The Importance of Study Cryptographic Properties of H Boolean Function with Hamming Weight of 2n-1 + 2n-2

Abstract The notion of the confusion coefficient is a property that attempts to characterize confusion property of cryptographic algorithms against differential power analysis. In this article, we establish a relationship between the confusion coefficient and the autocorrelation function for any Boolean function and give a tight upper bound and a tight lower bound on the confusion coefficient for any (balanced) Boolean function. We also deduce some deep relationships between the sum-of-squares of the confusion coefficient and other cryptographic indicators (the sum-of-squares indicator, hamming weight, algebraic immunity and correlation immunity), respectively. Moreover, we obtain some trade-offs among the sum-of-squares of the confusion coefficient, the signal-to-noise ratio and the redefined transparency order for a Boolean function.

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

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.774-776.1721 ◽

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.

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FKN theorem for the multislice, with applications

Combinatorics Probability Computing ◽

10.1017/s0963548319000361 ◽

2019 ◽

Vol 29 (2) ◽

pp. 200-212

Author(s):

Yuval Filmus

Keyword(s):

Boolean Function ◽

Isoperimetric Inequality ◽

Hamming Weight ◽

Boolean Cube ◽

Optimal Set ◽

Single Coordinate

AbstractThe Friedgut–Kalai–Naor (FKN) theorem states that if ƒ is a Boolean function on the Boolean cube which is close to degree one, then ƒ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice.As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.

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Relationship between Correlation Immune Order and Algebraic Immunity Order of Boolean Functions

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.740.273 ◽

2013 ◽

Vol 740 ◽

pp. 273-278

Author(s):

Jing Lian Huang ◽

Zhuo Wang ◽

Ya Jing Liu

Keyword(s):

Boolean Function ◽

Internal Structure ◽

Boolean Functions ◽

Necessary Conditions ◽

Algebraic Immunity ◽

Hamming Weight ◽

Analytic Combinatorics ◽

Sufficient And Necessary Conditions ◽

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. Additionally, by the methods of cascade calculations and analytic combinatorics, cryptographic properties such as correlation immune and algebraic immunity of H Boolean functions with Hamming weight of with diffusibility are studied. Then we prove the existing of m order correlation immune H Boolean functions ,and get the result of the sufficient and necessary conditions of algebraic immunity order is 1 of Boolean function with correlation immune order is m.

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Some Cryptographic Properties of H Boolean Functions

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.740.279 ◽

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.

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Effects of e-Derivative on Algebraic Immunity, Correlation Immunity and Algebraic Degree of H Boolean Functions

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.411-414.45 ◽

2013 ◽

Vol 411-414 ◽

pp. 45-48 ◽

Cited By ~ 1

Author(s):

Jing Lian Huang ◽

Zhuo Wang ◽

Jing Zhang

Keyword(s):

Boolean Function ◽

Boolean Functions ◽

Algebraic Degree ◽

Algebraic Immunity ◽

Hamming Weight ◽

Correlation Immunity ◽

Research Tools

Using the derivative of the Boolean function and the e-derivative defined by ourselves as research tools, we study the Effects of e-derivative on algebraic immunity, correlation immunity and algebraic degree of H Boolean functions with the Hamming weight . We get some theorems which relevance together algebraic immunity, annihilators, correlation immunity and algebraic degree of H Boolean functions by the e-derivative. Besides, we also get the results that algebraic immunity, correlation immunity and algebraic degree of Boolean functions can be linked together by the e-derivative of H Boolean functions.

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Relations among E-Derivative, Derivative, Algebraic Degree, Correlation Immunity and Annihilators for H Boolean Functions

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.989-994.2599 ◽

2014 ◽

Vol 989-994 ◽

pp. 2599-2604

Author(s):

Jing Lian Huang ◽

Zhuo Wang ◽

Juan Li

Keyword(s):

Boolean Function ◽

Boolean Functions ◽

Algebraic Degree ◽

Hamming Weight ◽

Application Range ◽

Correlation Immunity ◽

Degree Correlation ◽

Cryptographic Primitive ◽

Relationship Of ◽

Formula Method

Using the derivative of the Boolean function and the e-derivative defined by ourselves as research tools, going deep into the internal structure of Boolean functions, we study relationship of algebraic degree, correlation immunity and annihilators for H Boolean functions with a specific Hamming weight. We obtain the algebraic degree of the e-derivative which is a component of H Boolean functions decide the algebraic degree of H Boolean functions. Besides, we describe the characteristics of the algebraic degree of e-derivative for the correlation immune H Boolean functions. We also check the e-derivative of H Boolean functions can put annihilators, correlation immunity and algebraic degree of H Boolean functions together. Meanwhile, we also deduce a formula method to solving annihilators of H Boolean functions. Such researches are important in cryptographic primitive designs, and have significance and role in the theory and application range of cryptosystems.

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