scholarly journals Thickness distribution of Boolean functions in 4 and 5 variables and a comparison with other cryptographic properties

2020 ◽  
Vol Accepted manuscript ◽  
Author(s):  
Mathias Hopp ◽  
Pål Ellingsen ◽  
Constanza Riera ◽  
Pantelimon Stănică
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Thickness Distribution ◽  
Complete Analysis ◽  
Affine Transformations ◽  
Open Questions ◽  
Minimum Number

This paper explores the distribution of algebraic thickness of Boolean functions (that is, the minimum number of terms in the ANF of the functions in the orbit of a Boolean function, through all affine transformations), in four and five variables, and the complete distribution is presented. Additionally, a complete analysis of some complexity properties (e.g., nonlinearity, balancedness, etc.) of all relevant orbits of Boolean functions is presented. Some properties of our notion of rigid function (which enabled us to reduce significantly the computation) are shown and some open questions are proposed, providing some further explanation of one of these questions.

scholarly journals Periodic Fourier representation of Boolean functions

2019 ◽  
Vol 19 (5&6) ◽  
pp. 392-412
Author(s):  
Ryuhei Mori
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Bell Inequalities ◽  
Polynomial Representation ◽  
Symmetric Boolean Function ◽  
Fourier Representation ◽  
Minimum Number ◽  
New Type ◽  
Adaptive Measurement ◽  
Small Periodic

In this work, we consider a new type of Fourier-like representation of Boolean function f\colon\{+1,-1\}^n\to\{+1,-1\}: f(x) = \cos(\pi\sum_{S\subseteq[n]}\phi_S \prod_{i\in S} x_i). This representation, which we call the periodic Fourier representation, of Boolean function is closely related to a certain type of multipartite Bell inequalities and non-adaptive measurement-based quantum computation with linear side-processing NMQCp. The minimum number of non-zero coefficients in the above representation, which we call the periodic Fourier sparsity, is equal to the required number of qubits for the exact computation of f by \NMQCp. Periodic Fourier representations are not unique, and can be directly obtained both from the Fourier representation and the F_2-polynomial representation. In this work, we first show that Boolean functions related to ZZZZ-polynomial have small periodic Fourier sparsities. Second, we show that the periodic Fourier sparsity is at least 2^{\deg_{\mathbb{F}_2}(f)}-1, which means that NMQCp efficiently computes a Boolean function $f$ if and only if F_2-degree of f is small. Furthermore, we show that any symmetric Boolean function, e.g., AND_n, Mod^3_n, Maj_n, etc, can be exactly computed by depth-2 NMQCp using a polynomial number of qubits, that implies exponential gaps between NMQCp and depth-2 NMQCp.

scholarly journals Sectorization for Water Distribution Systems with Multiple Sources: A Performance Indices Comparison

Water ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 131
Author(s):  
Jezabel D. Bianchotti ◽  
Melina Denardi ◽  
Mario Castro-Gama ◽  
Gabriel D. Puccini
Keyword(s):  
Distribution Systems ◽  
Water Distribution ◽  
Distribution Networks ◽  
Performance Criteria ◽  
Performance Indices ◽  
Multiple Sources ◽  
Multiobjective Problems ◽  
Open Questions ◽  
Minimum Number ◽  
And Performance

Sectorization is an effective technique for reducing the complexities of analyzing and managing of water systems. The resulting sectors, called district metering areas (DMAs), are expected to meet some requirements and performance criteria such as minimum number of intervention, pressure uniformity, similarity of demands, water quality and number of districts. An efficient methodology to achieve all these requirements together and the proper choice of a criteria governing the sectorization is one of the open questions about optimal DMAs design. This question is addressed in this research by highlighting the advantages of three different criteria when applied to real-word water distribution networks (WDNs). To this, here it is presented a two-stage approach for optimal design of DMAs. The first stage, the clustering of the system, is based on a Louvain-type greedy algorithm for the generalized modularity maximization. The second stage, the physical dividing of the system, is stated as a two-objective optimization problem that utilises the SMOSA version of simulated annealing for multiobjective problems. One objective is the number of isolation valves whereas for the second objective three different performance indices (PIs) are analyzed and compared: (a) standard deviation, (b) Gini coefficient and (c) loss of resilience. The methodology is applied to two real case studies where the first two PIs are optimized to address similar demands among DMAs. The results demonstrate that the proposed method is effective for sectorization into independent DMAs with similar demands. Surprisingly, it found that for the real studied systems, loss of resilience achieves better performance for each district in terms of pressure uniformity and demand similarity than the other two specific performance criteria.

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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