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Journal articles on the topic 'Curves over finite fields'

Author: Grafiati

Published: 4 June 2021

Last updated: 11 February 2022

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

JEON, DAEYEOL. "POINTS ON MODULAR CURVES OVER FINITE FIELDS." Journal of the Chungcheong Mathematical Society 28, no. 3 (August 15, 2015): 443–49. http://dx.doi.org/10.14403/jcms.2015.28.3.443.

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12

Stöhr, Karl-Otto, and José Felipe Voloch. "Weierstrass Points and Curves Over Finite Fields." Proceedings of the London Mathematical Society s3-52, no. 1 (January 1986): 1–19. http://dx.doi.org/10.1112/plms/s3-52.1.1.

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13

Hirschfeld, J. W. P. "Book Review: Algebraic curves over finite fields." Bulletin of the American Mathematical Society 27, no. 2 (October 1, 1992): 327–33. http://dx.doi.org/10.1090/s0273-0979-1992-00321-x.

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14

von zur Gathen, Joachim, Igor Shparlinski, and Alistair Sinclair. "Finding Points on Curves over Finite Fields." SIAM Journal on Computing 32, no. 6 (January 2003): 1436–48. http://dx.doi.org/10.1137/s0097539799351018.

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15

Garcia, Arnaldo, and Luciane Quoos. "A construction of curves over finite fields." Acta Arithmetica 98, no. 2 (2001): 181–95. http://dx.doi.org/10.4064/aa98-2-8.

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16

ACHTER, JEFFREY D., and SIMAN WONG. "QUOTIENTS OF ELLIPTIC CURVES OVER FINITE FIELDS." International Journal of Number Theory 09, no. 06 (September 2013): 1395–412. http://dx.doi.org/10.1142/s1793042113500334.

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Fix a prime ℓ, and let 𝔽q be a finite field with q ≡ 1 (mod ℓ) elements. If ℓ > 2 and q ≫ℓ 1, we show that asymptotically (ℓ - 1)2/2ℓ2 of the elliptic curves E/𝔽q with complete rational ℓ-torsion are such that E/〈P〉 does not have complete rational ℓ-torsion for any point P ∈ E(𝔽q) of order ℓ. For ℓ = 2 the asymptotic density is 0 or 1/4, depending whether q ≡ 1 (mod 4) or 3 (mod 4). We also show that for any ℓ, if E/𝔽q has an 𝔽q-rational point R of order ℓ2, then E/〈ℓR〉 always has complete rational ℓ-torsion.

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17

Aubry, Yves, and Marc Perret. "Coverings of singular curves over finite fields." Manuscripta Mathematica 88, no. 1 (December 1995): 467–78. http://dx.doi.org/10.1007/bf02567835.

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18

Fukshansky, Lenny, and Hiren Maharaj. "Lattices from elliptic curves over finite fields." Finite Fields and Their Applications 28 (July 2014): 67–78. http://dx.doi.org/10.1016/j.ffa.2014.01.007.

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19

Padmanabhan, R., and Alok Shukla. "Orchards in elliptic curves over finite fields." Finite Fields and Their Applications 68 (December 2020): 101756. http://dx.doi.org/10.1016/j.ffa.2020.101756.

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20

Montanucci, Maria, and Giovanni Zini. "Generalized Artin–Mumford curves over finite fields." Journal of Algebra 485 (September 2017): 310–31. http://dx.doi.org/10.1016/j.jalgebra.2017.05.020.

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21

Schoof, René. "Nonsingular plane cubic curves over finite fields." Journal of Combinatorial Theory, Series A 46, no. 2 (November 1987): 183–211. http://dx.doi.org/10.1016/0097-3165(87)90003-3.

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22

Huang, Ming-Deh, and Doug Ierardi. "Counting Points on Curves over Finite Fields." Journal of Symbolic Computation 25, no. 1 (January 1998): 1–21. http://dx.doi.org/10.1006/jsco.1997.0164.

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23

Nestler, Andrew. "SK1 of Affine Curves over Finite Fields." Journal of Algebra 225, no. 2 (March 2000): 943–46. http://dx.doi.org/10.1006/jabr.1999.8187.

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24

Galbraith, Steven D. "Constructing Isogenies between Elliptic Curves Over Finite Fields." LMS Journal of Computation and Mathematics 2 (1999): 118–38. http://dx.doi.org/10.1112/s1461157000000097.

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AbstractLet E1 and E2 be ordinary elliptic curves over a finite field Fp such that #E1(Fp) = #E2(Fp). Tate's isogeny theorem states that there is an isogeny from E1 to E2 which is defined over Fp. The goal of this paper is to describe a probabilistic algorithm for constructing such an isogeny.The algorithm proposed in this paper has exponential complexity in the worst case. Nevertheless, it is efficient in certain situations (that is, when the class number of the endomorphism ring is small). The significance of these results to elliptic curve cryptography is discussed.

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25

Hu, Yong. "Weak approximation over function fields of curves over large or finite fields." Mathematische Annalen 348, no. 2 (January 22, 2010): 357–77. http://dx.doi.org/10.1007/s00208-010-0481-y.

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26

Koike, Masao. "Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields." Hiroshima Mathematical Journal 25, no. 1 (1995): 43–52. http://dx.doi.org/10.32917/hmj/1206127824.

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27

NAJMAN, FILIP. "EXCEPTIONAL ELLIPTIC CURVES OVER QUARTIC FIELDS." International Journal of Number Theory 08, no. 05 (July 6, 2012): 1231–46. http://dx.doi.org/10.1142/s1793042112500716.

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We study the number of elliptic curves, up to isomorphism, over a fixed quartic field K having a prescribed torsion group T as a subgroup. Let T = ℤ/mℤ⊕ℤ/nℤ, where m|n, be a torsion group such that the modular curve X1(m, n) is an elliptic curve. Let K be a number field such that there is a positive and finite number of elliptic curves E over K having T as a subgroup. We call such pairs (T, K)exceptional. It is known that there are only finitely many exceptional pairs when K varies through all quadratic or cubic fields. We prove that when K varies through all quartic fields, there exist infinitely many exceptional pairs when T = ℤ/14ℤ or ℤ/15ℤ and finitely many otherwise.

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28

HUMPHRIES, PETER. "ON THE MERTENS CONJECTURE FOR ELLIPTIC CURVES OVER FINITE FIELDS." Bulletin of the Australian Mathematical Society 89, no. 1 (February 28, 2013): 19–32. http://dx.doi.org/10.1017/s0004972712001116.

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AbstractWe introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms of the size of the finite field and the trace of the Frobenius endomorphism acting on the curve.

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29

Schoof, René. "Counting points on elliptic curves over finite fields." Journal de Théorie des Nombres de Bordeaux 7, no. 1 (1995): 219–54. http://dx.doi.org/10.5802/jtnb.142.

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30

Bogomolov, Fedor, Mikhail Korotiaev, and Yuri Tschinkel. "A Torelli Theorem for Curves over Finite Fields." Pure and Applied Mathematics Quarterly 6, no. 1 (2010): 245–94. http://dx.doi.org/10.4310/pamq.2010.v6.n1.a7.

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31

Voloch, J. F. "A note on elliptic curves over finite fields." Bulletin de la Société mathématique de France 116, no. 4 (1988): 455–58. http://dx.doi.org/10.24033/bsmf.2107.

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32

Cheon, J., and S. Hahn. "Division polynomials of elliptic curves over finite fields." Proceedings of the Japan Academy, Series A, Mathematical Sciences 72, no. 10 (1996): 226–27. http://dx.doi.org/10.3792/pjaa.72.226.

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33

Gutierrez, J. "Recovering zeroes of hyperelliptic curves over finite fields." ACM Communications in Computer Algebra 49, no. 2 (August 14, 2015): 58. http://dx.doi.org/10.1145/2815111.2815140.

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34

Chandee, Vorrapan, Chantal David, Dimitris Koukoulopoulos, and Ethan Smith. "Group Structures of Elliptic Curves Over Finite Fields." International Mathematics Research Notices 2014, no. 19 (June 24, 2013): 5230–48. http://dx.doi.org/10.1093/imrn/rnt120.

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35

Top, Jaap. "Curves of genus 3 over small finite fields." Indagationes Mathematicae 14, no. 2 (June 2003): 275–83. http://dx.doi.org/10.1016/s0019-3577(03)90011-5.

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36

Hirschfeld, J. W. P., and J. F. Voloch. "The characterization of elliptic curves over finite fields." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 45, no. 2 (October 1988): 275–86. http://dx.doi.org/10.1017/s1446788700030172.

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AbstractIn a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.

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37

R{ück, Hans-Georg. "A note on elliptic curves over finite fields." Mathematics of Computation 49, no. 179 (September 1, 1987): 301. http://dx.doi.org/10.1090/s0025-5718-1987-0890272-3.

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38

Ruck, Hans-Georg. "A Note on Elliptic Curves Over Finite Fields." Mathematics of Computation 49, no. 179 (July 1987): 301. http://dx.doi.org/10.2307/2008268.

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39

Katz, Nicholas M. "Corrections to: Space Filling Curves Over Finite Fields." Mathematical Research Letters 8, no. 5 (2001): 689–91. http://dx.doi.org/10.4310/mrl.2001.v8.n5.a10.

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40

Vlăduţ, S. G. "Cyclicity Statistics for Elliptic Curves over Finite Fields." Finite Fields and Their Applications 5, no. 1 (January 1999): 13–25. http://dx.doi.org/10.1006/ffta.1998.0225.

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41

Castro, Francis N., and Carlos J. Moreno. "L-functions of Singular Curves over Finite Fields." Journal of Number Theory 84, no. 1 (September 2000): 136–55. http://dx.doi.org/10.1006/jnth.2000.2530.

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42

Wittmann, Christian. "Group Structure of Elliptic Curves over Finite Fields." Journal of Number Theory 88, no. 2 (June 2001): 335–44. http://dx.doi.org/10.1006/jnth.2000.2622.

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43

Carlin, Matthew L., and José Felipe Voloch. "Plane Curves with Many Points over Finite Fields." Rocky Mountain Journal of Mathematics 34, no. 4 (December 2004): 1255–59. http://dx.doi.org/10.1216/rmjm/1181069798.

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44

Chudnovsky, D. V., and G. V. Chudnovsky. "Algebraic complexities and algebraic curves over finite fields." Proceedings of the National Academy of Sciences 84, no. 7 (April 1, 1987): 1739–43. http://dx.doi.org/10.1073/pnas.84.7.1739.

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45

Cao, Wei, Shanmeng Han, and Ruyun Wang. "Rational points on Fermat curves over finite fields." Journal of Algebra and Its Applications 16, no. 03 (March 2017): 1750046. http://dx.doi.org/10.1142/s0219498817500463.

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Let [Formula: see text] be the [Formula: see text]-rational point on the Fermat curve [Formula: see text] with [Formula: see text]. It has recently been proved that if [Formula: see text] then each [Formula: see text] is a cube in [Formula: see text]. It is natural to wonder whether there is a generalization to [Formula: see text]. In this paper, we show that the result cannot be extended to [Formula: see text] in general and conjecture that each [Formula: see text] is a cube in [Formula: see text] if and only if [Formula: see text].

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46

Lee, Jong Won. "Isomorphism Classes of Picard Curves over Finite Fields." Applicable Algebra in Engineering, Communication and Computing 16, no. 1 (May 27, 2005): 33–44. http://dx.doi.org/10.1007/s00200-003-0145-1.

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47

Meagher, Stephen, and Jaap Top. "Twists of genus three curves over finite fields." Finite Fields and Their Applications 16, no. 5 (September 2010): 347–68. http://dx.doi.org/10.1016/j.ffa.2010.06.001.

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48

Alekseenko, E., S. Aleshnikov, N. Markin, and A. Zaytsev. "Optimal curves over finite fields with discriminant −19." Finite Fields and Their Applications 17, no. 4 (July 2011): 350–58. http://dx.doi.org/10.1016/j.ffa.2011.02.006.

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49

Rezaeian Farashahi, Reza, Dustin Moody, and Hongfeng Wu. "Isomorphism classes of Edwards curves over finite fields." Finite Fields and Their Applications 18, no. 3 (May 2012): 597–612. http://dx.doi.org/10.1016/j.ffa.2011.12.004.

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50

Zaytsev, Alexey. "Optimal curves of low genus over finite fields." Finite Fields and Their Applications 37 (January 2016): 203–24. http://dx.doi.org/10.1016/j.ffa.2015.09.008.

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