scholarly journals 1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Gang Han ◽  
Yu Yu ◽  
Xiangxue Li ◽  
Qifeng Zhou ◽  
Dong Zheng ◽  
...  
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Algebraic Immunity ◽  
Immune Functions ◽  
Cryptographic Primitives ◽  
Algebraic Attacks ◽  
First Order ◽  
New Class ◽  
Resilient Function ◽  
Fast Algebraic Attacks

Several factors (e.g., balancedness, good correlation immunity) are considered as important properties of Boolean functions for using in cryptographic primitives. A Boolean function is perfect algebraic immune if it is with perfect immunity against algebraic and fast algebraic attacks. There is an increasing interest in construction of Boolean function that is perfect algebraic immune combined with other characteristics, like resiliency. A resilient function is a balanced correlation-immune function. This paper uses bivariate representation of Boolean function and theory of finite field to construct a generalized and new class of Boolean functions on even variables by extending the Carlet-Feng functions. We show that the functions generated by this construction support cryptographic properties of 1-resiliency and (sub)optimal algebraic immunity and further propose the sufficient condition of achieving optimal algebraic immunity. Compared experimentally with Carlet-Feng functions and the functions constructed by the method of first-order concatenation existing in the literature on even (from 6 to 16) variables, these functions have better immunity against fast algebraic attacks. Implementation results also show that they are almost perfect algebraic immune functions.

A CLASS OF 1-RESILIENT BOOLEAN FUNCTIONS WITH OPTIMAL ALGEBRAIC IMMUNITY AND GOOD BEHAVIOR AGAINST FAST ALGEBRAIC ATTACKS

2014 ◽  
Vol 25 (06) ◽  
pp. 763-780 ◽  
Author(s):  
DENG TANG ◽  
CLAUDE CARLET ◽  
XIAOHU TANG
Keyword(s):  
Boolean Functions ◽  
Algebraic Degree ◽  
Stream Ciphers ◽  
Algebraic Immunity ◽  
Infinite Class ◽  
Algebraic Attacks ◽  
Correlation Attack ◽  
Fast Algebraic Attacks ◽  
First Time ◽  
Class Of Functions

Recently, Tang, Carlet and Tang presented a combinatorial conjecture about binary strings, allowing proving that all balanced functions in some infinite class they introduced have optimal algebraic immunity. Later, Cohen and Flori completely proved that the conjecture is true. These functions have good (provable or at least observable) cryptographic properties but they are not 1-resilient, which represents a drawback for their use as filter functions in stream ciphers. We propose a construction of an infinite class of 1-resilient Boolean functions with optimal algebraic immunity by modifying the functions in this class. The constructed functions have optimal algebraic degree, that is, meet the Siegenthaler bound, and high nonlinearity. We prove a lower bound on their nonlinearity, but as for the Carlet-Feng functions and for the functions mentioned above, this bound is not enough for ensuring a nonlinearity sufficient for allowing resistance to the fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for small numbers of variables. Our computations show that the functions in this class have very good nonlinearity and also good immunity to fast algebraic attacks. This is the first time that an infinite class of functions gathers all of the main criteria allowing these functions to be used as filters in stream ciphers.

Construction of Odd-Variable Boolean Functions with Optimum Algebraic Immunity

2013 ◽  
Vol 347-350 ◽  
pp. 2952-2956
Author(s):  
Zhi Chao Zhang ◽  
Zheng Huang ◽  
Jie Zhang ◽  
Qiao Yan Wen
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Decomposition Method ◽  
Algebraic Immunity ◽  
Algebraic Attacks ◽  
Cryptographic Property

Recently, algebraic attacks becomes a major attack method to threat to cryptography security. In order to resist algebraic attacks, algebraic immunity as a Boolean function cryptographic property has been put out. This makes that Boolean functions should have high algebraic immunity to resist algebraic attacks. In this paper, a specific decomposition method of the space is proposed. By the method, we construct a class of odd number of variables Boolean functions with optimal algebraic immunity.

Construction of Boolean Function with Maximum Algebraic Immunity

2012 ◽  
Vol 546-547 ◽  
pp. 387-392
Author(s):  
Xiao Wen Xiong ◽  
Ai Guo Wei ◽  
Kai Yin
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Stream Ciphers ◽  
Algebraic Immunity ◽  
Algebraic Attacks ◽  
Survey Paper

To resist algebraic attacks, a high algebraic immunity is now an important criteria for Boolean functions used in stream ciphers. In recent years, several constructions of Boolean functions with maximum algebraic immunity (MAI) have been investigated. In this survey paper, we review the recent constructions of Boolean functions with MAI, and classify those into several different classes by the construction idea. Further, we also present some results and developments of these methods, including some results of the authors.

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.

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