scholarly journals On the confusion coefficient of Boolean functions

2021 ◽  
Vol 16 (1) ◽  
pp. 1-13
Author(s):  
Yu Zhou ◽  
Jianyong Hu ◽  
Xudong Miao ◽  
Yu Han ◽  
Fuzhong Zhang
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Power Analysis ◽  
Signal To Noise Ratio ◽  
Sum Of Squares ◽  
Hamming Weight ◽  
Signal To Noise ◽  
Trade Offs ◽  
Cryptographic Algorithms ◽  
Transparency Order

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.

scholarly journals On the Modified Transparency Order of n , m -Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Yu Zhou ◽  
Yongzhuang Wei ◽  
Hailong Zhang ◽  
Wenzheng Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Power Analysis ◽  
Sum Of Squares ◽  
Walsh Transform ◽  
Algebraic Immunity ◽  
Hamming Weight ◽  
Absolute Value ◽  
Transparency Order ◽  
Weight Model

The concept of transparency order is introduced to measure the resistance of n , m -functions against multi-bit differential power analysis in the Hamming weight model, including the original transparency order (denoted by TO ), redefined transparency order (denoted by RTO ), and modified transparency order (denoted by MTO ). In this paper, we firstly give a relationship between MTO and RTO and show that RTO is less than or equal to MTO for any n , m -functions. We also give a tight upper bound and a tight lower bound on MTO for balanced n , m -functions. Secondly, some relationships between MTO and the maximal absolute value of the Walsh transform (or the sum-of-squares indicator, algebraic immunity, and the nonlinearity of its coordinates) for n , m -functions are obtained, respectively. Finally, we give MTO and RTO for (4,4) S-boxes which are commonly used in the design of lightweight block ciphers, respectively.

On the Signal-to-Noise Ratio for Boolean Functions

2020 ◽  
Vol E103.A (12) ◽  
pp. 1659-1665
Author(s):  
Yu ZHOU ◽  
Wei ZHAO ◽  
Zhixiong CHEN ◽  
Weiqiong WANG ◽  
Xiaoni DU
Keyword(s):  
Boolean Functions ◽  
Signal To Noise Ratio ◽  
Signal To Noise ◽  
Noise Ratio

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.

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

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.

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.

ONE SUFFICIENT AND NECESSARY CONDITION ON BALANCED BOOLEAN FUNCTIONS WITH σf = 22n + 2n+3(n ≥ 3)

2014 ◽  
Vol 25 (03) ◽  
pp. 343-353 ◽  
Author(s):  
YU ZHOU ◽  
LIN WANG ◽  
WEIQIONG WANG ◽  
XINFENG DONG ◽  
XIAONI DU
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Sum Of Squares ◽  
Necessary Condition ◽  
Resilient Function ◽  
The Absolute ◽  
Global Avalanche Characteristics ◽  
Sufficient And Necessary Condition

The Global Avalanche Characteristics (including the sum-of-squares indicator and the absolute indicator) measure the overall avalanche characteristics of a cryptographic Boolean function. Son et al. (1998) gave the lower bound on the sum-of-squares indicator for a balanced Boolean function. In this paper, we give a sufficient and necessary condition on a balanced Boolean function reaching the lower bound on the sum-of-squares indicator. We also analyze whether these balanced Boolean functions exist, and if they reach the lower bounds on the sum-of-squares indicator or not. Our result implies that there does not exist a balanced Boolean function with n-variable for odd n(n ≥ 5). We conclude that there does not exist a m(m ≥ 1)-resilient function reaching the lower bound on the sum-of-squares indicator with n-variable for n ≥ 7.

scholarly journals Measuring the Sum-of-Squares Indicator of Boolean Functions in Encryption Algorithm for Internet of Things

2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
Yu Zhou ◽  
Yongzhuang Wei ◽  
Fengrong Zhang
Keyword(s):  
Internet Of Things ◽  
Boolean Function ◽  
Boolean Functions ◽  
Search Algorithm ◽  
Sum Of Squares ◽  
Encryption Algorithm ◽  
Correlation Distribution ◽  
Cryptographic Algorithm ◽  
Global Avalanche Characteristics ◽  
The Internet Of Things

Encryption algorithm has an important application in ensuring the security of the Internet of Things. Boolean function is the basic component of symmetric encryption algorithm, and its many cryptographic properties are important indicators to measure the security of cryptographic algorithm. This paper focuses on the sum-of-squares indicator of Boolean function; an upper bound and a lower bound of the sum-of-squares on Boolean functions are obtained by the decomposition Boolean functions; some properties and a search algorithm of Boolean functions with the same autocorrelation (or cross-correlation) distribution are given. Finally, a construction method to obtain a balanced Boolean function with small sum-of-squares indicator is derived by decomposition Boolean functions. Compared with the known balanced Boolean functions, the constructed functions have the higher nonlinearity and the better global avalanche characteristics property.

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

2013 ◽  
Vol 411-414 ◽  
pp. 45-48 ◽  
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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