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Primality test for numbers M with a large power of 5 dividing M4−1

2003
*
Theoretical Computer Science
*

The Quintic Reciprocity Law is used to produce an algorithm, that runs in polynomial time, and that determines the

doi:10.1016/s0304-3975(02)00617-5
fatcat:kj5kamp5pfgijp5hv5iqi2ovje
*primality**of**numbers**M*, such that*M*4 −*1*, is divisible by*a**power**of**5*which is larger ... The same*test*equations are used*for*all such*M*.*A*su ciency*test*, together*with*its probability*of*succeeding in determining*primality*is given when the condition on*M*modulo p is omitted. (P. ...*A*preliminary version*of*this paper was published In the proceeding*of*Latin 2000: Lecture Notes in Computer Science 1776, pp. 269-279. ...##
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Implementation of a new primality test

1987
*
Mathematics of Computation
*

An implementation

doi:10.1090/s0025-5718-1987-0866102-2
fatcat:5xcu4xzikbal3d4allxc5qtf7a
*of*the Cohen-Lenstra version*of*the Adleman-Pomerance-Rumely*primality**test*is presented. ...*Primality**of*prime*numbers**of*up to 213 decimal digits can now routinely be proved within approximately ten minutes. ... Lenstra, Jr.*for*his great help in writing this paper, and to D. T. Winter*for*writing the multiprecision routines. Université de Bordeaux I Talence, France ...##
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Implementation of a New Primality Test

1987
*
Mathematics of Computation
*

An implementation

doi:10.2307/2007877
fatcat:wg2fbdrmcvac7pnpu23ej6k3zi
*of*the Cohen-Lenstra version*of*the Adleman-Pomerance-Rumely*primality**test*is presented. ...*Primality**of*prime*numbers**of*up to 213 decimal digits can now routinely be proved within approximately ten minutes. ... Lenstra, Jr.*for*his great help in writing this paper, and to D. T. Winter*for*writing the multiprecision routines. Université de Bordeaux I Talence, France ...##
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Lucas Pseudoprimes

1980
*
Mathematics of Computation
*

Since these

doi:10.2307/2006406
fatcat:2kwhtrylm5at7nc3unxvvffnzu
*tests*require the evaluation*of*the least*number*f(n)*for*which the Jacobi symbol (f(n)/n) is less than*1*, we evaluate the average order*of*the function /. ... We suggest some*powerful*new*primality**tests*which combine Lucas pseudoprimes*with*ordinary pseudoprimes. ... Since*m*< 2/iß we have Rik2hQ) >Qi-*5*)/2*for*all odd n > 7. This inequality is enough to prove the theorem.*5*.*Powerful**Tests**for**Primality*. ...##
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Lucas pseudoprimes

1980
*
Mathematics of Computation
*

Since these

doi:10.1090/s0025-5718-1980-0583518-6
fatcat:ccqsbpl4wbchdffeucfnnxwbvy
*tests*require the evaluation*of*the least*number*f(n)*for*which the Jacobi symbol (f(n)/n) is less than*1*, we evaluate the average order*of*the function /. ... We suggest some*powerful*new*primality**tests*which combine Lucas pseudoprimes*with*ordinary pseudoprimes. ... Since*m*< 2/iß we have Rik2hQ) >Qi-*5*)/2*for*all odd n > 7. This inequality is enough to prove the theorem.*5*.*Powerful**Tests**for**Primality*. ...##
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Lattice-based Key Sharing Schemes - A Survey
[article]

2020
*
IACR Cryptology ePrint Archive
*

However, most if not all

dblp:journals/iacr/RaviHCB20
fatcat:gwfp7xfzbbgxnldzbngfc4ru7q
*of*it is predominantly built upon hardness guarantees*of**number*theoretic problems that can be broken by*large*scale quantum computers in the future. ... Sensing the imminent threat from continued advances in quantum computing, NIST has recently initiated*a*global level standardization process*for*quantum resistant public-key cryptographic primitives such ... [84] proposed to perform*a**number**of*statistical*tests*such as 𝑡-*tests*and chi-squared*tests*to*test*the validity*of*the generated secrets and errors. ...##
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Analytic Deep Learning-based Surrogate Model for Operational Planning with Dynamic TTC Constraints
[article]

2020
*
arXiv
*
pre-print

The increased penetration

arXiv:2006.16186v1
fatcat:d7upvtdazrf3fcpc3rirzd56ym
*of*wind*power*introduces more operational changes*of*critical corridors and the traditional time-consuming transient stability constrained total transfer capability (TTC) operational ... The key idea is to resort to the deep learning*for*developing*a*computationally cheap surrogate model to replace the original time-consuming differential-algebraic constraints related to TTC. ... 0.5; 2)*for*SLNN and DLNN*with**a*different*number**of*layers, the settings are in Table II . ...##
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Design and phenotyping procedures for recording wool, skin, parasite resistance, growth, carcass yield and quality traits of the SheepGENOMICS mapping flock

2012
*
Animal Production Science
*

*A*major aim

*of*the research program known as SheepGENOMICS was to deliver DNA markers

*for*commercial breeding programs. ... To that end,

*a*resource flock was established, comprehensively phenotyped and genotyped

*with*DNA markers. ... Without the input

*of*all the above, the work described here would not have taken place. ...

##
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A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics

2021
*
Mathematics
*

*For*the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach

*number*limit

*of*the equations, hence

*a*consistent hybrid FV/FE method

*for*the incompressible ... numerical simulations

*of*turbulent incompressible flows to high Mach

*number*compressible flows

*with*shock waves. ... Acknowledgments: The authors would like to acknowledge support from the Leibniz Rechenzentrum (LRZ) in Garching, Germany,

*for*granting access to the SuperMUC-NG supercomputer under project

*number*pr63qo ...

##
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A fuzzy bi-level optimization model for multi-period post-disaster relief distribution in sustainable humanitarian supply chains

2021
*
International Journal of Production Economics
*

method

doi:10.1016/j.ijpe.2021.108081
fatcat:ysdzdsztavd3jiw3vbsut6jjra
*A*B S T R*A*C T In the aftermath*of**large*-scale natural disasters, supply shortage and inequitable distribution cause various losses, hindering humanitarian supply chains' performance. ... If the decision-agents*with*higher authorities act as the leaders*with*dominant*power*in SHSCs, the optimal decisions, respectively taking hierarchical and horizontal relationships into account would result ... In general,*large*-scale natural disasters such as the Wenchuan earthquake prefers*a*decentralized mode. Yet centralized mode is considered*for**a*subset*of**a**large*-scale or small one. ...##
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Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

2021
*
Mathematics
*

The rheology

doi:10.3390/math9222972
fatcat:glsktwbntvdztnvuc7cyks5req
*for*calculating the laminar viscosity coefficient under consideration in this work is the one*of**a*non-Newtonian Herschel–Bulkley (*power*-law) fluid*with*yield stress, which includes the Bingham ... The implicit discretization*of*the viscous terms is mandatory*for*non-Newtonian fluids, since the apparent viscosity can tend to infinity*for*fluids*with*yield stress and certain*power*-law fluids. ... Acknowledgments: The authors would like to acknowledge support from the Leibniz Rechenzentrum (LRZ) in Garching, Germany,*for*granting access to the SuperMUC-NG supercomputer under project*number*pr63qo ...##
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Numerical simulations of shear dependent viscoelastic flows with a combined finite element–finite volume method

2007
*
Computers and Mathematics with Applications
*

creeping Oldroyd-B flows,

doi:10.1016/j.camwa.2006.02.035
fatcat:gkjsk4a27ffmrml2qy34udfgvm
*for**a*range*of*Weissenberg*numbers*. ... The method is applied to the 4:*1*planar contraction benchmark problem, to investigate the influence*of*the viscosity effects on the flow and results are compared*with*those found in the literature*for*... Fig.*5*.*5*Evolution*of*the stress tensor τ along the line y =*1*on the 4:*1*contraction geometry,*for*different meshes,*with*Weissenberg*number*We = 1.5. component*of*the stress tensor in*a*channel and ...##
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Stochastic optimization for power system configuration with renewable energy in remote areas

2012
*
Annals of Operations Research
*

Given the computational complexity

doi:10.1007/s10479-012-1110-9
fatcat:p5xs22bqv5atvpmfyzwrbfm5oi
*of*the model, we developed*a*Benders' decomposition algorithm*with*Pareto-optimal cuts. ... This paper presents the first stochastic mixed integer programming model*for**a*comprehensive hybrid*power*system design, including renewable energy generation, storage device, transmission network, and ... Gordon Gillette at Tampa Electric Company*for*their support to this work. ...##
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Cross-Layer Optimal Rate Allocation for Heterogeneous Wireless Multicast

2009
*
EURASIP Journal on Wireless Communications and Networking
*

In this paper, we address the problem

doi:10.1155/2009/467182
fatcat:kgjyt4fkpfdxphr72xwfcpckea
*of*resource allocation*for**a*set*of*heterogeneous multicast sessions over multihop wireless networks. ... We propose an iterative algorithm that achieves the optimal rates*for**a*set*of*heterogeneous multicast sessions such that the aggregate utility*for*all sessions is maximized. ... Acknowledgment This research was supported by*a*collaborative R&D grant from the Qatar National Research Fund Project no. NPRP 26-6-7-10. ...##
###
Speeding the Pollard and Elliptic Curve Methods of Factorization

1987
*
Mathematics of Computation
*

Since 1974, several algorithms have been developed that attempt to factor

doi:10.2307/2007888
fatcat:6zfknzpvu5acflxq5gfz6vmkiy
*a**large**number*N by doing extensive computations modulo N and occasionally taking GCDs*with*N. ...*A*parametrization*of*elliptic curves lets Step*1**of*ECM compute the x-coordinate*of*nP from that*of*P in about 9.3 log2 n multiplications*for*arbitrary P. ... Lehmer provided me*with**a*copy*of*his multiprecision arithmetic package. H. C. Williams and Robert Silverman verified the*primality**of*many*of*the cofactors found. ...
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