scholarly journals On Unbalanced Boolean Functions with Best Correlation Immunity

10.37236/8557 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Denis S. Krotov ◽  
Konstantin V. Vorob'ev
Keyword(s):  
Boolean Function ◽  
Infinite Series ◽  
Boolean Functions ◽  
Orthogonal Arrays ◽  
Equivalence Classes ◽  
Correlation Immunity

It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in $n$ variables cannot exceed $2n/3-1$; moreover, it is $2n/3-1$ if and only if the function corresponds to an equitable $2$-partition of the $n$-cube with an eigenvalue $-n/3$ of the quotient matrix. The known series of such functions have proportion $1:3$, $3:5$, or $7:9$ of the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean function attains the correlation-immunity bound and has ratio $C:B$ of the number of ones and zeros, then $CB$ is divisible by $3$. In particular, this proves the nonexistence of equitable partitions for an infinite series of putative quotient matrices.  We also establish that there are exactly $2$ equivalence classes of the equitable partitions of the $12$-cube with quotient matrix $[[3,9],[7,5]]$ and $16$ classes, with $[[0,12],[4,8]]$. These parameters correspond to the Boolean functions in $12$ variables with correlation immunity $7$ and proportion $7:9$ and $1:3$, respectively (the case $3:5$ remains unsolved). This also implies the characterization of the orthogonal arrays OA$(1024,12,2,7)$ and  OA$(512,11,2,6)$.

On 1-stable perfectly balanced Boolean functions

2017 ◽  
Vol 27 (2) ◽  
Author(s):  
Stanislav V. Smyshlyaev
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
The Other ◽  
Correlation Immunity ◽  
Other Hand

AbstractThe paper is concerned with relations between the correlation-immunity (stability) and the perfectly balancedness of Boolean functions. It is shown that an arbitrary perfectly balanced Boolean function fails to satisfy a certain property that is weaker than the 1-stability. This result refutes some assertions by Markus Dichtl. On the other hand, we present new results on barriers of perfectly balanced Boolean functions which show that any perfectly balanced function such that the sum of the lengths of barriers is smaller than the length of variables, is 1-stable.

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.

Classification of correlation-immune and minimal correlation-immune Boolean functions of 4 and 5 variables

2015 ◽  
Vol 25 (4) ◽  
Author(s):  
Evgeniy. K. Alekseev ◽  
Ekaterina K. Karelina
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Equivalence Classes ◽  
Constant Function ◽  
Immune Functions

AbstractA classification of correlation-immune and minimal corelation-immune Boolean function of 4 and 5 variables with respect to the Jevons group is given. Representatives of the equivalence classes of correlationimmune functions of 4 and 5 variables are decomposed into minimal correlation-immune functions. Characteristics of various decompositions of the constant function 1 into minimal correlation-immune functions are presented.

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.

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

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.

On Relationship of some Cryptographic Properties for Weight Symmetric H Boolean Functions

2013 ◽  
Vol 774-776 ◽  
pp. 1762-1765 ◽  
Author(s):  
Jing Lian Huang ◽  
Zhuo Wang ◽  
Jie Hu
Keyword(s):  
Boolean Function ◽  
Internal Structure ◽  
Research Methods ◽  
Boolean Functions ◽  
Theoretical Basis ◽  
Algebraic Immunity ◽  
Correlation Immunity ◽  
New Research ◽  
Relationship Of ◽  
The Relationship

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,and discuss the relationship of 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 get the results of the weight symmetric H Boolean function should have the same algebraic immunity order, correlation immunity order, the degree of diffusion and nonlinearity. The results provide a theoretical basis to reduce nearly half workload for studying the cryptographic properties of H Boolean function, and provides a new research methods for the study of the properties of cryptographic security of Boolean functions.

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

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.

scholarly journals ON THE CAYLEY GRAPHS OF BOOLEAN FUNCTIONS

2020 ◽  
pp. 873
Author(s):  
Lotfallah Pourfaray ◽  
Modjtaba Ghorbani
Keyword(s):  
Automorphism Group ◽  
Boolean Function ◽  
Vector Space ◽  
Cayley Graph ◽  
Boolean Functions ◽  
Cayley Graphs ◽  
Complete Characterization ◽  
Walsh Spectrum

A Boolean function is a function $f:\Bbb{Z}_n^2 \rightarrow \{0,1\}$ and we denote the set of all $n$-variable Boolean functions by $BF_n$. For $f\in BF_n$ the vector $[{\rm W}_f(a_0),\ldots,{\rm W}_f(a_{2n-1})]$ is called the Walsh spectrum of $f$, where ${\rm W}_f(a)= \sum_{x\in V} (-1)^{f(x) \oplus ax}$, where $V_n$ is the vector space of dimension $n$ over the two-element field $F_2$. In this paper, we shall consider the Cayley graph $\Gamma_f$ associated with a Boolean function $f$. We shall also find a complete characterization of the bent Boolean functions of order $16$ and determine the spectrum of related Cayley graphs.In addition, we shall enumerate all orbits of the action of automorphism group on the set $BF_n$. 

On inherited properties of restricted Boolean functions

2002 ◽  
Vol 12 (3) ◽  
Author(s):  
O.A. Logachev ◽  
A.A. Salnikov ◽  
V.V. Yashchenko
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Correlation Immunity

AbstractFor a property P of Boolean functions, a Boolean function f(x), x ∈ VIn this paper, this approach is applied to the following property of Boolean functions: the value f̂(α)/2We give convenient criteria for (H, α)-stability in terms of zeros of the Walsh-Hadamard coefficients, and establish relations between the (H, α)-stability, correlation immunity, and m-resiliency.

Clique Here: On the Distributed Complexity in Fully-Connected Networks

2016 ◽  
Vol 26 (01) ◽  
pp. 1650004 ◽  
Author(s):  
Benny Applebaum ◽  
Dariusz R. Kowalski ◽  
Boaz Patt-Shamir ◽  
Adi Rosén
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Message Passing ◽  
Constant Number ◽  
Running Time ◽  
Message Size ◽  
Explicit Function ◽  
Randomized Protocols ◽  
Fully Connected ◽  
Fully Connected Networks

We consider a message passing model with n nodes, each connected to all other nodes by a link that can deliver a message of B bits in a time unit (typically, B = O(log n)). We assume that each node has an input of size L bits (typically, L = O(n log n)) and the nodes cooperate in order to compute some function (i.e., perform a distributed task). We are interested in the number of rounds required to compute the function. We give two results regarding this model. First, we show that most boolean functions require ‸ L/B ‹ − 1 rounds to compute deterministically, and that even if we consider randomized protocols that are allowed to err, the expected running time remains [Formula: see text] for most boolean function. Second, trying to find explicit functions that require superconstant time, we consider the pointer chasing problem. In this problem, each node i is given an array Ai of length n whose entries are in [n], and the task is to find, for any [Formula: see text], the value of [Formula: see text]. We give a deterministic O(log n/ log log n) round protocol for this function using message size B = O(log n), a slight but non-trivial improvement over the O(log n) bound provided by standard “pointer doubling.” The question of an explicit function (or functionality) that requires super constant number of rounds in this setting remains, however, open.

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