) is equal to the size of its largest subfunction that takes the value 1 on an odd number of input strings. We then present some properties of odd functions, i.e., functions that take the value 1 on an odd number of strings, and analyze the connections between the problem of the existence of odd functions with very low maximal sensitivity and the long standing open problem of the relationship between the maximal sensitivity and the block sensitivity of Boolean functions.

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