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

Pedro Berrizbeitia, Mauricio Odremán, Juan Tena Ayuso
2003 Theoretical Computer Science  
The Quintic Reciprocity Law is used to produce an algorithm, that runs in polynomial time, and that determines the 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.  ... 
doi:10.1016/s0304-3975(02)00617-5 fatcat:kj5kamp5pfgijp5hv5iqi2ovje

Implementation of a new primality test

H. Cohen, A. K. Lenstra
1987 Mathematics of Computation  
An implementation 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  ... 
doi:10.1090/s0025-5718-1987-0866102-2 fatcat:5xcu4xzikbal3d4allxc5qtf7a

Implementation of a New Primality Test

H. Cohen, A. K. Lenstra
1987 Mathematics of Computation  
An implementation 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  ... 
doi:10.2307/2007877 fatcat:wg2fbdrmcvac7pnpu23ej6k3zi

Lucas Pseudoprimes

Robert Baillie, Samuel S. Wagstaff
1980 Mathematics of Computation  
Since these 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.  ... 
doi:10.2307/2006406 fatcat:2kwhtrylm5at7nc3unxvvffnzu

Lucas pseudoprimes

Robert Baillie, Samuel S. Wagstaff
1980 Mathematics of Computation  
Since these 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.  ... 
doi:10.1090/s0025-5718-1980-0583518-6 fatcat:ccqsbpl4wbchdffeucfnnxwbvy

Lattice-based Key Sharing Schemes - A Survey [article]

Prasanna Ravi, James Howe, Anupam Chattopadhyay, Shivam Bhasin
2020 IACR Cryptology ePrint Archive  
However, most if not all 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.  ... 
dblp:journals/iacr/RaviHCB20 fatcat:gwfp7xfzbbgxnldzbngfc4ru7q

Analytic Deep Learning-based Surrogate Model for Operational Planning with Dynamic TTC Constraints [article]

Gao Qiu, Youbo Liu, Junyong Liu, Junbo Zhao, Lingfeng Wang, Tingjian Liu, Hongjun Gao
2020 arXiv   pre-print
The increased penetration 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 .  ... 
arXiv:2006.16186v1 fatcat:d7upvtdazrf3fcpc3rirzd56ym

Design and phenotyping procedures for recording wool, skin, parasite resistance, growth, carcass yield and quality traits of the SheepGENOMICS mapping flock

Jason D. White, Peter G. Allingham, Chris M. Gorman, David L. Emery, Philip Hynd, John Owens, Amy Bell, Jason Siddell, Greg Harper, Ben J. Hayes, Hans D. Daetwyler, Jonathan Usmar (+13 others)
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.  ... 
doi:10.1071/an11085 fatcat:yizjzrup2ngf3doj6ozeyoli7a

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

Laura Río-Martín, Saray Busto, Michael Dumbser
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  ... 
doi:10.3390/math9182316 fatcat:vrjmxtsocjcvzfcegl5zpwdf5e

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

Cejun Cao, Yang Liu, Ou Tang, Xuehong Gao
2021 International Journal of Production Economics  
method 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.  ... 
doi:10.1016/j.ijpe.2021.108081 fatcat:ysdzdsztavd3jiw3vbsut6jjra

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

Saray Busto, Michael Dumbser, Laura Río-Martín
2021 Mathematics  
The rheology 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  ... 
doi:10.3390/math9222972 fatcat:glsktwbntvdztnvuc7cyks5req

Numerical simulations of shear dependent viscoelastic flows with a combined finite element–finite volume method

Lionel Nadau, Adélia Sequeira
2007 Computers and Mathematics with Applications  
creeping Oldroyd-B flows, 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  ... 
doi:10.1016/j.camwa.2006.02.035 fatcat:gkjsk4a27ffmrml2qy34udfgvm

Stochastic optimization for power system configuration with renewable energy in remote areas

Ludwig Kuznia, Bo Zeng, Grisselle Centeno, Zhixin Miao
2012 Annals of Operations Research  
Given the computational complexity 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.  ... 
doi:10.1007/s10479-012-1110-9 fatcat:p5xs22bqv5atvpmfyzwrbfm5oi

Cross-Layer Optimal Rate Allocation for Heterogeneous Wireless Multicast

Amr Mohamed, Hussein Alnuweiri
2009 EURASIP Journal on Wireless Communications and Networking  
In this paper, we address the problem 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.  ... 
doi:10.1155/2009/467182 fatcat:kgjyt4fkpfdxphr72xwfcpckea

Speeding the Pollard and Elliptic Curve Methods of Factorization

Peter L. Montgomery
1987 Mathematics of Computation  
Since 1974, several algorithms have been developed that attempt to factor 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.  ... 
doi:10.2307/2007888 fatcat:6zfknzpvu5acflxq5gfz6vmkiy
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