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.

scholarly journals A Method of Two-Level Simplification of Boolean Functions

1967 ◽  
Vol 29 ◽  
pp. 201-210
Author(s):  
Toshio Umezawa
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
The Other ◽  
Product Form ◽  
Automatic Computation ◽  
Prime Implicants ◽  
Other Hand ◽  
Group 1

There are a number of methods to find minimal two-level forms for a given Boolean function, e g. Harvard’s group [1], Veitch [2], Quine [3], [4], Karnaugh [5], Nelson [6], [7] etc,. This paper presents an approach which is suitable for mechanical or automatic computation, as the Harvard method and the Quine method are so. On the other hand, it shares the same property as the Veitch method in the sense that some of essential prime implicants may be found before all prime implicants are computed. It also adopts the procedure to reduce the necessary steps for computation which is shown in Lawler [8]. The method described is applicable to the interval of Boolean functions f, g such that f implies g where for simplification of sum form the variables occurring in g also occur in f and for product form the variables in f also occur in g.

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.

scholarly journals A heuristic method for bi-decomposition of partial Boolean functions

Informatics ◽  
2020 ◽  
Vol 17 (3) ◽  
pp. 44-53
Author(s):  
Yu. V. Pottosin
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Heuristic Method ◽  
Disjunctive Normal Form ◽  
The Other ◽  
Output Function ◽  
Minimal Rank ◽  
Boolean Space ◽  
Definitional Domain ◽  
The Given

The problem of decomposition of a Boolean function is to represent a given Boolean function in the form of a superposition of some Boolean functions whose number of arguments are less than the number of given function. The bi-decomposition represents a given function as a logic algebra operation, which is also given, over two Boolean functions. The task is reduced to specification of those two functions. A method for bi-decomposition of incompletely specified (partial) Boolean function is suggested. The given Boolean function is specified by two sets, one of which is the part of the Boolean space of the arguments of the function where its value is 1, and the other set is the part of the space where the function has the value 0. The complete graph of orthogonality of Boolean vectors that constitute the definitional domain of the given function is considered. In the graph, the edges are picked out, any of which has its ends corresponding the elements of Boolean space where the given function has different values. The problem of bi-decomposition is reduced to the problem of a weighted two-block covering the set of picked out edges of considered graph by its complete bipartite subgraphs (bicliques). Every biclique is assigned with a disjunctive normal form (DNF) in definite way. The weight of a biclique is a pair of certain parameters of   assigned DNF. According to each biclique of obtained cover, a Boolean function is constructed whose arguments are the variables from the term of minimal rank on the DNF. A technique for constructing the mentioned cover for two kinds of output function is described.

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 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 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.

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

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