scholarly journals An Extended Quadratic Frobenius Primality Test with Average and Worst Case Error Estimates

2003 ◽  
Vol 10 (9) ◽  
Author(s):  
Ivan B. Damgård ◽  
Gudmund Skovbjerg Frandsen
Keyword(s):  
Error Probability ◽  
Upper Bounds ◽  
Worst Case ◽  
Average Case ◽  
Closed Expression ◽  
Incremental Search ◽  
Starting Point ◽  
Primality Test ◽  
Algebraic Properties ◽  
Random Starting Point

We present an Extended Quadratic Frobenius Primality Test (EQFT), which is related to the Miller-Rabin test and the Quadratic Frobenius test (QFT) by Grantham. EQFT takes time about equivalent to 2 Miller-Rabin tests, but has much smaller error probability, namely 256/331776^t for t iterations of the test in the worst case. EQFT extends QFT by verifying additional algebraic properties related to the existence of elements of order dividing 24. We also give bounds on the average-case behaviour of the test: consider the algorithm that repeatedly chooses random odd k bit numbers, subjects them to t iterations of our test and outputs the first one found that passes all tests. We obtain numeric upper bounds for the error probability of this algorithm as well as a general closed expression bounding the error. For instance, it is at most 2^{-143} for k=500, t = 2. Compared to earlier similar results for the Miller-Rabin test, the results indicate that our test in the average case has the effect of 9 Miller-Rabin tests, while only taking time equivalent to about 2 such tests. We also give bounds for the error in case a prime is sought by incremental search from a random starting point.

scholarly journals An Extended Quadratic Frobenius Primality Test with Average Case Error Estimates

2001 ◽  
Vol 8 (45) ◽  
Author(s):  
Ivan B. Damgård ◽  
Gudmund Skovbjerg Frandsen
Keyword(s):  
Error Probability ◽  
Prime Numbers ◽  
Upper Bounds ◽  
Worst Case ◽  
Average Case ◽  
Closed Expression ◽  
Incremental Search ◽  
Starting Point ◽  
Primality Test ◽  
Small Set

We present an Extended Quadratic Frobenius Primality Test (EQFT), which is related to the Miller-Rabin test and the Quadratic Frobenius test (QFT) by Grantham. EQFT is well-suited for generating large, random prime numbers since on a random input number, it takes time about equivalent to 2 Miller-Rabin tests, but has much smaller error probability. EQFT extends QFT by verifying additional algebraic properties related to the existence of elements of order 3 and 4. We obtain a simple closed expression that upper bounds the probability of acceptance for any input number. This in turn allows us to give strong bounds on the average-case behaviour of the test: consider the algorithm that repeatedly chooses random odd k bit numbers, subjects them to t iterations of our test and outputs the first one found that passes all tests. We obtain numeric upper bounds for the error probability of this algorithm as well as a general closed expression bounding the error. For instance, it is at most 2^{-143} for k=500, t=2 . Compared to earlier similar results for the Miller-Rabin test, the results indicates that our test in the average case has the effect of 9 Miller-Rabin tests, while only taking time equivalent to about 2 such tests. We also give bounds for the error in case a prime is sought by incremental search from a random starting point. While EQFT is slower than the average case on a small set of inputs, we present a variant that is always fast, i.e. takes time about 2 Miller-Rabin tests. The variant has slightly larger worst case error probability than EQFT, but still improves on previous proposed tests.

scholarly journals A new algorithm for fixed point quantum search

2006 ◽  
Vol 6 (6) ◽  
pp. 483-494
Author(s):  
T. Tulsi ◽  
L.K. Grover ◽  
A. Patel
Keyword(s):  
Fixed Point ◽  
Error Probability ◽  
Search Algorithm ◽  
Quantum Search ◽  
Worst Case ◽  
Average Case ◽  
Standard Quantum ◽  
Quantum Search Algorithm ◽  
Monotonic Convergence ◽  
Worst Case Behavior

The standard quantum search lacks a feature, enjoyed by many classical algorithms, of having a fixed point, i.e. monotonic convergence towards the solution. Recently a fixed point quantum search algorithm has been discovered, referred to as the Phase-\pi/3 search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from \epsilon to \epsilon^{2q+1} using q oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase-\pi/3 search algorithm but much better average-case behavior. Furthermore the new algorithm gives \epsilon^{2q+1} convergence for all integral q, whereas the Phase-\pi/3 search algorithm requires q to be (3^{n}-1)/2 with n a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.

AVERAGE-CASE "MESSY" BROADCASTING

2008 ◽  
Vol 09 (04) ◽  
pp. 487-505 ◽  
Author(s):  
CHENKUAN LI ◽  
THOMAS E. HART ◽  
KEVIN J. HENRY ◽  
IAN A. NEUFELD
Keyword(s):  
Probability Theory ◽  
Complete Graph ◽  
Recursive Formula ◽  
Upper Bounds ◽  
Worst Case ◽  
Average Case ◽  
Occupancy Problem ◽  
Network Topologies

Current studies of "messy" broadcasting have so far concentrated on finding worst-case times. However, such worst-case scenarios are extremely unlikely to occur in general. Hence, determining average-case times or tight upper bounds for completing "messy" broadcasting in various network topologies is both necessary and meaningful in practice. In this paper, we focus on seeking the average-case "messy" broadcast times of stars, paths, cycles, and d-ary trees, and finding good upper bounds for hypercubes. Finally, we derive a recursive formula to express the average-case time for a specific "messy" broadcast model on a complete graph using a classical occupancy problem in probability theory, and provide a nice simulation result which indicates that this model behaves like classical broadcasting.

scholarly journals The Complexity of Malign Ensembles

1990 ◽  
Vol 19 (335) ◽  
Author(s):  
Peter Bro Miltersen
Keyword(s):  
Polynomial Time ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Worst Case ◽  
Average Case ◽  
Exponential Time ◽  
Running Time ◽  
Exponential Time Algorithms

We analyze the concept of <em> malignness</em>, which is the property of probability ensembles of making the average case running time equal to the worst case running time for a class of algorithms. We derive lower and upper bounds on the complexity of malign ensembles, which are tight for exponential time algorithms, and which show that no polynomial time computable malign ensemble exists for the class of superlinear algorithms. Furthermore, we show that for no class of superlinear algorithms a polynomial time computable malign ensemble exists, unless every language in P has an expected polynomial time constructor.

Rule Based Classifiers for Suspect Detection from CCTV Footages

2020 ◽  
Vol 13 ◽  
Author(s):  
Sunil Pathak
Keyword(s):  
Answer Sheet ◽  
Normal Activity ◽  
Contact Detection ◽  
Accuracy Rate ◽  
Hand Detection ◽  
Worst Case ◽  
Average Case ◽  
The Face ◽  
Case Simulation ◽  
Material Exchange

Background: The significant work has been present to identify suspects, gathering information and examining any videos from CCTV Footage. This exploration work expects to recognize suspicious exercises, i.e. object trade, passage of another individual, peeping into other's answer sheet and individual trade from the video caught by a reconnaissance camera amid examinations. This requires the procedure of face acknowledgment, hand acknowledgment and distinguishing the contact between the face and hands of a similar individual and that among various people. Methods: Segmented frames has given as input to obtain foreground image with the help of Gaussian filtering and background modeling method. Suh foreground images has given to Activity Recognition model to detect normal activity or suspicious activity. Results: Accuracy rate, Precision and Recall are calculate for activities detection, contact detection for Best Case, Average Case and Worst Case. Simulation results are compare with performance parameter such as Material Exchange, Position Exchange, and Introduction of a new person, Face and Hand Detection and Multi Person Scenario. Conclusion: In this paper, a framework is prepared for suspect detection. This framework will absolutely realize an unrest in the field of security observation in the training area.

scholarly journals Ratio-based multi-level resistive memory cells

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Miguel Angel Lastras-Montaño ◽  
Osvaldo Del Pozo-Zamudio ◽  
Lev Glebsky ◽  
Meiran Zhao ◽  
Huaqiang Wu ◽  
...  
Keyword(s):  
Error Probability ◽  
Resistance Switching ◽  
Closed Form Expression ◽  
Memory Cells ◽  
Resistive Memory ◽  
Worst Case ◽  
Information Encoding ◽  
Bit Error Probability ◽  
Multi Level ◽  
Switching Devices

AbstractRatio-based encoding has recently been proposed for single-level resistive memory cells, in which the resistance ratio of a pair of resistance-switching devices, rather than the resistance of a single device (i.e. resistance-based encoding), is used for encoding single-bit information, which significantly reduces the bit error probability. Generalizing this concept for multi-level cells, we propose a ratio-based information encoding mechanism and demonstrate its advantages over the resistance-based encoding for designing multi-level memory systems. We derive a closed-form expression for the bit error probability of ratio-based and resistance-based encodings as a function of the number of levels of the memory cell, the variance of the distribution of the resistive states, and the ON/OFF ratio of the resistive device, from which we prove that for a multi-level memory system using resistance-based encoding with bit error probability x, its corresponding bit error probability using ratio-based encoding will be reduced to $$x^2$$ x 2 at the best case and $$x^{\sqrt{2}}$$ x 2 at the worst case. We experimentally validated these findings on multiple resistance-switching devices and show that, compared to the resistance-based encoding on the same resistive devices, our approach achieves up to 3 orders of magnitude lower bit error probability, or alternatively it could reduce the cell’s programming time and programming energy by up 5–10$$\times$$ × , while achieving the same bit error probability.

scholarly journals Interregional Transfers for Pandemic Surges

2020 ◽  
Author(s):  
Kenneth A Michelson ◽  
Chris A Rees ◽  
Jayshree Sarathy ◽  
Paige VonAchen ◽  
Michael Wornow ◽  
...  
Keyword(s):  
The United States ◽  
Upper Bounds ◽  
Hospital Inpatient ◽  
Worst Case ◽  
Icu Patients ◽  
Interregional Transfers ◽  
Number Of Patients ◽  
Bed Capacity ◽  
Model Scenarios ◽  
Do So

Abstract Background Hospital inpatient and intensive care unit (ICU) bed shortfalls may arise due to regional surges in volume. We sought to determine how interregional transfers could alleviate bed shortfalls during a pandemic. Methods We used estimates of past and projected inpatient and ICU cases of coronavirus disease 2019 (COVID-19) from 4 February 2020 to 1 October 2020. For regions with bed shortfalls (where the number of patients exceeded bed capacity), transfers to the nearest region with unused beds were simulated using an algorithm that minimized total interregional transfer distances across the United States. Model scenarios used a range of predicted COVID-19 volumes (lower, mean, and upper bounds) and non–COVID-19 volumes (20%, 50%, or 80% of baseline hospital volumes). Scenarios were created for each day of data, and worst-case scenarios were created treating all regions’ peak volumes as simultaneous. Mean per-patient transfer distances were calculated by scenario. Results For the worst-case scenarios, national bed shortfalls ranged from 669 to 58 562 inpatient beds and 3208 to 31 190 ICU beds, depending on model volume parameters. Mean transfer distances to alleviate daily bed shortfalls ranged from 23 to 352 miles for inpatient and 28 to 423 miles for ICU patients, depending on volume. Under all worst-case scenarios except the highest-volume ICU scenario, interregional transfers could fully resolve bed shortfalls. To do so, mean transfer distances would be 24 to 405 miles for inpatients and 73 to 476 miles for ICU patients. Conclusions Interregional transfers could mitigate regional bed shortfalls during pandemic hospital surges.

scholarly journals A Splay Tree-Based Approach for Efficient Resource Location in P2P Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Wei Zhou ◽  
Zilong Tan ◽  
Shaowen Yao ◽  
Shipu Wang
Keyword(s):  
Routing Algorithm ◽  
Adaptive Routing ◽  
Upper And Lower Bounds ◽  
Resource Location ◽  
Hop Count ◽  
Worst Case ◽  
Average Case ◽  
Splay Tree ◽  
Efficient Resource ◽  
P2p System

Resource location in structured P2P system has a critical influence on the system performance. Existing analytical studies of Chord protocol have shown some potential improvements in performance. In this paper a splay tree-based new Chord structure called SChord is proposed to improve the efficiency of locating resources. We consider a novel implementation of the Chord finger table (routing table) based on the splay tree. This approach extends the Chord finger table with additional routing entries. Adaptive routing algorithm is proposed for implementation, and it can be shown that hop count is significantly minimized without introducing any other protocol overheads. We analyze the hop count of the adaptive routing algorithm, as compared to Chord variants, and demonstrate sharp upper and lower bounds for both worst-case and average case settings. In addition, we theoretically analyze the hop reducing in SChord and derive the fact that SChord can significantly reduce the routing hops as compared to Chord. Several simulations are presented to evaluate the performance of the algorithm and support our analytical findings. The simulation results show the efficiency of SChord.

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