scholarly journals Classification of Boolean Functions Where Affine Functions Are Uniformly Distributed

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ranjeet Kumar Rout ◽  
Pabitra Pal Choudhury ◽  
Sudhakar Sahoo
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Cartesian Product ◽  
Affine Function ◽  
Recursive Procedure ◽  
Symmetry Properties ◽  
Affine Functions

The present paper on classification of -variable Boolean functions highlights the process of classification in a coherent way such that each class contains a single affine Boolean function. Two unique and different methods have been devised for this classification. The first one is a recursive procedure that uses the Cartesian product of sets starting from the set of one variable Boolean functions. In the second method, the classification is done by changing some predefined bit positions with respect to the affine function belonging to that class. The bit positions which are changing also provide us information concerning the size and symmetry properties of the classes/subclasses in such a way that the members of classes/subclasses satisfy certain similar properties.

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.

Mean and variance of the number of subfunctions of random Boolean function which are close to the affine functions set

2017 ◽  
Vol 27 (1) ◽  
Author(s):  
Aleksandr A. Serov
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Affine Functions ◽  
Mean And Variance

AbstractFor random equiprobable Boolean functions we investigate the distribution of the number of subfunctions which have a given number of variables and are close to the set of affine Boolean functions. It is shown, for example, that for Boolean functions of

scholarly journals Behavior of Analogical Inference w.r.t. Boolean Functions

2018 ◽  
Author(s):  
Miguel Couceiro ◽  
Nicolas Hug ◽  
Henri Prade ◽  
Gilles Richard
Keyword(s):  
Normal Form ◽  
Empirical Study ◽  
Boolean Functions ◽  
Hamming Distance ◽  
Affine Function ◽  
Analogical Inference ◽  
Wrong Prediction ◽  
Affine Functions ◽  
Classification Tasks ◽  
Algebraic Normal Form

It has been observed that a particular form of analogical inference, based on analogical proportions, yields competitive results in classification tasks. Using the algebraic normal form of Boolean functions, it has been shown that analogical prediction is always exact iff the labeling function is affine. We point out that affine functions are also meaningful when using another view of analogy. We address the accuracy of analogical inference for arbitrary Boolean functions and show that if a function is epsilon-close to an affine function, then the probability of making a wrong prediction is upper bounded by 4 epsilon. This result is confirmed by an empirical study showing that the upper bound is tight. It highlights the specificity of analogical inference, also characterized in terms of the Hamming distance.

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.

On a new classification of Boolean functions

2019 ◽  
Vol 10 (2) ◽  
pp. 159-168
Author(s):  
Sergei N Fedorov
Keyword(s):  
Boolean Functions ◽  
New Classification

Рассматривается недавно предложенный подход к исследованию булевых функций, в основе которого лежит понятие класса $\Delta$-эквивалентности: множества булевых функций с одной и той же функцией автокорреляции. Такая классификация представляется полезной, поскольку многие криптографические характеристики булевых функций, принадлежащих одному и тому же классу $\Delta$-эквивалентности, одинаковы.

scholarly journals Exact quantum algorithms have advantage for almost all Boolean functions

2015 ◽  
pp. 435-452
Author(s):  
Andris Ambainis ◽  
Jozef Gruska ◽  
Shenggen Zheng
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Almost All ◽  
Quantum Query

It has been proved that almost all n-bit Boolean functions have exact classical query complexity n. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all n-bit Boolean functions can be computed by an exact quantum algorithm with less than n queries. More exactly, we prove that ANDn is the only n-bit Boolean function, up to isomorphism, that requires n queries.

scholarly journals On the confusion coefficient of Boolean functions

2021 ◽  
Vol 16 (1) ◽  
pp. 1-13
Author(s):  
Yu Zhou ◽  
Jianyong Hu ◽  
Xudong Miao ◽  
Yu Han ◽  
Fuzhong Zhang
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Power Analysis ◽  
Signal To Noise Ratio ◽  
Sum Of Squares ◽  
Hamming Weight ◽  
Signal To Noise ◽  
Trade Offs ◽  
Cryptographic Algorithms ◽  
Transparency Order

Abstract The notion of the confusion coefficient is a property that attempts to characterize confusion property of cryptographic algorithms against differential power analysis. In this article, we establish a relationship between the confusion coefficient and the autocorrelation function for any Boolean function and give a tight upper bound and a tight lower bound on the confusion coefficient for any (balanced) Boolean function. We also deduce some deep relationships between the sum-of-squares of the confusion coefficient and other cryptographic indicators (the sum-of-squares indicator, hamming weight, algebraic immunity and correlation immunity), respectively. Moreover, we obtain some trade-offs among the sum-of-squares of the confusion coefficient, the signal-to-noise ratio and the redefined transparency order for a Boolean function.

scholarly journals Crypto-Archaeology: unearthing design methodology of DES s-boxes

2017 ◽  
Author(s):  
Sankhanil Dey ◽  
Ranjan Ghosh
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Design Methodology ◽  
Block Cipher ◽  
Block Ciphers ◽  
Public Domain ◽  
Differential Cryptanalysis ◽  
Linear Cryptanalysis ◽  
Strict Avalanche Criterion ◽  
Independence Criterion

US defence sponsored the DES program in 1974 and released it in 1977. It remained as a well-known and well accepted block cipher until 1998. Thirty-two 4-bit DES S-Boxes are grouped in eight each with four and are put in public domain without any mention of their design methodology. S-Boxes, 4-bit, 8-bit or 32-bit, find a permanent seat in all future block ciphers. In this paper, while looking into the design methodology of DES S-Boxes, we find that S-Boxes have 128 balanced and non-linear Boolean Functions, of which 102 used once, while 13 used twice and 92 of 102 satisfy the Boolean Function-level Strict Avalanche Criterion. All the S-Boxes satisfy the Bit Independence Criterion. Their Differential Cryptanalysis exhibits better results than the Linear Cryptanalysis. However, no S-Boxes satisfy the S-Box-level SAC analyses. It seems that the designer emphasized satisfaction of Boolean-Function-level SAC and S-Box-level BIC and DC, not the S-Box-level LC and SAC.

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