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.

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 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 Binary and Multiclass Text Classification by Means of Separable Convolutional Neural Network

Inventions ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 70
Author(s):  
Elena Solovyeva ◽  
Ali Abdullah
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Convolutional Neural Network ◽  
Recurrent Neural Networks ◽  
Low Cost ◽  
Computational Cost ◽  
High Accuracy ◽  
Activation Functions ◽  
Fully Connected ◽  
Fully Connected Networks

In this paper, the structure of a separable convolutional neural network that consists of an embedding layer, separable convolutional layers, convolutional layer and global average pooling is represented for binary and multiclass text classifications. The advantage of the proposed structure is the absence of multiple fully connected layers, which is used to increase the classification accuracy but raises the computational cost. The combination of low-cost separable convolutional layers and a convolutional layer is proposed to gain high accuracy and, simultaneously, to reduce the complexity of neural classifiers. Advantages are demonstrated at binary and multiclass classifications of written texts by means of the proposed networks under the sigmoid and Softmax activation functions in convolutional layer. At binary and multiclass classifications, the accuracy obtained by separable convolutional neural networks is higher in comparison with some investigated types of recurrent neural networks and fully connected networks.

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 Improving data transfer for model coupling

2015 ◽  
Vol 8 (10) ◽  
pp. 8981-9020 ◽  
Author(s):  
C. Zhang ◽  
L. Liu ◽  
G. Yang ◽  
R. Li ◽  
B. Wang
Keyword(s):  
Performance Improvement ◽  
Message Passing ◽  
Message Passing Interface ◽  
Data Transfer ◽  
The Public ◽  
Message Size ◽  
Abstract Data ◽  
Point To Point ◽  
Component Models ◽  
Size Variable

Abstract. Data transfer, which means transferring data fields between two component models or rearranging data fields among processes of the same component model, is a fundamental operation of a coupler. Most of state-of-the-art coupler versions currently use an implementation based on the point-to-point (P2P) communication of the Message Passing Interface (MPI) (call such an implementation "P2P implementation" for short). In this paper, we reveal the drawbacks of the P2P implementation, including low communication bandwidth due to small message size, variable and big number of MPI messages, and jams during communication. To overcome these drawbacks, we propose a butterfly implementation for data transfer. Although the butterfly implementation can outperform the P2P implementation in many cases, it degrades the performance in some cases because the total message size transferred by the butterfly implementation is larger than that by the P2P implementation. To make the data transfer completely improved, we design and implement an adaptive data transfer library that combines the advantages of both butterfly implementation and P2P implementation. Performance evaluation shows that the adaptive data transfer library significantly improves the performance of data transfer in most cases and does not decrease the performance in any cases. Now the adaptive data transfer library is open to the public and has been imported into a coupler version C-Coupler1 for performance improvement of data transfer. We believe that it can also improve other coupler versions.

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.

Sign in / Sign up

Export Citation Format

Share Document