Relevant bibliographies by topics / Curves over finite fields

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Curves over finite fields'

Author: Grafiati
Published: 4 June 2021
Last updated: 11 February 2022

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Journal articles on the topic "Curves over finite fields"

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van der Geer, Gerard. "Counting curves over finite fields." Finite Fields and Their Applications 32 (March 2015): 207–32. http://dx.doi.org/10.1016/j.ffa.2014.09.008.

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Garcia, A., and J. F. Voloch. "Fermat curves over finite fields." Journal of Number Theory 30, no. 3 (November 1988): 345–56. http://dx.doi.org/10.1016/0022-314x(88)90007-8.

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Auer, R. "Legendre Elliptic Curves over Finite Fields." Journal of Number Theory 95, no. 2 (August 2002): 303–12. http://dx.doi.org/10.1016/s0022-314x(01)92760-x.

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Voloch, José Felipe. "Jacobians of Curves over Finite Fields." Rocky Mountain Journal of Mathematics 30, no. 2 (June 2000): 755–59. http://dx.doi.org/10.1216/rmjm/1022009294.

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Katz, Nicholas M. "Space filling curves over finite fields." Mathematical Research Letters 6, no. 6 (1999): 613–24. http://dx.doi.org/10.4310/mrl.1999.v6.n6.a2.

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Auer, Roland, and Jaap Top. "Legendre Elliptic Curves over Finite Fields." Journal of Number Theory 95, no. 2 (August 2002): 303–12. http://dx.doi.org/10.1006/jnth.2001.2760.

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Morain, François, Charlotte Scribot, and Benjamin Smith. "Computing cardinalities of -curve reductions over finite fields." LMS Journal of Computation and Mathematics 19, A (2016): 115–29. http://dx.doi.org/10.1112/s1461157016000267.

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We present a specialized point-counting algorithm for a class of elliptic curves over $\mathbb{F}_{p^{2}}$ that includes reductions of quadratic $\mathbb{Q}$-curves modulo inert primes and, more generally, any elliptic curve over $\mathbb{F}_{p^{2}}$ with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof–Elkies–Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.
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Skałba, M. "Points on elliptic curves over finite fields." Acta Arithmetica 117, no. 3 (2005): 293–301. http://dx.doi.org/10.4064/aa117-3-7.

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Shparlinski, Igor E., and José Felipe Voloch. "Visible Points on Curves over Finite Fields." Bulletin of the Polish Academy of Sciences Mathematics 55, no. 3 (2007): 193–99. http://dx.doi.org/10.4064/ba55-3-1.

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Bridy, Andrew. "Automatic sequences and curves over finite fields." Algebra & Number Theory 11, no. 3 (May 6, 2017): 685–712. http://dx.doi.org/10.2140/ant.2017.11.685.

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Dissertations / Theses on the topic "Curves over finite fields"

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Voloch, J. F. "Curves over finite fields." Thesis, University of Cambridge, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.355283.

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Rovi, Carmen. "Algebraic Curves over Finite Fields." Thesis, Linköping University, Department of Mathematics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-56761.

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This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa's construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a nite eld and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~geer, which to the time of writing this Thesis appear as "no information available". In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of Nq(g) is now known.

At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.

 

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Thuen, Øystein Øvreås. "Constructing elliptic curves over finite fields using complex multiplication." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2006. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-9434.

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We study and improve the CM-method for the creation of elliptic curves with specified group order over finite fields. We include a thorough review of the mathematical theory needed to understand this method. The ability to construct elliptic curves with very special group order is important in pairing-based cryptography.

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Cam, Vural. "Drinfeld Modular Curves With Many Rational Points Over Finite Fields." Phd thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613118/index.pdf.

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In our study Fq denotes the finite field with q elements. It is interesting to construct curves of given genus over Fq with many Fq -rational points. Drinfeld modular curves can be used to construct that kind of curves over Fq . In this study we will use reductions of the Drinfeld modular curves X_{0} (n) to obtain curves over finite fields with many rational points. The main idea is to divide the Drinfeld modular curves by an Atkin-Lehner involution which has many fixed points to obtain a quotient with a better #{rational points} /genus ratio. If we divide the Drinfeld modular curve X_{0} (n) by an involution W, then the number of rational points of the quotient curve WX_{0} (n) is not less than half of the original number. On the other hand, if this involution has many fixed points, then by the Hurwitz-Genus formula the genus of the curve WX_{0} (n) is much less than half of the g (X_{0}(n)).
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Kirlar, Baris Bulent. "Isomorphism Classes Of Elliptic Curves Over Finite Fields Of Characteristic Two." Master's thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/2/12606489/index.pdf.

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In this thesis, the work of Menezes on the isomorphism classes of elliptic curves over finite fields of characteristic two is studied. Basic definitions and some facts of the elliptic curves required in this context are reviewed and group structure of elliptic curves are constructed. A fairly detailed investigation is made for the isomorphism classes of elliptic curves due to Menezes and Schoof. This work plays an important role in Elliptic Curve Digital Signature Algorithm. In this context, those isomorphism classes of elliptic curves recommended by National Institute of Standards and Technology are listed and their properties are discussed.
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Ducet, Virgile. "Construction of algebraic curves with many rational points over finite fields." Thesis, Aix-Marseille, 2013. http://www.theses.fr/2013AIXM4043/document.

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L'étude du nombre de points rationnels d'une courbe définie sur un corps fini se divise naturellement en deux cas : lorsque le genre est petit (typiquement g<=50), et lorsqu'il tend vers l'infini. Nous consacrons une partie de cette thèse à chacun de ces cas. Dans la première partie de notre étude nous expliquons comment calculer l'équation de n'importe quel revêtement abélien d'une courbe définie sur un corps fini. Nous utilisons pour cela la théorie explicite du corps de classe fournie par les extensions de Kummer et d'Artin-Schreier-Witt. Nous détaillons également un algorithme pour la recherche de bonnes courbes, dont l'implémentation fournit de nouveaux records de nombre de points sur les corps finis d'ordres 2 et 3. Nous étudions dans la seconde partie une formule de trace d'opérateurs de Hecke sur des formes modulaires quaternioniques, et montrons que les courbes de Shimura associées forment naturellement des suites récursives de courbes asymptotiquement optimales sur une extension quadratique du corps de base. Nous prouvons également qu'alors la contribution essentielle en points rationnels est fournie par les points supersinguliers
The study of the number of rational points of a curve defined over a finite field naturally falls into two cases: when the genus is small (typically g<=50), and when it tends to infinity. We devote one part of this thesis to each of these cases. In the first part of our study, we explain how to compute the equation of any abelian covering of a curve defined over a finite field. For this we use explicit class field theory provided by Kummer and Artin-Schreier-Witt extensions. We also detail an algorithm for the search of good curves, whose implementation provides new records of number of points over the finite fields of order 2 and 3. In the second part, we study a trace formula of Hecke operators on quaternionic modular forms, and we show that the associated Shimura curves of the form naturally form recursive sequences of asymptotically optimal curves over a quadratic extension of the base field. Moreover, we then prove that the essential contribution to the rational points is provided by supersingular points
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Riquelme, Faúndez Edgardo. "Algorithms for l-sections on genus two curves over finite fields and applications." Doctoral thesis, Universitat de Lleida, 2016. http://hdl.handle.net/10803/393881.

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We study \ell-section algorithms for Jacobian of genus two over finite fields. We provide trisection (division by \ell=3) algorithms for Jacobians of genus 2 curves over finite fields \F_q of odd and even characteristic. In odd characteristic we obtain a symbolic trisection polynomial whose roots correspond (bijectively) to the set of trisections of the given divisor. We also construct a polynomial whose roots allow us to calculate the 3-torsion divisors. We show the relation between the rank of the 3-torsion subgroup and the factorization of this 3-torsion polynomial, and describe the factorization of the trisection polynomials in terms of the galois structure of the 3- torsion subgroup. We generalize these ideas and we determine the field of definition of an \ell-section with \ell \in {3, 5, 7}. In characteristic two for non-supersingular hyperelliptic curves we characterize the 3-torsion divisors and provide a polynomial whose roots correspond to the set of trisections of the given divisor. We also present a generalization of the known algorithms for the computation of the 2-Sylow subgroup to the case of the \ell-Sylow subgroup in general and we present explicit algorithms for the computation of the 3-Sylow subgroup. Finally we show some examples where we can obtain the central coefficients of the characteristic polynomial of the Frobenius endomorphism reduced modulo 3 using the generators obtained with the 3-Sylow algorithm.
En esta tesis se estudian algoritmos de \ell-división para Jacobianas de curvas de género 2. Se presentan algoritmos de trisección (división por \ell=3) para Jacobianas de curvas de género 2 definidas sobre cuerpos finitos \F_q de característica par o impar indistintamente. En característica impar se obtiene explícitamente un polinomio de trisección, cuyas raíces se corresponden biyectivamente con el conjunto de trisecciones de un divisor cualquiera de la Jacobiana. Asimismo se proporciona otro polinomio a partir de cuyas raíces se calcula el conjunto de los divisores de orden 3. Se muestra la relación entre el rango del subgrupo de 3-torsión y la factorización del polinomio de la 3- torsión, y se describe la factorización del polinomio de trisección en términos de las órbitas galoisianas de la 3- torsión. Se generalizan estas ideas para otros valores de \ell y se determina el cuerpo de definición de una \ell-sección para \ell=3,5,7. Para curvas no-supersingulares en característica par también se da una caracterización de la 3-torsión y se proporciona un polinomio de trisección para un divisor cualquiera. Se da una generalización, para \ell arbitraria, de los algoritmos conocidos para el cómputo explícito del subgrupo de 2-Sylow, y se detalla explícitamente el algoritmo para el cómputo del subgrupo de 3-Sylow. Finalmente, se dan ejemplos de cómo obtener los valores de la reducción módulo 3 de los coeficientes centrales del polinomio característico del endomorfismo de Frobenius mediante los generadores proporcionados por el algoritmo de cálculo del 3-Sylow.
En aquesta tesi s'estudien algoritmes de \ell-divisió per a grups de punts de Jacobianes de corbes de gènere 2. Es presenten algoritmes de trisecció (divisió per \ell=3) per a Jacobianes de corbes de gènere 2 definides sobre cossos finits \F_q de característica parell o senar indistintament. En característica parell s'obté explícitament un polinomi de trisecció, les arrels del qual estan en bijecció amb el conjunt de triseccions d'un divisor de la Jacobiana qualsevol. De manera semblant, es proporciona un altre polinomi amb les arrels del qual es calcula el conjunt dels divisors d'ordre 3. Es mostra la relació entre el rang del subgrup de 3-torsió i la factorització del polinomi de la 3-torsió, i es descriu la factorització del polinomi de trisecció en termes de les òrbites galoisianes de la 3-torsió. Es generalitzen aquestes idees a altres valors de \ell i es determina el cos de definició d'una \ell-secció per a \ell=3,5,7. Per a corbes nosupersingulars en característica 2 també es proporciona una caracterització de la 3-torsió i un polinomi de trisecció per a un divisor qualsevol. Es dóna una generalització, per a \ell arbitrària, dels algoritmes coneguts per al càlcul explícit del subgrup de 2-Sylow, i es detalla explícitament en el cas del 3-Sylow. Finalment es mostren exemples de com obtenir els valors de la reducció mòdul 3 dels coeficients centrals del polinomi característic de l'endomorfisme de Frobenius fent servir els generadors proporcionats per l'algoritme de càlcul del 3-Sylow.
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Cai, Zhi, and 蔡植. "A study on parameters generation of elliptic curve cryptosystem over finite fields." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2001. http://hub.hku.hk/bib/B31225639.

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Fuselier, Jenny G. "Hypergeometric functions over finite fields and relations to modular forms and elliptic curves." [College Station, Tex. : Texas A&M University, 2007. http://hdl.handle.net/1969.1/ETD-TAMU-1547.

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Hoshi, Yuichiro. "Absolute anabelian cuspidalizations of configuration spaces of proper hyperbolic curves over finite fields." 京都大学 (Kyoto University), 2009. http://hdl.handle.net/2433/126568.

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Kyoto University (京都大学)
0048
新制・論文博士
博士(理学)
乙第12377号
論理博第1509号
新制||理||1507(附属図書館)
27312
UT51-2009-K686
京都大学大学院理学研究科数学・数理解析専攻
(主査)教授 望月 新一, 教授 玉川 安騎男, 教授 向井 茂
学位規則第4条第2項該当
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Books on the topic "Curves over finite fields"

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Moreno, Carlos J. Algebraic curves over finite fields. Cambridge [England]: Cambridge University Press, 1991.

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Fried, Michael D., ed. Applications of Curves over Finite Fields. Providence, Rhode Island: American Mathematical Society, 1999. http://dx.doi.org/10.1090/conm/245.

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Hansen, Søren Have. Rational points on curves over finite fields. [Aarhus, Denmark: Aarhus Universitet, Matematisk Institut, 1995.

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Alam, Shajahan. Zeta-functions of curves over finite fields. Manchester: UMIST, 1996.

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AMS-IMS-SIAM Joint Summer Research Conference on Applications of Curves over Finite Fields (1997 University of Washington). Applications of curves over finite fields: 1997 AMS-IMS-SIAM Joint Summer Research Conference on Applications of Curves over Finite Fields, July 27-31, 1997, University of Washington, Seattle. Edited by Fried Michael D. 1942-. Providence, R.I: American Mathematical Society, 1999.

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Algebraic curves and cryptography. Providence, R.I: American Mathematical Society, 2010.

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Shparlinski, Igor E., and David R. Kohel. Frobenius distributions: Lang-Trotter and Sato-Tate conjectures : Winter School on Frobenius Distributions on Curves, February 17-21, 2014 [and] Workshop on Frobenius Distributions on Curves, February 24-28, 2014, Centre International de Rencontres Mathematiques, Marseille, France. Providence, Rhode Island: American Mathematical Society, 2016.

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Moreno, Carlos. Algebraic curvesover finite fields. Cambridge: Cambridge University Press, 1991.

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Projective geometries over finite fields. 2nd ed. Oxford: Clarendon Press, 1998.

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Winterhof, Arne, Harald Niederreiter, Alina Ostafe, and Daniel Panario. Algebraic curves and finite fields: Cryptography and other applications. Berlin: De Gruyter, 2014.

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Book chapters on the topic "Curves over finite fields"

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Husemöller, Dale. "Elliptic Curves over Finite Fields." In Elliptic Curves, 242–61. New York, NY: Springer New York, 1987. http://dx.doi.org/10.1007/978-1-4757-5119-2_14.

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Tsfasman, Michael, Serge Vlǎduţ, and Dmitry Nogin. "Curves over finite fields." In Mathematical Surveys and Monographs, 133–89. Providence, Rhode Island: American Mathematical Society, 2007. http://dx.doi.org/10.1090/surv/139/03.

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Blake, Ian F., XuHong Gao, Ronald C. Mullin, Scott A. Vanstone, and Tomik Yaghoobian. "Elliptic Curves over Finite Fields." In Applications of Finite Fields, 139–50. Boston, MA: Springer US, 1993. http://dx.doi.org/10.1007/978-1-4757-2226-0_7.

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Silverman, Joseph H. "Elliptic Curves over Finite Fields." In The Arithmetic of Elliptic Curves, 137–56. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-0-387-09494-6_5.

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Silverman, Joseph H., and John T. Tate. "Cubic Curves over Finite Fields." In Rational Points on Elliptic Curves, 117–66. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-18588-0_4.

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Silverman, Joseph H. "Elliptic Curves over Finite Fields." In The Arithmetic of Elliptic Curves, 130–45. New York, NY: Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4757-1920-8_6.

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Silverman, Joseph H., and John Tate. "Cubic Curves over Finite Fields." In Rational Points on Elliptic Curves, 107–44. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-4252-7_5.

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Enge, Andreas. "Elliptic Curves Over Finite Fields." In Elliptic Curves and Their Applications to Cryptography, 45–107. Boston, MA: Springer US, 1999. http://dx.doi.org/10.1007/978-1-4615-5207-9_3.

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Sury, B. "Elliptic Curves over Finite Fields." In Elliptic Curves, Modular Forms and Cryptography, 33–47. Gurgaon: Hindustan Book Agency, 2003. http://dx.doi.org/10.1007/978-93-86279-15-6_3.

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Stepanov, Serguei A. "Counting Points on Curves over Finite Fields." In Codes on Algebraic Curves, 143–72. Boston, MA: Springer US, 1999. http://dx.doi.org/10.1007/978-1-4615-4785-3_6.

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Conference papers on the topic "Curves over finite fields"

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Voight, John. "Curves over finite fields with many points: an introduction." In Computational Aspects of Algebraic Curves. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812701640_0010.

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Shparlinski, Igor E. "Pseudorandom Points on Elliptic Curves over Finite Fields." In Proceedings of the First SAGA Conference. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812793430_0006.

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Shankar, B. R., and Kamath K. Karuna. "(2,1)-Lagged Fibonacci Generators Using Elliptic Curves over Finite Fields." In 2009 International Conference on Computer Engineering and Technology (ICCET). IEEE, 2009. http://dx.doi.org/10.1109/iccet.2009.103.

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Buchmann, Johannes, and Volker Müller. "Computing the number of points of elliptic curves over finite fields." In the 1991 international symposium. New York, New York, USA: ACM Press, 1991. http://dx.doi.org/10.1145/120694.120718.

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Cohen, Ran. "Group Law Algorithms for Jacobian Varieties of Curves over Finite Fields." In Proceedings of the First SAGA Conference. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812793430_0011.

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Daikpor, Michael Naseimo, and Oluwole Adegbenro. "Arithmetic Operations on Elliptic Curves Defined over Un-conventional Element Finite Fields." In 2012 International Conference on Cyber-Enabled Distributed Computing and Knowledge Discovery (CyberC). IEEE, 2012. http://dx.doi.org/10.1109/cyberc.2012.29.

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Zhang, Yuhong, Meng Zhang, and Maozhi Xu. "Finding vulnerable curves over finite fields of characteristic 2 by pairing reduction." In 2017 IEEE/ACIS 16th International Conference on Computer and Information Science (ICIS). IEEE, 2017. http://dx.doi.org/10.1109/icis.2017.7960080.

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Freeman, David, and Kristin Lauter. "Computing endomorphism rings of Jacobians of genus 2 curves over finite fields." In Proceedings of the First SAGA Conference. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812793430_0002.

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McEliece, Robert J., and M. C. Rodriguez-palanquex. "AG Goppa Codes from Maximal Curves over determined Finite Fields of characteristic 2." In 2006 IEEE International Symposium on Information Theory. IEEE, 2006. http://dx.doi.org/10.1109/isit.2006.261891.

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Izu, Tetsuya, Masahiko Takenaka, and Masaya Yasuda. "Time Estimation of Cheon's Algorithm over Elliptic Curves on Finite Fields with Characteristic 3." In 2011 Fifth International Conference on Innovative Mobile and Internet Services in Ubiquitous Computing (IMIS). IEEE, 2011. http://dx.doi.org/10.1109/imis.2011.113.

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Reports on the topic "Curves over finite fields"

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Wei, Fulu, Ce Wang, Xiangxi Tian, Shuo Li, and Jie Shan. Investigation of Durability and Performance of High Friction Surface Treatment. Purdue University, 2021. http://dx.doi.org/10.5703/1288284317281.

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The Indiana Department of Transportation (INDOT) completed a total of 25 high friction surface treatment (HFST) projects across the state in 2018. This research study attempted to investigate the durability and performance of HFST in terms of its HFST-pavement system integrity and surface friction performance. Laboratory tests were conducted to determine the physical and mechanical properties of epoxy-bauxite mortar. Field inspections were carried out to identify site conditions and common early HFST distresses. Cyclic loading test and finite element method (FEM) analysis were performed to evaluate the bonding strength between HFST and existing pavement, in particular chip seal with different pretreatments such as vacuum sweeping, shotblasting, and scarification milling. Both surface friction and texture tests were undertaken periodically (generally once every 6 months) to evaluate the surface friction performance of HFST. Crash records over a 5-year period, i.e., 3 years before installation and 2 years after installation, were examined to determine the safety performance of HFST, crash modification factor (CMF) in particular. It was found that HFST epoxy-bauxite mortar has a coefficient of thermal expansion (CTE) significantly higher than those of hot mix asphalt (HMA) mixtures and Portland cement concrete (PCC), and good cracking resistance. The most common early HFST distresses in Indiana are reflective cracking, surface wrinkling, aggregate loss, and delamination. Vacuum sweeping is the optimal method for pretreating existing pavements, chip seal in particular. Chip seal in good condition is structurally capable of providing a sound base for HFST. On two-lane highway curves, HFST is capable of reducing the total vehicle crash by 30%, injury crash by 50%, and wet weather crash by 44%, and providing a CMF of 0.584 in Indiana. Great variability may arise in the results of friction tests on horizontal curves by the use of locked wheel skid tester (LWST) due both to the nature of vehicle dynamics and to the operation of test vehicle. Texture testing, however, is capable of providing continuous texture measurements that can be used to calculate a texture height parameter, i.e., mean profile depth (MPD), not only for evaluating friction performance but also implementing quality control (QC) and quality assurance (QA) plans for HFST.
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You might also be interested in the extended bibliographies on the topic 'Curves over finite fields' for particular source types:
Journal articles Dissertations / Theses Books
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