CONSTRUCTING 2m-VARIABLE BOOLEAN FUNCTIONSWITH OPTIMAL ALGEBRAIC IMMUNITY BASED ON POLAR DECOMPOSITION OF $\mathbb{F}^\ast_{2^{2m}}$

2014 ◽  
Vol 25 (05) ◽  
pp. 537-551 ◽  
Author(s):  
JIA ZHENG ◽  
BAOFENG WU ◽  
YUFU CHEN ◽  
ZHUOJUN LIU
Keyword(s):  
Boolean Functions ◽  
Polar Decomposition ◽  
Multiplicative Group ◽  
Slight Modification ◽  
Additive Group ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
New Construction ◽  
Fast Algebraic Attacks ◽  
Class Of Functions

Constructing 2m-variable Boolean functions with optimal algebraic immunity based on decomposition of additive group of the finite field [Formula: see text] seems to be a promising approach since Tu and Deng's work. In this paper, we consider the same problem in a new way. Based on polar decomposition of the multiplicative group of [Formula: see text], we propose a new construction of Boolean functions with optimal algebraic immunity. By a slight modification of it, we obtain a class of balanced Boolean functions achieving optimal algebraic immunity, which also have optimal algebraic degree and high nonlinearity. Computer investigations imply that this class of functions also behaves well against fast algebraic attacks.

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.

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.

scholarly journals New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

2020 ◽  
Vol 17 (7) ◽  
pp. 639-654
Author(s):  
Dheeraj Kumar SHARMA ◽  
Rajoo PANDEY
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Boolean Functions ◽  
Primitive Element ◽  
Galois Field ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
Greatest Common Divisor ◽  
Primitive Elements ◽  
High Nonlinearity

This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field,  used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and  is 1. The constructed balanced  variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of   for odd ,  for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.

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.

NEW CONSTRUCTIONS OF VECTORIAL BOOLEAN FUNCTIONS WITH GOOD CRYPTOGRAPHIC PROPERTIES

2012 ◽  
Vol 23 (03) ◽  
pp. 749-760
Author(s):  
DESHUAI DONG ◽  
LONGJIANG QU ◽  
SHAOJING FU ◽  
CHAO LI
Keyword(s):  
Boolean Functions ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
High Nonlinearity ◽  
Vectorial Boolean Functions ◽  
Very High

Vectorial Boolean functions play an important role in cryptography. How to construct vectorial Boolean functions with good cryptographic properties is a nice problem that worth to be investigated. In this paper we present several constructions of balanced vectorial Boolean functions with high algebraic immunity, high(or optimum) algebraic degree, and very high nonlinearity. In some cases, the constructed functions also achieve optimum algebraic immunity.

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