Global avalanche characteristics and propagation criterion of balanced Boolean functions

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.

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

Security and Communication Networks ◽

10.1155/2019/1348639 ◽

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.

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A New Family of Boolean Functions with Good Cryptographic Properties

Axioms ◽

10.3390/axioms10020042 ◽

2021 ◽

Vol 10 (2) ◽

pp. 42

Author(s):

Guillermo Sosa-Gómez ◽

Octavio Paez-Osuna ◽

Omar Rojas ◽

Evaristo José Madarro-Capó

Keyword(s):

Boolean Functions ◽

Linear Codes ◽

Bent Functions ◽

New Family ◽

Propagation Criterion ◽

Daunting Task ◽

Wide Range ◽

Algebraic Codes ◽

New Construction

In 2005, Philippe Guillot presented a new construction of Boolean functions using linear codes as an extension of the Maiorana–McFarland’s (MM) construction of bent functions. In this paper, we study a new family of Boolean functions with cryptographically strong properties, such as non-linearity, propagation criterion, resiliency, and balance. The construction of cryptographically strong Boolean functions is a daunting task, and there is currently a wide range of algebraic techniques and heuristics for constructing such functions; however, these methods can be complex, computationally difficult to implement, and not always produce a sufficient variety of functions. We present in this paper a construction of Boolean functions using algebraic codes following Guillot’s work.

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Relationship between Algebraic Immunity, Correlation Immune and Propagation of Boolean Functions

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.321-324.2649 ◽

2013 ◽

Vol 321-324 ◽

pp. 2649-2652

Author(s):

Jing Lian Huang ◽

Zhuo Wang ◽

Chun Ling Zhang

Keyword(s):

Boolean Function ◽

Boolean Functions ◽

Algebraic Immunity ◽

Correlation Immunity ◽

Propagation Criterion ◽

Function Correlation ◽

The Relationship ◽

Research Tools

Using the derivative of the Boolean function and thee-derivative defined by ourselves as research tools, we study the problem of relationship between algebraic immunity,correlation immunity and propagation of H Boolean functions with weight of and satisfying the 1st-order propagation criterion togetherwith the problem of their compatibility. We get the results , suchas the relationship between the number of annihilators and correlation immunityorder, the relationship between the number of correlation immunity order and algebraic immune degree together with theircompatibility and the largest propagations of H Boolean function, the relationships between propagationsof Boolean function, correlation immunity order and algebraic immune degree.

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