VINNICHUK S.D., ZHILIN A.V., MISKO V.N.

ABSTRACT

The authors have proposed a method for factoring the number of the form N = pq, where p and q are simple, as the solution to the problem of determining the exponent in the equation α^xmodn=b. It is shown that the proposed method and the method of Fermat are equivalent in terms of computational complexity, but the number of iterations of the discrete logarithm is [0,5log2N] times less than for the method of Fermat.

KEYWORDS

factorization, method of Fermat, RSA algorithm, computational complexity. 

REFERENCES

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8. Vasilenko, O.N. (2003), Teoretiko-chislovye algoritmy v kriptografii [Theoretical and numerical algorithms in cryptography], MTsNMO, Moscow, Russia. 
9. Shnayyer, B. (2002), Prikladnaya kriptografiya. Protokoly, algoritmy, iskhodnye teksty na yazyke Si Applied Cryptography. Protocols, algorithms, source code in C language], Triumf, Moscow, Russia.
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13. Brown, D.R.L. (2005), “Breaking RSA May Be As Difficult As Factoring”, available at: http://www.pgpru.com/novosti/2005/1026vzlomrsabezfaktoriza ciirealennoneeffektiven.
14. The GNU Multiple Precision Arithmetic Library (2013),  Edition 5.1.1.11 February 2013, available at: http://gmplib.org/gmp-man-5.1.1.pdf.

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SAUKH S.E.

ABSTRACT

A mathematical description of equilibrium states of the electric power markets as the matrix-vector form of the system of complementary relationships and algebraic equations is offered. For the system of semismooth algebraic equations generated by the Fischer-Burmeister complementary function we have obtained the formulas for calculation of the block-elements of the Jackobian and the Clarcke generalized Jackobian, which allow us to apply effectively the modern algorithms of numerical solutions of such systems for the computer modelling of the equilibrium states of the electric power markets.

KEYWORDS

the balance of the energy market, non-smooth systems of equations, matrix-vector shapes, Jacobi matrix, generalized  jacobian Clark. 

REFERENCES

1. Hobbs, B. and  Helman, U. (2004), Complementarity-Based Equilibrium Modeling for Electric Power Markets. Modeling Prices in Competitive Electricity Markets. Series in Financial Economics, Wiley, Chichester.
2. Murphy, F. and  Smeers, Y. (2005), “Generation capacity expansion in imperfectly competitive restructured electricity markets”,  Operations Research, Vol. 53, no. 4, pp. 646-661.
3. Murphy, F. and  Smeers, Y. (2007), “On the Impact of Forward Markets on Investments in Oligopolistic Markets with Reference to Electricity”,  Part 2. Uncertain Demand, Harvard Electricity Policy Group Research Paper, available at: http://www.hks.harvard.edu/hepg/Papers/Murphy_and_Smeers_June_18_07.pdf.
4. Pineau, P.-O. (2000), “Electricity market reforms: Industrial developments, investment dynamics and game modelling”, Ph. D. Thesis, Montreal, available at:  http://www.irec.net/upload/File/memoires_et_theses/260.pdf
5. Saukh, S.E. and Borisenko, A.V. (2010), “Equilibrium models of the functioning and development of generating capacities of Ukraine in market conditions”,  Energetika Rossii v XXI veke: strategiya razvitiya — vostochnyy vektor. Energeticheskaya kooperatsiya v Azii: chto posle krizisa? Sb. dokl. ob"yedinennogo simpoziuma  [Energy of Russia in XXI Century: Development Strategy - Eastern vector. Asian Energy Cooperation: after the crisis? Coll. rep. Joint Symposium], Irkutsk, ISEM SO RAN, August 30-September 3, 2010, pp. 413-419.
6. Saukh, S.E. and  Semagina, E.P. (2011),  “Determination of the equilibrium state of the electricity market in Ukraine methods of mathematical modeling”, Elektronnoe modelirovanie,  Vol.  33, no. 4,  pp. 3-14.
7. Saukh, S.E., Borisenko, A.V., Podkovalnikov, S.V. and Khamisov, O.V. (2012), “Mathematical modelling of competitive balance at electric power markets of the Russian Federation and Ukraine. II. Mathematical models of oligopolistic electric power markets and their application”, Ibid., Vol. 34, no. 3, pp. 3-24.
8. Facchinei, F. and Pang, J.-S. (2003), Finite-dimensional  Variational Inequalities and Complementarity Problems, Vol. 1, Springer.
9. Delarue, E., Bekaert, D., Belmans, R. and  D'haeseleer, W. (2007), “Development of a comprehensive electricity generation simulation model using a mixed integer programming approach”,  Proc. of the Intern. Conf. on Computer, Electrical, and Systems Science, and Engineering, Prague, July 27-29, 2007, pp. 99-104.
10. Billups, S.C., Dirkse, S.P. and  Ferris, M.C. (1997), “A comparison of large scale mixed complementarity problem solvers”, Computational Optimization and Applications, no. 7, pp. 3-25.
11. Petra, S. (2008), “Semismooth Least Squares Methods for Complementarity Problems”, Ph .D. Thesis, Wurzburg, available at:  http://www.opus-bayern.de/uni-wuerzburg/volltexte/2006/1866/pdf/dissertation_petra.pdf.
12. Ruggiero, V. and Tinti, F. (2006), “Apreconditioner for solving large scale variational inequality problems by a semismooth inexact approach”, Intern. Journal of Computer Mathematics, no. 10, pp. 723-739. 

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KRAVTSOV G.A.

ABSTRACT

The author proposes a new classification of mathematical models used in modeling of different processes in the Smart Grid. The mathematical apparatus of the model has been shown as the basic classification feature. However, as it is shown, the use of only mathematical apparatus as the classification feature has some disadvantages. It is required to involve a new classification parameter such as paradigm to determine the choice of the mathematical apparatus.

KEYWORDS

mathematical model, mathematical apparatus, classification feature, paradigm. 

REFERENCES

1. Johnson, A.P. (2010), “The history of the Smart Grid evolution at Sourthern California Edison. Innovative Smart Grid Technologies (ISGT)”, Conf. Publications, Gaithersburg, MD, pp. 1-3.

2. Massoud S. Amin  and  Wollenberg, B.F. (2005), “Toward a Smart Grid”, IEEE P&EMagazine, Vol. 3 (5), pp. 34-41.

3. Stogniy, B.S., Kyrylenko, O.V. and  Denysyuk, S.P. (2010), “Intelligent electric network of electric power systems and their technological support”,  Tekhnichna elektrodynamyka, no.  6, available at: http://techned.org.ua/article/10-6/st7.pdf.
4. Kravtsov, G.O. (2013), “Simulation of energy smart systems”, Problemy matematychnogo modelyuvannya. Tezy dopovidey Mizhderzhavnoyi naukovo-metodychnoyi konf.. [Problems of mathematical modelling. Proceedings of Interstate scientific and methodological conference], Dneprodzerzhynsk, DDTU, June  5-7, 2013, pp. 52-53.
5. Gjorgjieva, B.J., Rieke, F., Shea-Brown, E. (2010), “When are feedforward microcircuits well-modeled by maximum entropy methods?”, available at: http://arxiv.org/pdf/1011.2797v3.pdf.
6. Sommerstad, T. (2012), A framework and theory for cyber security assessment.  Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Royal Institute of Technology, Stockholm, Sweden.
7. Pelqim Spahiu and  Evans, Ian R. (2011), “Protection Systems that verify and supervise themselves”,  IEEE ISGT Innovative Smart Grid Technologies Europe, available at: http://www.ieee-isgt-2011.eu/wordpress/wp-content/uploads/2012/01/ID9_Self-Healing-Grids_Protection_Systems1.pdf.
8. Filatrella, G., Nielsen, A.H. and  Pedersen, N.F. “Analysis of a power grid using a Kuramoto-like model”, available at: http://arxiv.org/ftp/arxiv/papers/0705/0705.1305.pdf.
9. Equation Kuramoto-Sivashynskogo, available at: http://www.d-dm.ru/kse/common/.
10. Tarasevich, Yu.Yu. (2002), Perkolyatsiya: teoriya, prilozheniya, algoritmy  [Percolation: theory, applications, algorithms],  Editorial URSS, Moscow,  Russia.
11. Khaykin, S. (2006), Neyronnye seti: polnyy kurs. 2-e izd., Perevod s angl.  [Neural networks: a full course. 2nd ed.  Translation from English], OOO I.D. Vilyams, Moscow, Russia.
12. Werbos, Р.J. (2006), “Using Adaptive Dynamic Programming to Understand and Replicate Brain Intelligence: the Next Level Design”, available at: http://arxiv.org/ftp/q-bio/papers/0612/0612045.pdf.
13. Borisyuk, G.N. and et al. (1992), “Oscillatory neural network. Mathematic  results  and applications”, Matematicheskoe modelirovanie, Vol. 4, no. 1, available at: http://masters.donntu.edu.ua/2001/fvti/kuznetsov/diss/lib/neuroosc/index.htm.
14. Vvedenov, A.A., Ezhov, A.A., Knizhnikova, L.A. and et al. (1987), “Nonlinear systems  with memory functions and simulation of functions of neural ensembles”,   Intellektualnye protsessy i ikh modelirovanie,  Nauka, Moscow, available at: http://masters.donntu.edu.ua/2001/fvti/kuznetsov/diss/lib/neuroans/index.htm.
15. Guberman, Sh.A. (1987), “About the relation between perception and thinking in the problems of  artificial intelligence”,  Intellektualnye protsessy i ikh modelirovanie,  Nauka, Moscow, available at: http://masters.donntu.edu.ua/2001/fvti/kuznetsov/diss/lib/aiconcep/index.htm.
16. Vayntsvayg, M.N. and  Polyakova, M.P. (1987), “The mechanism of thinking and modeling his work in real time”,  Intellektualnye protsessy i ikh modelirovanie,  Nauka, Moscow, available at:http://masters.donntu.edu.ua/2001/fvti/kuznetsov/diss/lib/brainmdl/index.htm.
17. Andryukhin, A.I. and  Nedbaylo, S.V. (2000), “Modeling of processes in network structures”, Iskusstvennyy intellect, no. 1, available at: http://masters.donntu.edu.ua/2001/fvti/kuznetsov/diss/lib/netmodel/index.htm.
18. Andryukhin, A.I. and  Kuznetsov, A.V. (2001), “Boolean models of self-diagnosis of discrete systems”, available at:  http://masters.donntu.edu.ua/2001/fvti/kuznetsov/diss/lib/introspe/index.htm.
19. He, M., Murugesan, S. and  Zhang, J. (2010), “Multiple Timescale Dispatch and Scheduling for Stochastic Reliability in Smart Grids with Wind Generation Integration”, available at: http://arxiv.org/pdf/1008.3932v2.pdf.
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22. Kleynrok, L. (1990), Teoriya massovogo obsluzhivaniya [Queueing theory], Mashinostroenie, Moscow, Russia.
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24. Snityuk, V.E. (2006), “Evolutionary modeling and programming lifecycle of technical systems in deterministic conditions”, Iskusstvennyy intellect,  no. 4,  pp. 10-15.

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KRASNOBAYEV V.A., KOSHMAN S.A., MAVRINA M.A.

ABSTRACT

The method of correction of single errors in the residue class (RC) is considered in the article. The results of analysis of arithmetic code correcting possibilities showed high efficiency of the use of position-independent code structures in RC. Examples of correction of the data single errors presented by the RC code are given in the article.

KEYWORDS

REFERENCES

Full text: PDF (in Russian)