The relationship between the level of affinity and cryptographic parameters of Boolean functions

2008 ◽  
Vol 18 (3) ◽  
Author(s):  
M. L. Buryakov
Keyword(s):  
Boolean Functions ◽  
The Relationship

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.

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.

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.

The Judgment of Algebraic Immunity and Resilience of Balanced H Boolean Functions

2013 ◽  
Vol 459 ◽  
pp. 195-200
Author(s):  
Jing Lian Huang ◽  
Zhuo Wang ◽  
Ya Jing Liu
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Algebraic Immunity ◽  
Main Research ◽  
The Relationship ◽  
Research Tools

Using the derivative of the Boolean function and the e-derivative defined by ourselves as main research tools, we study the relationship among e-derivative, algebraic immunity and resilience of balanced H Boolean functions.We get some theorems which connect algebraic immunity, annihilators, resilience, derivative and e-derivative of balanced H Boolean functions together. Besides, we also get the judgment of algebraic immunity and resilience for three classes of balanced Boolean functions by the e-derivative.

Some Cryptographic Properties of H Boolean Functions

2013 ◽  
Vol 740 ◽  
pp. 279-283
Author(s):  
Jing Lian Huang ◽  
Zhuo Wang ◽  
Ya Jing Liu
Keyword(s):  
Boolean Function ◽  
Internal Structure ◽  
Boolean Functions ◽  
Mathematical Expression ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
Hamming Weight ◽  
The Relationship ◽  
Research Tools

Using the derivative of the Boolean function and the e-derivative defined by ourselves as research tools, we go deep into the internal structure of the Boolean function values. Cryptographic properties such as algebraic immunity, correlation immune and algebraic degree of H Boolean functions with Hamming weight of with diffusibility and the relationship between these properties are studied. Then we get the results of the mathematical expression of linear annihilators, the values of algebraic degree and correlation immune order, and so on.

Matrix Structure and Cryptographic Properties of Rotation Symmetric H Boolean Functions

2013 ◽  
Vol 651 ◽  
pp. 955-960
Author(s):  
Jing Lian Huang ◽  
Zhuo Wang ◽  
Jing Zhang
Keyword(s):  
Boolean Functions ◽  
Structural Features ◽  
Algebraic Immunity ◽  
Matrix Structure ◽  
Third Order ◽  
Correlation Immunity ◽  
Pure Rotation ◽  
Rotation Symmetric ◽  
Relationship Of ◽  
The Relationship

Use the derivative of the Boolean function and the e-derivative defined by ourselves as research tools, Cryptographic properties such as structural features, and the balance, correlation immunity of diffused rotation symmetric H Boolean functions is studied. Then we get the results of the relationship of a matrix structure, correlation immunity, dimension and balance of rotationally symmetric H Boolean functions, etc. As well as the result of algebraic immunity and the algebraic immunity orders that related with the Third Order completely pure rotation symmetric Boolean functions.

scholarly journals Classes of bent functions identified by specific normal forms and generated using Boolean differential equations

2011 ◽  
Vol 24 (3) ◽  
pp. 357-383 ◽  
Author(s):  
Bernd Steinbach ◽  
Christian Posthoff
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Boolean Functions ◽  
Differential Calculus ◽  
Normal Forms ◽  
Direct Calculation ◽  
Bent Functions ◽  
Second Step ◽  
Complexity Classes ◽  
The Relationship

This paper aims at the identification of classes of bent functions in order to allow their construction without searching or sieving. In order to reach this aim, we studied first the relationship between bent functions and complexity classes defined by the Specific Normal Forms of all Boolean functions. As result of this exploration we found classes of bent functions which are embedded in different complexity classes defined by the Specific Normal Form. In the second step to reach our global aim, we utilized the found classes of bent functions in order to express bent functions in terms of derivative operations of the Boolean Differential Calculus. In detail, we studied bent functions of two and four variables. This exploration leads finally to Boolean differential equations that will allow the direct calculation of all bent functions of two and four variables. A given generalization allows to calculate subsets of bent functions for each even number of Boolean variables.

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