scholarly journals On the Relation between Boolean Curve Fitting and the Inverse Problem of Boolean Equations

2017 ◽  
Vol 28 (2) ◽  
pp. 3-9
Author(s):  
Ali M. Rushdi Ali M. Rushdi
Keyword(s):  
Inverse Problem ◽  
Boolean Function ◽  
Curve Fitting ◽  
Boolean Functions ◽  
Consistency Condition ◽  
Similarities And Differences ◽  
Special Case

This paper explores the similarities and differences between two prominent problems in the mathematics of Boolean functions. The first of these problems is that of Boolean curve fitting (BCF), also known as Boolean interpolation, which deals with constructing a curve 𝑧 = 𝑓(𝐗) through a number of points z𝑘 = 𝑓(𝐗𝑘 ) where 𝑘 = 1,2, … , 𝑚. The second problem is the Inverse Problem of Boolean equations (IPBE), which constructs a Boolean function whose zeroes are all known. While the problem of Boolean curve fitting might require a consistency condition for its solution, the Inverse Problem of Boolean equations might use a consistency condition as an input. Without a consistency condition, the Inverse Problem of Boolean equations can be viewed as a special case of the problem of Boolean curve fitting, provided the specified points z𝑘 are the only zeros of 𝑓(𝐗). Our findings are illustrated via a detailed typical example.

scholarly journals Boolean Curve Fitting with the Aid of Variable-Entered Karnaugh Maps

2019 ◽  
Vol 4 (6) ◽  
pp. 1287-1306
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Ahmed Said Balamesh
Keyword(s):  
Boolean Algebra ◽  
Boolean Function ◽  
Unique Solution ◽  
Curve Fitting ◽  
Multiple Solutions ◽  
Consistency Condition ◽  
Visual Perspective ◽  
Boolean Space ◽  
Potential Applications ◽  
Karnaugh Map

The Variable-Entered Karnaugh Map is utilized to grant a simpler view and a visual perspective to Boolean curve fitting (Boolean interpolation); a topic whose inherent complexity hinders its potential applications. We derive the function(s) through m points in the Boolean space B^(n+1) together with consistency and uniqueness conditions, where B is a general ‘big’ Boolean algebra of l≥1 generators, L atoms (2^(l-1)<L≤2^l) and 2^L elements. We highlight prominent cases in which the consistency condition reduces to the identity (0=0) with a unique solution or with multiple solutions. We conjecture that consistent (albeit not necessarily unique) curve fitting is possible if, and only if, m=2^n. This conjecture is a generalization of the fact that a Boolean function of n variables is fully and uniquely determined by its values in the {0,1}^n subdomain of its B^n domain. A few illustrative examples are used to clarify the pertinent concepts and techniques.

Exact Quantum 1-Query Algorithms and Complexity

SPIN ◽  
2021 ◽  
pp. 2140001
Author(s):  
Daowen Qiu ◽  
Guoliang Xu
Keyword(s):  
Boolean Function ◽  
Decision Trees ◽  
Future Study ◽  
Boolean Functions ◽  
Query Complexity ◽  
Symmetric Boolean Functions ◽  
Exact Quantum ◽  
Algorithms And Complexity ◽  
Special Case ◽  
Query Algorithms

Deutsch–Jozsa problem (D–J) has exact quantum 1-query complexity (“exact” means no error), but requires super-exponential queries for the optimal classical deterministic decision trees. D–J problem is equivalent to a symmetric partial Boolean function, and in fact, all symmetric partial Boolean functions having exact quantum 1-query complexity have been found out and these functions can be computed by D–J algorithm. A special case is that all symmetric Boolean functions with exact quantum 1-query complexity follow directly and these functions are also all total Boolean functions with exact quantum 1-query complexity obviously. Then there are pending problems concerning partial Boolean functions having exact quantum 1-query complexity and new results have been found, but some problems are still open. In this paper, we review these results regarding exact quantum 1-query complexity and in particular, we also obtain a new result that a partial Boolean function with exact quantum 1-query complexity is constructed and it cannot be computed by D–J algorithm. Further problems are pointed out for future study.

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.

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.

scholarly journals On self-correcting logic circuits of unreliable gates

2021 ◽  
pp. 1-18
Author(s):  
Kirill Andreevich Popkov
Keyword(s):  
Positive Integer ◽  
Boolean Function ◽  
Boolean Functions ◽  
Arbitrary Number ◽  
Logic Circuits ◽  
Logic Circuit ◽  
Complete Basis

The following statements are proved: 1) for any integer m ≥ 3 there is a basis consisting of Boolean functions of no more than m variables, in which any Boolean function can be implemented by a logic circuit of unreliable gates that self-corrects relative to certain faults in an arbitrary number of gates; 2) for any positive integer k there are bases consisting of Boolean functions of no more than two variables, in each of which any Boolean function can be implemented by a logic circuit of unreliable gates that self-correct relative to certain faults in no more than k gates; 3) there is a functionally complete basis consisting of Boolean functions of no more than two variables, in which almost no Boolean function can be implemented by a logic circuit of unreliable gates that self-correct relative to at least some faults in no more than one gate.

scholarly journals Linearization of multi-output logic functions by ordering of the autocorrelation values

2007 ◽  
Vol 20 (3) ◽  
pp. 479-498 ◽  
Author(s):  
Osnat Keren ◽  
Ilya Levin
Keyword(s):  
Computational Complexity ◽  
Boolean Function ◽  
Autocorrelation Function ◽  
Execution Time ◽  
Boolean Functions ◽  
Analytic Method ◽  
Implementation Cost ◽  
Initial Domain ◽  
Greedy Methods ◽  
Logic Functions

The paper deals with the problem of linear decomposition of a system of Boolean functions. A novel analytic method for linearization, by reordering the values of the autocorrelation function, is presented. The computational complexity of the linearization procedure is reduced by performing calculations directly on a subset of autocorrelation values rather than by manipulating the Boolean function in its initial domain. It is proved that unlike other greedy methods, the new technique does not increase the implementation cost. That is, it provides linearized functions with a complexity that is not greater than the complexity of the initial Boolean functions. Experimental results over standard benchmarks and random Boolean functions demonstrate the efficiency of the proposed procedure in terms of the complexity measure and the execution time.

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