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Journal articles on the topic 'Elliptic Curves over Finite Fields'

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Published: 4 June 2021
Last updated: 4 February 2022

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Luca, Florian, and Igor E. Shparlinski. "Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields." Canadian Mathematical Bulletin 50, no. 3 (September 1, 2007): 409–17. http://dx.doi.org/10.4153/cmb-2007-039-2.

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AbstractWe show that, for most of the elliptic curves E over a prime finite field p of p elements, the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over p is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over p.
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12

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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13

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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14

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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15

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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16

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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17

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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18

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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19

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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20

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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21

Gekeler, Ernst-Ulrich. "Statistics about elliptic curves over finite prime fields." manuscripta mathematica 127, no. 1 (May 20, 2008): 55–67. http://dx.doi.org/10.1007/s00229-008-0192-9.

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22

Bruin, Peter, and Filip Najman. "Hyperelliptic modular curves and isogenies of elliptic curves over quadratic fields." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 578–602. http://dx.doi.org/10.1112/s1461157015000157.

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We study elliptic curves over quadratic fields with isogenies of certain degrees. Let $n$ be a positive integer such that the modular curve $X_{0}(n)$ is hyperelliptic of genus ${\geqslant}2$ and such that its Jacobian has rank $0$ over $\mathbb{Q}$. We determine all points of $X_{0}(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, every elliptic curve over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by a quadratic extension $L$ of $K$. We determine $d$ and $L$ explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, all elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb{Q}$-curves.
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23

Friedlander, John B., Carl Pomerance, and Igor E. Shparlinski. "Finding the group structure of elliptic curves over finite fields." Bulletin of the Australian Mathematical Society 72, no. 2 (October 2005): 251–63. http://dx.doi.org/10.1017/s0004972700035048.

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We show that an algorithm of V. Miller to compute the group structure of an elliptic curve over a prime finite field runs in probabilistic polynomial time for almost all curves over the field. Important to our proof are estimates for some divisor sums.
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24

Hakuta, Keisuke. "Metrics on the Sets of Nonsupersingular Elliptic Curves in Simplified Weierstrass Form over Finite Fields of Characteristic Two." International Journal of Mathematics and Mathematical Sciences 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/597849.

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Elliptic curves have a wide variety of applications in computational number theory such as elliptic curve cryptography, pairing based cryptography, primality tests, and integer factorization. Mishra and Gupta (2008) have found an interesting property of the sets of elliptic curves in simplified Weierstrass form (or short Weierstrass form) over prime fields. The property is that one can induce metrics on the sets of elliptic curves in simplified Weierstrass form over prime fields of characteristic greater than three. Later, Vetro (2011) has found some other metrics on the sets of elliptic curves in simplified Weierstrass form over prime fields of characteristic greater than three. However, to our knowledge, no analogous result is known in the characteristic two case. In this paper, we will prove that one can induce metrics on the sets of nonsupersingular elliptic curves in simplified Weierstrass form over finite fields of characteristic two.
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25

Saouter, Yannick. "Constructions of LDPCs From Elliptic Curves Over Finite Fields." IEEE Communications Letters 21, no. 12 (December 2017): 2558–61. http://dx.doi.org/10.1109/lcomm.2017.2750660.

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26

Sutherland, Andrew V. "Constructing elliptic curves over finite fields with prescribed torsion." Mathematics of Computation 81, no. 278 (August 4, 2011): 1131–47. http://dx.doi.org/10.1090/s0025-5718-2011-02538-x.

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27

Banks, William D., John B. Friedlander, Moubariz Z. Garaev, and Igor E. Shparlinski. "Double Character Sums over Elliptic Curves and Finite Fields." Pure and Applied Mathematics Quarterly 2, no. 1 (2006): 179–97. http://dx.doi.org/10.4310/pamq.2006.v2.n1.a8.

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28

Anema, A. S. I. "Elliptic curves maximal over extensions of finite base fields." Mathematics of Computation 88, no. 315 (May 4, 2018): 453–65. http://dx.doi.org/10.1090/mcom/3342.

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29

Bröker, Reinier, and Peter Stevenhagen. "Efficient CM-constructions of elliptic curves over finite fields." Mathematics of Computation 76, no. 260 (October 1, 2007): 2161–80. http://dx.doi.org/10.1090/s0025-5718-07-01980-1.

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30

Sha, Min. "On the lattices from elliptic curves over finite fields." Finite Fields and Their Applications 31 (January 2015): 84–107. http://dx.doi.org/10.1016/j.ffa.2014.10.004.

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31

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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32

David, Chantal, and Ethan Smith. "Elliptic curves with a given number of points over finite fields." Compositio Mathematica 149, no. 2 (November 1, 2012): 175–203. http://dx.doi.org/10.1112/s0010437x12000541.

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AbstractGiven an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over 𝔽p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.
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33

Uma Maheswari, A., and Prabha Durairaj. "Factoring Polynomials Using Elliptic Curves." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 112. http://dx.doi.org/10.14419/ijet.v7i4.10.20819.

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This paper presents a probabilistic algorithm to factor polynomials over finite fields using elliptic curves. The success of the algorithm depends on the initial choice of elliptic curve parameters. The algorithm is illustrated through numerical examples.
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34

Lauder, Alan G. B., and Daqing Wan. "Computing Zeta Functions of Artin–schreier Curves over Finite Fields." LMS Journal of Computation and Mathematics 5 (2002): 34–55. http://dx.doi.org/10.1112/s1461157000000681.

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AbstractThe authors present a practical polynomial-time algorithm for computing the zeta function of certain Artin–Schreier curves over finite fields. This yields a method for computing the order of the Jacobian of an elliptic curve in characteristic 2, and more generally, any hyperelliptic curve in characteristic 2 whose affine equation is of a particular form. The algorithm is based upon an efficient reduction method for the Dwork cohomology of one-variable exponential sums.
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35

KINJO, KENSAKU, and YUKEN MIYASAKA. "HYPERGEOMETRIC SERIES AND ARITHMETIC–GEOMETRIC MEAN OVER 2-ADIC FIELDS." International Journal of Number Theory 08, no. 03 (April 7, 2012): 831–44. http://dx.doi.org/10.1142/s1793042112500480.

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Dwork proved that the Gaussian hypergeometric function on p-adic numbers can be extended to a function which takes values of the unit roots of ordinary elliptic curves over a finite field of characteristic p ≥ 3. We present an analogous theory in the case p = 2. As an application, we give a relation between the canonical lift and the unit root of an elliptic curve over a finite field of characteristic 2 by using the 2-adic arithmetic–geometric mean.
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36

Hu, Jian Jun. "Constructing Elliptic Curves over Ramanujan's Class Invariants." Advanced Materials Research 915-916 (April 2014): 1336–40. http://dx.doi.org/10.4028/www.scientific.net/amr.915-916.1336.

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The Complex Multiplication (CM) method is a widely used technique for constructing elliptic curves over finite fields. The key point in this method is parameter generation of the elliptic curve and root compution of a special type of class polynomials. However, there are several class polynomials which can be used in the CM method, having much smaller coefficients, and fulfilling the prerequisite that their roots can be easily transformed to the roots of the corresponding Hilbert polynomials.In this paper, we provide a method which can construct elliptic curves by Ramanujan's class invariants. We described the algorithm for the construction of elliptic curves (ECs) over imaginary quadratic field and given the transformation from their roots to the roots of the corresponding Hilbert polynomials. We compared the efficiency in the use of this method and other methods.
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37

Maurer, Markus, Alfred Menezes, and Edlyn Teske. "Analysis of the GHS Weil Descent Attack on the ECDLP over Characteristic Two Finite Fields of Composite Degree." LMS Journal of Computation and Mathematics 5 (2002): 127–74. http://dx.doi.org/10.1112/s1461157000000723.

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AbstractIn this paper, the authors analyze the Gaudry-Hess-Smart (GHS) Weil descent attack on the elliptic curve discrete logarithm problem (ECDLP) for elliptic curves defined over characteristic two finite fields of composite extension degree. For each such field F2N, where N is in [100,600], elliptic curve parameters are identified such that: (i) there should exist a cryptographically interesting elliptic curve E over F2N with these parameters; and (ii) the GHS attack is more efficient for solving the ECDLP in E(F2N) than for solving the ECDLP on any other cryptographically interesting elliptic curve over F2N. The feasibility of the GHS attack on the specific elliptic curves is examined over F2176, F2208, F2272, F2304 and F2368, which are provided as examples in the ANSI X9.62 standard for the elliptic curve signature scheme ECDSA. Finally, several concrete instances are provided of the ECDLP over F2N, N composite, of increasing difficulty; these resist all previously known attacks, but are within reach of the GHS attack.
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38

Diem, Claus. "On the discrete logarithm problem in elliptic curves." Compositio Mathematica 147, no. 1 (October 15, 2010): 75–104. http://dx.doi.org/10.1112/s0010437x10005075.

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AbstractWe study the elliptic curve discrete logarithm problem over finite extension fields. We show that for any sequences of prime powers (qi)i∈ℕand natural numbers (ni)i∈ℕwithni⟶∞andni/log (qi)⟶0 fori⟶∞, the elliptic curve discrete logarithm problem restricted to curves over the fields 𝔽qniican be solved in subexponential expected time (qnii)o(1). We also show that there exists a sequence of prime powers (qi)i∈ℕsuch that the problem restricted to curves over 𝔽qican be solved in an expected time ofe𝒪(log (qi)2/3).
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39

Enge, Andreas, and Reinhard Schertz. "Constructing elliptic curves over finite fields using double eta-quotients." Journal de Théorie des Nombres de Bordeaux 16, no. 3 (2004): 555–68. http://dx.doi.org/10.5802/jtnb.460.

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40

Kim, Yong-Tae. "Fast Factorization Methods based on Elliptic Curves over Finite Fields." Journal of the Korea institute of electronic communication sciences 10, no. 10 (October 31, 2015): 1093–100. http://dx.doi.org/10.13067/jkiecs.2015.10.10.1093.

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41

Chen, Yen-Mei J., and Jing Yu. "On a density problem for elliptic curves over finite fields." Asian Journal of Mathematics 4, no. 4 (2000): 737–56. http://dx.doi.org/10.4310/ajm.2000.v4.n4.a2.

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42

Adj, Gora, Omran Ahmadi, and Alfred Menezes. "On isogeny graphs of supersingular elliptic curves over finite fields." Finite Fields and Their Applications 55 (January 2019): 268–83. http://dx.doi.org/10.1016/j.ffa.2018.10.002.

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43

Shparlinski, Igor E. "Small discriminants of complex multiplication fields of elliptic curves over finite fields." Czechoslovak Mathematical Journal 65, no. 2 (June 2015): 381–88. http://dx.doi.org/10.1007/s10587-015-0183-4.

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44

Hasegawa, Takehiro, Miyoko Inuzuka, and Takafumi Suzuki. "Towers of function fields over finite fields corresponding to elliptic modular curves." Finite Fields and Their Applications 18, no. 1 (January 2012): 1–18. http://dx.doi.org/10.1016/j.ffa.2011.04.003.

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45

Kamthawee, Krissanee, and Bhichate Chiewthanakul. "The Construction of ElGamal over Koblitz Curve." Advanced Materials Research 931-932 (May 2014): 1441–46. http://dx.doi.org/10.4028/www.scientific.net/amr.931-932.1441.

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Recently elliptic curve cryptosystems are widely accepted for security applications key generation, signature and verification. Cryptographic mechanisms based on elliptic curves depend on arithmetic involving the points of the curve. it is possible to use smaller primes, or smaller finite fields, with elliptic curves and achieve a level of security comparable to that for much larger integers. Koblitz curves, also known as anomalous binary curves, are elliptic curves defined over F2. The primary advantage of these curves is that point multiplication algorithms can be devised that do not use any point doublings. The ElGamal cryptosystem, which is based on the Discrete Logarithm problem can be implemented in any group. In this paper, we propose the ElGamal over Koblitz Curve Scheme by applying the arithmetic on Koblitz curve to the ElGamal cryptosystem. The advantage of this scheme is that point multiplication algorithms can be speeded up the scalar multiplication in the affine coodinate of the curves using Frobenius map. It has characteristic two, therefore it’s arithmetic can be designed in any computer hardware. Moreover, it has more efficient to employ the TNAF method for scalar multiplication on Koblitz curves to decrease the number of nonzero digits. It’s security relies on the inability of a forger, who does not know a private key, to compute elliptic curve discrete logarithm.
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46

CONCEIÇÃO, RICARDO. "ON INTEGRAL POINTS ON ISOTRIVIAL ELLIPTIC CURVES OVER FUNCTION FIELDS." Bulletin of the Australian Mathematical Society 102, no. 2 (March 27, 2020): 177–85. http://dx.doi.org/10.1017/s0004972720000155.

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Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in terms of $g$ and the size of $S$. In the second part we assume that $L$ is the function field of a hyperelliptic curve $C_{A}:s^{2}=A(t)$, where $A(t)$ is a square-free $k$-polynomial of odd degree. If $\infty$ is the place of $L$ associated to the point at infinity of $C_{A}$, then we prove that the set of separable $\{\infty \}$-points can be bounded solely in terms of $g$ and does not depend on the Mordell–Weil group $E(L)$. This is done by bounding the number of separable integral points over $k(t)$ on elliptic curves of the form $E_{A}:A(t)y^{2}=f(x)$, where $f(x)$ is a polynomial over $k$. Additionally, we show that, under an extra condition on $A(t)$, the existence of a separable integral point of ‘small’ height on the elliptic curve $E_{A}/k(t)$ determines the isomorphism class of the elliptic curve $y^{2}=f(x)$.
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47

Pacheco, Amílcar. "Integral points on elliptic curves over function fields of positive characteristic." Bulletin of the Australian Mathematical Society 58, no. 3 (December 1998): 353–57. http://dx.doi.org/10.1017/s0004972700032329.

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Let K be a one variable function field of genus g defined over an algebraically closed field k of characteristic p > 0. Let E/K be a non-constant elliptic curve. Denote by MK the set of places of K and let S ⊂ MK be a non-empty finite subset.Mason in his paper “Diophantine equations over function fields” Chapter VI, Theorem 14 and Voloch in “Explicit p-descent for elliptic curves in characteristic p” Theorem 5.3 proved that the number of S-integral points of a Weiertrass equation of E/K defined over RS is finite. However, no explicit upper bound for this number was given. In this note, under the extra hypotheses that E/K is semi-stable and p > 3, we obtain an explicit upper bound for this number for a certain class of Weierstrass equations called S-minimal.
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48

Hakuta, Keisuke. "Distance functions on the sets of ordinary elliptic curves in short Weierstrass form over finite fields of characteristic three." Mathematica Slovaca 68, no. 4 (August 28, 2018): 749–66. http://dx.doi.org/10.1515/ms-2017-0142.

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Abstract We study distance functions on the set of ordinary (or non-supersingular) elliptic curves in short Weierstrass form (or simplified Weierstrass form) over a finite field of characteristic three. Mishra and Gupta (2008) firstly construct distance functions on the set of elliptic curves in short Weierstrass form over any prime field of characteristic greater than three. Afterward, Vetro (2011) constructs some other distance functions on the set of elliptic curves in short Weierstrass form over any prime field of characteristic greater than three. Recently, Hakuta (2015) has proposed distance functions on the set of ordinary elliptic curves in short Weierstrass form over any finite field of characteristic two. However, to our knowledge, no analogous result is known in the characteristic three case. In this paper, we shall prove that one can construct distance functions on the set of ordinary elliptic curves in short Weierstrass form over any finite field of characteristic three. A cryptographic application of our distance functions is also discussed.
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49

Nakazawa, Naoya. "Construction of elliptic curves with cyclic groups over prime fields." Bulletin of the Australian Mathematical Society 73, no. 2 (April 2006): 245–54. http://dx.doi.org/10.1017/s000497270003882x.

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The purpose of this article is to construct families of elliptic curves E over finite fields F so that the groups of F-rational points of E are cyclic, by using a representation of the modular invariant function by a generator of a modular function field associated with the modular group Γ0(N), where N = 5, 7 or 13.
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50

Silverman, Joseph. "Local-global aspects of (hyper)elliptic curves over (in)finite fields." Advances in Mathematics of Communications 4, no. 2 (May 2010): 101–14. http://dx.doi.org/10.3934/amc.2010.4.101.

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