On Walsh Spectrum of Cryptographic Boolean Function

2017 ◽  
Vol 67 (5) ◽  
pp. 536
Author(s):  
Shashi Kant Pandey ◽  
B. K. Dass
Keyword(s):  
Boolean Function ◽  
General Expression ◽  
Significant Role ◽  
Boolean Functions ◽  
Diophantine Equations ◽  
Bent Function ◽  
Bent Functions ◽  
Necessary Condition ◽  
Walsh Spectrum ◽  
Walsh Transformation

<p>Walsh transformation of a Boolean function ascertains a number of cryptographic properties of the Boolean function viz, non-linearity, bentness, regularity, correlation immunity and many more. The functions, for which the numerical value of Walsh spectrum is fixed, constitute a class of Boolean functions known as bent functions. Bent functions possess maximum possible non-linearity and therefore have a significant role in design of cryptographic systems. A number of generalisations of bent function in different domains have been proposed in the literature. General expression for Walsh transformation of generalised bent function (GBF) is derived. Using this condition, a set of Diophantine equations whose solvability is a necessary condition for the existence of GBF is also derived. Examples to demonstrate how these equations can be utilised to establish non-existence and regularity of GBFs is presented.</p>

scholarly journals Construction of an S-Box Based on Chaotic and Bent Functions

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 671
Author(s):  
Zijing Jiang ◽  
Qun Ding
Keyword(s):  
Chaos Theory ◽  
Boolean Functions ◽  
Security Analysis ◽  
Bent Function ◽  
Bent Functions ◽  
Encryption Algorithm ◽  
Symmetric Encryption ◽  
Differential Uniformity ◽  
Strict Avalanche Criterion ◽  
Independence Criterion

An S-box is the most important part of a symmetric encryption algorithm. Various schemes are put forward by using chaos theory. In this paper, a construction method of S-boxes with good cryptographic properties is proposed. The output of an S-box can be regarded as a group of Boolean functions. Therefore, we can use the different properties of chaos and Bent functions to generate a random Bent function with a high nonlinearity. By constructing a set of Bent functions as the output of an S-box, we can create an S-box with good cryptological properties. The nonlinearity, differential uniformity, strict avalanche criterion and the independence criterion of output bits are then analyzed and tested. A security analysis shows that the proposed S-box has excellent cryptographic properties.

ONE SUFFICIENT AND NECESSARY CONDITION ON BALANCED BOOLEAN FUNCTIONS WITH σf = 22n + 2n+3(n ≥ 3)

2014 ◽  
Vol 25 (03) ◽  
pp. 343-353 ◽  
Author(s):  
YU ZHOU ◽  
LIN WANG ◽  
WEIQIONG WANG ◽  
XINFENG DONG ◽  
XIAONI DU
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Sum Of Squares ◽  
Necessary Condition ◽  
Resilient Function ◽  
The Absolute ◽  
Global Avalanche Characteristics ◽  
Sufficient And Necessary Condition

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.

ON THE NONEXISTENCE of BENT FUNCTIONS

2011 ◽  
Vol 22 (06) ◽  
pp. 1431-1438 ◽  
Author(s):  
YIN ZHANG ◽  
MEICHENG LIU ◽  
DONGDAI LIN
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Normal Forms ◽  
Bent Functions

In this paper, we study the nonexistence of bent functions in the class of Boolean functions without monomials of degree less than d in their algebraic normal forms (ANF). We prove that n-variable Boolean functions in such class are not bent when there are not more than n + d - 3 monomials in their ANFs. We also show that an n-variable Boolean function is not bent if it has no monomial of degree less than ⌈3n/8 + 3/4⌉ in its ANF.

Research on Design and Construction Method of Bent Functions

2014 ◽  
Vol 571-572 ◽  
pp. 114-117
Author(s):  
Guang Xue Meng ◽  
Yan Guang Shen ◽  
Tao Jiang
Keyword(s):  
Projective Geometry ◽  
Boolean Functions ◽  
Construction Method ◽  
Bent Function ◽  
Bent Functions ◽  
C Language ◽  
Walsh Spectra ◽  
Algebra Method ◽  
Design And Construction

Bent function is a class of the highest nonlinear Boolean functions. In this paper three methods of design and construction are discussed with examples, which are algebra method, the character function in projective geometry and random researching method. Also, the Bent function of class is implemented with C language. At last, the concatenate construction from m = 2n-k Bent functions of k variables to a Bent function of n variables is given and verified with Walsh spectra.

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 $BF_n$. 

scholarly journals Identification of Disjoint Bi-decompositions in Boolean Functions through Walsh Spectrum

VLSI Design ◽  
2002 ◽  
Vol 14 (3) ◽  
pp. 307-313
Author(s):  
Bogdan J. Falkowski ◽  
Sudha Kannurao
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Type A ◽  
Walsh Spectrum ◽  
Spectral Coefficients ◽  
Walsh Spectra ◽  
Filtering Procedure

A new algorithm for the identification of disjoint bi-decomposition in Boolean functions from its Walsh spectrum is proposed. The type of bi-decomposition and its existence is derived from the knowledge of a subset of Walsh spectrum for a Boolean function. All three types of bi-decomposition are considered including OR, AND and EXOR type. A filtering procedure that uses just few Walsh spectral coefficients (SC) is applied to quickly eliminate the functions that are not bi-decomposable and hence the algorithm is very efficient. The type of bi-decomposition and affirmation/negation of variables in its logic sub-functions are directly identified by manipulation on the reduced cubical representation of Boolean functions and their corresponding Walsh spectra. The presented algorithm has been implemented in C and tested on the standard benchmark functions. The number of Boolean functions having various disjoint bi-decompositions has also been enumerated.

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.

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.

scholarly journals On diagnostic tests of contact break for contact circuits

2020 ◽  
Vol 30 (2) ◽  
pp. 103-116 ◽  
Author(s):  
Kirill A. Popkov
Keyword(s):  
Boolean Function ◽  
Diagnostic Test ◽  
Boolean Functions ◽  
Diagnostic Tests ◽  
Almost All

AbstractWe prove that, for n ⩾ 2, any n-place Boolean function may be implemented by a two-pole contact circuit which is irredundant and allows a diagnostic test with length not exceeding n + k(n − 2) under at most k contact breaks. It is shown that with k = k(n) ⩽ 2n−4, for almost all n-place Boolean functions, the least possible length of such a test is at most 2k + 2.

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