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Journal articles on the topic 'Elliptic curve points group'

Author: Grafiati

Published: 30 May 2022

Last updated: 31 July 2025

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1

Shchur, Nataliia, Oleksandra Pokotylo, and Yelyzaveta Bailiuk. "ELLIPTIC CURVE CRYPTOGRAPHY AND ITS PRACTICAL APPLICATION." Cybersecurity: Education, Science, Technique 1, no. 21 (2023): 48–64. http://dx.doi.org/10.28925/2663-4023.2023.21.4864.

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Elliptic curves are one of the most promising tools for constructing modern cryptographic algorithms. The security of elliptic curve cryptography is based on the complexity of solving the discrete logarithm problem in the group of points of the elliptic curve over a finite field. Elliptic curve cryptography enables two parties communicating over public channel using elliptic curve encryption and signing algorithms. Elliptic curves allow to achieve the same level of security with small key sizes than other asymmetric cryptographic algorithms. The article describes the mathematical apparatus of

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2

Skuratovskii, Ruslan. "SUPERSINGULAR EDWARDS CURVES AND EDWARDS CURVE POINTS COUNTING METHOD OVER FINITE FIELD." Journal of Numerical and Applied Mathematics, no. 1 (133) (2020): 68–88. http://dx.doi.org/10.17721/2706-9699.2020.1.06.

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We consider problem of order counting of algebraic affine and projective curves of Edwards [2, 8] over the finite field $F_{p^n}$. The complexity of the discrete logarithm problem in the group of points of an elliptic curve depends on the order of this curve (ECDLP) [4, 20] depends on the order of this curve [10]. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve over a finite field. It should be noted that thi

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3

Wohlfahrt, K. "Macbeath's Curve and the Modular Group." Glasgow Mathematical Journal 28, no. 2 (1986): 241. http://dx.doi.org/10.1017/s0017089500006583.

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On p. 244 of Glasgow Math. J.27 (1985) on the right hand side of one of the 6 equations characterizing the 4 fixed points of the involution v a sign error has occurred.The relevant equation should ready0y3y5y6=–1,or the points would not lie on the curve.Correcting the error unfortunately invalidates the model of an elliptic curve given in §6, which therefore has to be re-evaluated. First we find, in the notation of the paper,2 f (x) = ((r + 1)/(R + 2))2.

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4

Fan, Jing, Xuejun Fan, Ningning Song, and Long Wang. "Hyperelliptic Covers of Different Degree for Elliptic Curves." Mathematical Problems in Engineering 2022 (July 4, 2022): 1–11. http://dx.doi.org/10.1155/2022/9833393.

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In elliptic curve cryptography (ECC) and hyperelliptic curve cryptography (HECC), the size of cipher-text space defined by the cardinality of Jacobian is a significant factor to measure the security level. Counting problems on Jacobians of elliptic curve can be solved in polynomial time by Schoof–Elkies–Atkin (SEA) algorithm. However, counting problems on Jacobians of hyperelliptic curves are solved less satisfactorily than those on elliptic curves. So, we consider the construction of the cover map from the hyperelliptic curves to the elliptic curves to convert point counting problems on hyper

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5

CONCEIÇÃO, RICARDO. "ON INTEGRAL POINTS ON ISOTRIVIAL ELLIPTIC CURVES OVER FUNCTION FIELDS." Bulletin of the Australian Mathematical Society 102, no. 2 (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 i

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6

Tama, Yanuar Bhakti Wira, and Muhammad Firdhausi Fahmi. "Sistem Kriptografi Klasik Dengan Memanfaatkan Orde Dari Grup Titik Pada Kurva Eliptik Bentuk Montgomery." Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi 11, no. 2 (2023): 361–71. http://dx.doi.org/10.37905/euler.v11i2.23009.

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Elliptic curve cryptography is one of the application fields of algebra and number theory concepts. One form of elliptic curve cryptography is Montgomery elliptic curve cryptography. In this paper, a method for a classical cryptographic system be formulated, consisting of encryption and decryption involving twenty-six alphabetical letters which are mapped to points on an elliptic curve by utilizing the order of the point group on the Montgomery elliptic curve. Several examples of implementation in simple cases are given to verify the results.

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7

Jayanti, Sravani, K. Chittibabu, and Chandra Sekhar Akkapeddi. "A Cryptosystem of Skewed Affine Cipher of Multiple Keys." ECS Transactions 107, no. 1 (2022): 15071–80. http://dx.doi.org/10.1149/10701.15071ecst.

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In this era, where communication over technology has become vital, the reliability of the same is of utmost need. Cryptography ensures confidentiality, user authentication, and integrity of data. One of the techniques is the Elliptic Curve Cryptography (ECC). Several classical ciphers are designed based on mathematical backgrounds. In this paper, we focus on combining Affine Cipher and ECC to magnify the security provided by an Affine cipher. Hence a skewed Affine cipher that uses multiple keys over Elliptic curves is proposed. The keys chosen are derived from the points on the specified Ellip

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8

Dalkılıç, Şeyda, and Ercan Altınışık. "Mestre's Finite Field Method for Searching Elliptic Curves with High Ranks." Journal of New Theory, no. 47 (June 30, 2024): 20–27. http://dx.doi.org/10.53570/jnt.1467401.

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The theory of elliptic curves is one of the popular topics of recent times with its unsolved problems and interesting conjectures. In 1922, Mordell proved that the group of $\mathbb{Q}$-rational points on an elliptic curve is finitely generated. However, the rank of this group, signifying the number of independent generators, can be arbitrarily high for certain curves, a fact yet to be definitively proven. This study leverages the computer algebra system Magma to investigate curves with potentially high ranks using a technique developed by Mestre.

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9

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

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10

Carita, Sa'aadah Sajjana, and Herman Kabetta. "MODIFICATION OF POLLARD RHO ALGORITHM USING NEGATION MAPPING." BAREKENG: Jurnal Ilmu Matematika dan Terapan 16, no. 4 (2022): 1159–66. http://dx.doi.org/10.30598/barekengvol16iss4pp1159-1166.

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El Gamal encryption was introduced in 1985 and is still commonly used today. Its hardness is based on a discrete logarithm problem defined over the finite abelian cyclic group group chosen in the original paper was but later it was proven that using the group of Elliptic Curve points could significantly reduce the key size required. The modified El Gamal encryption is dubbed its analog version. This analog encryption bases its hardness on Elliptic Curve Discrete Logarithm Problem (ECDLP). One of the fastest attacks in cracking ECDLP is the Pollard Rho algorithm, with the expected number of ite

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11

DELAUNAY, CHRISTOPHE, and CHRISTIAN WUTHRICH. "SELF-POINTS ON ELLIPTIC CURVES OF PRIME CONDUCTOR." International Journal of Number Theory 05, no. 05 (2009): 911–32. http://dx.doi.org/10.1142/s1793042109002456.

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Let E be an elliptic curve of conductor p. Given a cyclic subgroup C of order p in E[p], we construct a modular point PC on E, called self-point, as the image of (E,C) on X0(p) under the modular parametrization X0(p) → E. We prove that the point is of infinite order in the Mordell–Weil group of E over the field of definition of C. One can deduce a lower bound on the growth of the rank of the Mordell–Weil group in its PGL 2(ℤp)-tower inside ℚ(E[p∞]).

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12

Akbary, Amir, and V. Kumar Murty. "Descending Rational Points on Elliptic Curves to Smaller Fields." Canadian Journal of Mathematics 53, no. 3 (2001): 449–69. http://dx.doi.org/10.4153/cjm-2001-019-5.

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AbstractIn this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galo

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13

Jafari, Mohammad Mahdi. "On Selmer Ranks of Elliptic Curves With a Rational 2-Torsion." Kazakh Mathematical Journal 25, no. 3 (2025): 13–22. https://doi.org/10.70474/sqw8ys05.

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This study investigates the asymptotic behavior of the ranks of Selmer groups associated with elliptic curves possessing a rational 2-torsion point defined over the integers. The Selmer group plays a central role in understanding the Mordell–Weil group and the Birch and Swinnerton-Dyer conjecture. The arithmetic of elliptic curves with torsion points has long attracted significant interest, with foundational results tracing back to the work of Mordell, Selmer, and later refinements by Cassels and others. In particular, the behavior of 2-Selmer groups provides insights into the distribution of

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14

Jetchev, Dimitar. "Global divisibility of Heegner points and Tamagawa numbers." Compositio Mathematica 144, no. 4 (2008): 811–26. http://dx.doi.org/10.1112/s0010437x08003497.

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AbstractWe improve Kolyvagin’s upper bound on the order of the p-primary part of the Shafarevich–Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely that predicted by the Birch and Swinnerton-Dyer conjectural formula.

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15

Skuratovskii, Ruslan, and Volodymyr Osadchyy. "Criterions of Supersinguliarity and Groups of Montgomery and Edwards Curves in Cryptography." WSEAS TRANSACTIONS ON MATHEMATICS 19 (March 1, 2021): 709–22. http://dx.doi.org/10.37394/23206.2020.19.77.

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We consider the algebraic affine and projective curves of Edwards over the finite field Fpn. It is well known that many modern cryptosystems can be naturally transformed into elliptic curves. The criterions of the supersingularity of Montgomery and Edwards curves are found. In this paper, we extend our previous research into those Edwards algebraic curves over a finite field and we construct birational isomorphism of them with cubic in Weierstrass normal form. One class of twisted Edwards is researched too. We propose a novel effective method of point counting for both Edwards and elliptic cur

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16

Burhanuddin, Iftikhar A., and Ming-Deh A. Huang. "On the Equation y2=x3-pqx." Journal of Numbers 2014 (July 16, 2014): 1–5. http://dx.doi.org/10.1155/2014/825634.

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We consider certain quartic twists of an elliptic curve. We establish the rank of these curves under the Birch and Swinnerton-Dyer conjecture and obtain bounds on the size of Shafarevich-Tate group of these curves. We also establish a reduction between the problem of factoring integers of a certain form and the problem of computing rational points on these twists.

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17

Ulas, Maciej. "Rational Points in Arithmetic Progressions on y2 = xn + k." Canadian Mathematical Bulletin 55, no. 1 (2012): 193–207. http://dx.doi.org/10.4153/cmb-2011-058-1.

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AbstractLet C be a hyperelliptic curve given by the equation y2 = f(x) for f ∈ ℤ[x] without multiple roots. We say that points Pi = (xi, yi) ∈ C(ℚ) for i = 1, 2, … , m are in arithmetic progression if the numbers xi for i = 1, 2, … , m are in arithmetic progression.In this paper we show that there exists a polynomial k ∈ ℤ[t] with the property that on the elliptic curve ε′ : y2 = x3+k(t) (defined over the field ℚ(t)) we can find four points in arithmetic progression that are independent in the group of all ℚ(t)-rational points on the curve Ε′. In particular this result generalizes earlier resu

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18

Gepner, David, and Lennart Meier. "On equivariant topological modular forms." Compositio Mathematica 159, no. 12 (2023): 2638–93. http://dx.doi.org/10.1112/s0010437x23007509.

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Following ideas of Lurie, we give a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Specializing to the universal elliptic curve we obtain, in particular, equivariant spectra of topological modular forms. We compute the fixed points of these spectra for the circle group and more generally for tori.

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19

Mina, R. J. S., and J. B. Bacani. "Elliptic Curves of Type y2=x3−3pqx Having Ranks Zero and One." Malaysian Journal of Mathematical Sciences 17, no. 1 (2023): 67–76. http://dx.doi.org/10.47836/mjms.17.1.06.

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The group of rational points on an elliptic curve over Q is always a finitely generated Abelian group, hence isomorphic to Zr×G with G a finite Abelian group. Here, r is the rank of the elliptic curve. In this paper, we determine sufficient conditions that need to be set on the prime numbers p and q so that the elliptic curve E:y2=x3−3pqx over Q would possess a rank zero or one. Specifically, we verify that if distinct primes p and q satisfy the congruence p≡q≡5(mod24), then E has rank zero. Furthermore, if p≡5(mod12) is considered instead of a modulus of 24, then E has rank zero or one. Lastl

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20

Pomykała, Jacek, and Sławomir Barabasz. "Eliptic Curve Based Threshold Proxy Signature Scheme with Known Signers." Fundamenta Informaticae 69, no. 4 (2006): 411–25. https://doi.org/10.3233/fun-2006-69403.

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In the article we present a new (t,n) threshold proxy signature scheme with known signers. It is based on the elliptic curve cryptosystem whose security refers to the discrete logarithm problem (DLP) in the group E(E _p ) of rational points of elliptic curve over the finite field. In comparision to similar schemes based on the RSA or DSS systems our solution requires application of significantly shorter cryptographic keys. The scheme is relatively simple in construction, has the property of unforgeability, non-repudation and admits the proactive security.

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21

Ghioca, Dragos. "Elliptic Curves over the Perfect Closure of a Function Field." Canadian Mathematical Bulletin 53, no. 1 (2010): 87–94. http://dx.doi.org/10.4153/cmb-2010-019-9.

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22

Eid, Wesam, Turki F. Al-Somani, and Marius C. Silaghi. "Efficient Elliptic Curve Operators for Jacobian Coordinates." Electronics 11, no. 19 (2022): 3123. http://dx.doi.org/10.3390/electronics11193123.

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The speed up of group operations on elliptic curves is proposed using a new type of projective coordinate representation. These operations are the most common computations in key exchange and encryption for both current and postquantum technology. The boost this improvement brings to computational efficiency impacts not only encryption efforts but also attacks. For maintaining security, the community needs to take note of this development as it may need to operate changes in the key size of various algorithms. Our proposed projective representation can be viewed as a warp on the Jacobian proje

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23

Weng, Annegret. "On group orders of rational points of elliptic curves." Quaestiones Mathematicae 25, no. 4 (2002): 513–25. http://dx.doi.org/10.2989/16073600209486035.

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24

Buchholz, R. H., and S. M. Kelly. "On rational-derived quartics." Bulletin of the Australian Mathematical Society 51, no. 1 (1995): 121–32. http://dx.doi.org/10.1017/s0004972700013940.

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We present a characterisation of all quartic polynomials with exactly three distinct roots and the property that it and all its derivatives have rational roots. It turns out that there are an infinite number of distinct such quartics, each of which corresponds to a point on a related elliptic curve. Furthermore the collection of these points forms a proper subgroup of the group of rational points on the curve.

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25

Backhausz, Tibor, and Gergely Zábrádi. "Algebraic functional equations and completely faithful Selmer groups." International Journal of Number Theory 11, no. 04 (2015): 1233–57. http://dx.doi.org/10.1142/s1793042115500670.

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Let E be an elliptic curve — defined over a number field K — without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p∞-Selmer group of E over any strongly admissible p-adic Lie extension K∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K∞/K). Under some mild additional hypotheses, this gives an algebraic functional equation of the conjectured p-adic L-function. As an application, we construct completely faithful Selmer groups in case the p-adic L

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26

Chandee, Vorrapan, Chantal David, Dimitris Koukoulopoulos, and Ethan Smith. "The Frequency of Elliptic Curve Groups over Prime Finite Fields." Canadian Journal of Mathematics 68, no. 4 (2016): 721–61. http://dx.doi.org/10.4153/cjm-2015-013-1.

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AbstractLetting p vary over all primes and E vary over all elliptic curves over the finite field 𝔽p, we study the frequency to which a given group G arises as a group of points E(𝔽p). It is well known that the only permissible groups are of the form Gm,k:=ℤ/mℤ×ℤ/mkℤ. Given such a candidate group, we let M(Gm,k) be the frequency to which the group Gm,karises in this way. Previously, C.David and E. Smith determined an asymptotic formula for M(Gm,k) assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for M(Gm,k), pointwise and

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27

Challarapu, Neelima, and Suneetha Chivukula. "Elliptic Curve Cryptography Applied for (k,n) Threshold Secret Sharing Scheme." ECS Transactions 107, no. 1 (2022): 1021–28. http://dx.doi.org/10.1149/10701.1021ecst.

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Invention of Secret Sharing Scheme by Adi Shamir, along with the prevalent advancements offers strong protection of the secret key in communication network. Shamir’s scheme, which is established using Lagrange Interpolation polynomial. The group manager or dealer of the group splits the secret S to be communicated into n pieces allots all the n pieces to n participants. A subgroup of t or more participants of the group come together to reconstruct the secret key. Later, the cryptanalysis of secret sharing scheme came into picture in the direction of cheater detection whose motivation is to foo

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28

Lavryk, I., and O. Pryima. "FOR PSEUDORANDOM SEQUENCES BASED ON ELLIPTIC CURVE ISOGENIES GENERATING METHOD." Випробування та сертифікація, no. 1(3) (July 8, 2024): 119–24. http://dx.doi.org/10.37701/ts.03.2024.15.

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The pseudorandom sequences generation is a cryptographic systems fundamental aspect that affects cryptographic strength. One of these sequences advanced generating methods involves the use of elliptic curves (ECs), in particular by exploiting the isogeny properties of ECs. This approach not only improves the security features of cryptographic algorithms, but also ensures efficiency and reliability in the generation process. The use of isogenic transformations - morphisms between elliptic curves that preserve their group structure - further enriches the technique by introducing complex algebrai

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29

Grant, David, and Su-Ion Ih. "Integral division points on curves." Compositio Mathematica 149, no. 12 (2013): 2011–35. http://dx.doi.org/10.1112/s0010437x13007318.

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AbstractLet $k$ be a number field with algebraic closure $ \overline{k} $, and let $S$ be a finite set of primes of $k$ containing all the infinite ones. Let $E/ k$ be an elliptic curve, ${\mit{\Gamma} }_{0} $ be a finitely generated subgroup of $E( \overline{k} )$, and $\mit{\Gamma} \subseteq E( \overline{k} )$ the division group attached to ${\mit{\Gamma} }_{0} $. Fix an effective divisor $D$ of $E$ with support containing either: (i) at least two points whose difference is not torsion; or (ii) at least one point not in $\mit{\Gamma} $. We prove that the set of ‘integral division points on $

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30

SWINNERTON–DYER, PETER. "The effect of twisting on the 2-Selmer group." Mathematical Proceedings of the Cambridge Philosophical Society 145, no. 3 (2008): 513–26. http://dx.doi.org/10.1017/s0305004108001588.

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AbstractLet Γ be an elliptic curve defined over Q, all of whose 2-division points are rational, and let Γb be its quadratic twist by b. Subject to a mild additional condition on Γ, we find the limit of the probability distribution of the dimension of the 2-Selmer group of Γb as the number of prime factors of b increases; and we show that this distribution depends only on whether the 2-Selmer group of Γ has odd or even dimension.

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31

Wingberg, Kay. "Galois groups of number fields generated by torsion points of elliptic curves." Nagoya Mathematical Journal 104 (December 1986): 43–53. http://dx.doi.org/10.1017/s0027763000022662.

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Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Zp-extensions of number fields. In a special case they consider a non-cyclotomic Zp-extension F∞ defined via torsion points of E and a certain Iwasawa module attached to E/F, which can be interpreted as an abelian Galois group of an extension of F∞. We are interested in the corresponding non-abe

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32

Saikia, A. "Selmer Groups of Elliptic Curves with Complex Multiplication." Canadian Journal of Mathematics 56, no. 1 (2004): 194–208. http://dx.doi.org/10.4153/cjm-2004-009-7.

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AbstractSuppose K is an imaginary quadratic field and E is an elliptic curve over a number field F with complex multiplication by the ring of integers in K. Let p be a rational prime that splits as in K. Let Epn denote the pn-division points on E. Assume that F(Epn) is abelian over K for all n ≥ 0. This paper proves that the Pontrjagin dual of the -Selmer group of E over F(Ep∞) is a finitely generated free Λ-module, where Λ is the Iwasawa algebra of . It also gives a simple formula for the rank of the Pontrjagin dual as a Λ-module.

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33

ZYWINA, DAVID. "A REFINEMENT OF KOBLITZ'S CONJECTURE." International Journal of Number Theory 07, no. 03 (2011): 739–69. http://dx.doi.org/10.1142/s1793042111004411.

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Let E be an elliptic curve over the rationals. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of 𝔽p-points of E is prime. However, the constant occurring in his asymptotic does not take into account that the distributions of the |E(𝔽p)| need not be independent modulo distinct primes. We shall describe a corrected constant. We also take the opportunity to extend the scope of the original conjecture to ask how often |E(𝔽p)|/t is an integer and prime for a fixed positive integer t, and to consider elliptic curves over arbitrary number

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34

Pappalardi, Francesco. "On the exponent of the group of points of an elliptic curve over a finite field." Proceedings of the American Mathematical Society 139, no. 7 (2010): 2337–41. http://dx.doi.org/10.1090/s0002-9939-2010-10658-5.

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35

Heo, Donghoe, Suhri Kim, Kisoon Yoon, Young-Ho Park, and Seokhie Hong. "Optimized CSIDH Implementation Using a 2-Torsion Point." Cryptography 4, no. 3 (2020): 20. http://dx.doi.org/10.3390/cryptography4030020.

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The implementation of isogeny-based cryptography mainly use Montgomery curves, as they offer fast elliptic curve arithmetic and isogeny computation. However, although Montgomery curves have efficient 3- and 4-isogeny formula, it becomes inefficient when recovering the coefficient of the image curve for large degree isogenies. Because the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) requires odd-degree isogenies up to at least 587, this inefficiency is the main bottleneck of using a Montgomery curve for CSIDH. In this paper, we present a new optimization method for faster CSIDH prot

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Matar, Ahmed. "Fine Selmer groups, Heegner points and anticyclotomic ℤp-extensions". International Journal of Number Theory 14, № 05 (2018): 1279–304. http://dx.doi.org/10.1142/s179304211850080x.

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Let [Formula: see text] be an elliptic curve, [Formula: see text] a prime and [Formula: see text] the anticyclotomic [Formula: see text]-extension of a quadratic imaginary field [Formula: see text] satisfying the Heegner hypothesis. In this paper, we make a conjecture about the fine Selmer group over [Formula: see text]. We also make a conjecture about the structure of the module of Heegner points in [Formula: see text] where [Formula: see text] is the union of the completions of the fields [Formula: see text] at a prime of [Formula: see text] above [Formula: see text]. We prove that these con

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37

Collas, Benjamin, Michael Dettweiler, Stefan Reiter, and Will Sawin. "Monodromy of elliptic curve convolution, seven-point sheaves of G ₂ type, and motives of Beauville type." Journal für die reine und angewandte Mathematik (Crelles Journal) 2022, no. 784 (2022): 1–26. http://dx.doi.org/10.1515/crelle-2021-0070.

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Abstract We study Tannakian properties of the convolution product of perverse sheaves on elliptic curves. We establish that for certain sheaves with unipotent local monodromy over seven points the corresponding Tannaka group is isomorphic to G 2 {G_{2}} . This monodromy approach generalizes a result of Katz on the existence of G 2 {G_{2}} -motives in the middle cohomology of deformations of Beauville surfaces.

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Freitas, Nuno, Bartosz Naskręcki, and Michael Stoll. "The generalized Fermat equation with exponents 2, 3,." Compositio Mathematica 156, no. 1 (2019): 77–113. http://dx.doi.org/10.1112/s0010437x19007693.

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We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minimal list of twis

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39

Semaev, I. A. "Evaluation of discrete logarithms in a group of $p$-torsion points of an elliptic curve in characteristic $p$." Mathematics of Computation of the American Mathematical Society 67, no. 221 (1998): 353–56. http://dx.doi.org/10.1090/s0025-5718-98-00887-4.

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Weng, Jiang, Yunqi Dou, and Chuangui Ma. "Research on Attacking a Special Elliptic Curve Discrete Logarithm Problem." Mathematical Problems in Engineering 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/5361695.

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Cheon first proposed a novel algorithm for solving discrete logarithm problem with auxiliary inputs. Given some pointsP,αP,α2P,…,αdP∈G, an attacker can solve the secret key efficiently. In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs. We show that if some pointsP,αP,αkP,αk2P,αk3P,…,αkφ(d)-1P∈Gand a multiplicative cyclic groupK=〈k〉are given, wheredis a prime,φ(d)is the order ofK. The secret keyα∈Fp⁎can be solved inO((p-1)/d+d)group operations by usingO((p-1)/d)storage.

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41

Nakazawa, Naoya. "Construction of elliptic curves with cyclic groups over prime fields." Bulletin of the Australian Mathematical Society 73, no. 2 (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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42

Im, Bo-Hae, and Hansol Kim. "The automorphism group of the p-torsion points of an elliptic curve over a field of characteristic p ≥ 5." Finite Fields and Their Applications 106 (September 2025): 102631. https://doi.org/10.1016/j.ffa.2025.102631.

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Soleng, R. "Homomorphisms From the Group of Rational Points On Elliptic Curves to Class Groups of Quadratic Number Fields." Journal of Number Theory 46, no. 2 (1994): 214–29. http://dx.doi.org/10.1006/jnth.1994.1013.

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Frey, G. "On the Selmer Group of Twists of Elliptic Curves with Q-Rational Torsion Points." Canadian Journal of Mathematics 40, no. 3 (1988): 649–65. http://dx.doi.org/10.4153/cjm-1988-028-9.

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(1) The symbols p and q stand for prime numbers and throughout the paper we assume that p is fixed and contained in {3, 5, 7}. Let L be an algebraic number field (i.e., L is a finite extension of Q). Then prime divisors of L dividing p (resp. q) are denoted by (resp. ). The completion of L with respect to is denoted by . Let S be a finite set of prime numbers, and let M/L be a Galois extension with abelian Galois group of exponent p.Definition. M/L is said to be little ramified outside S if for primes q ∉ S and all one haswith k ∊ N and . Here ζp is a pth root of unity, u1, …, uk are elements

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Zelenova, Maria. "Counting the order of the group of points of an elliptic curve over a finite field based on Shanks’s algorithm." P-Adic Numbers, Ultrametric Analysis, and Applications 3, no. 2 (2011): 157–64. http://dx.doi.org/10.1134/s2070046611020075.

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Momot, Aleksander. "On Modular Ball-Quotient Surfaces of Kodaira Dimension One." ISRN Geometry 2011 (June 19, 2011): 1–5. http://dx.doi.org/10.5402/2011/214853.

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Let be a lattice which is not co-ompact, of finite covolume with respect to the Bergman metric and acting freely on the open unit ball . Then the toroidal compactification is a projective smooth surface with elliptic compactification divisor . In this short note we discover a new class of unramifed ball quotients . We consider ball quotients with kod and . We prove that each minimal surface with finite Mordell-Weil group in the class described admits an étale covering which is a pull-back of . Here denotes the elliptic modular surface parametrizing elliptic curves with 6-torsion points which g

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Roganov, Yu V., A. Stovas, and V. Yu Roganov. "Location of singular points in orthorhombic media." Geofizicheskiy Zhurnal 44, no. 3 (2022): 3–20. http://dx.doi.org/10.24028/gj.v44i3.261965.

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The dependence of the location of singular points of orthorhombic (ORT) media on the stiffness coefficients , and phase velocity at the singularity point is studied under the assumption that are larger than and . In this case, singular points appear only at the intersection of slowness surfaces of S1- and S2-waves. To simplify the presentation of the results, the values , are fixed and changed within the limits at which the stiffness matrix remains positive definite. We define the parameters, , , , which result in 0, 1, or 2 singular points in the symmetry planes of the ORT medium. The types o

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Cojocaru, Alina Carmen. "On the Cyclicity of the Group of Fp-Rational Points of Non-CM Elliptic Curves." Journal of Number Theory 96, no. 2 (2002): 335–50. http://dx.doi.org/10.1006/jnth.2002.2789.

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Verdure, Hugues. "A simple criterion for the m-cyclicity of the group of rational points on an elliptic curve defined over a finite field." Archiv der Mathematik 86, no. 2 (2006): 121–28. http://dx.doi.org/10.1007/s00013-005-1442-7.

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MATAR, AHMED. "Selmer Groups and Anticyclotomic Zp-extensions." Mathematical Proceedings of the Cambridge Philosophical Society 161, no. 3 (2016): 409–33. http://dx.doi.org/10.1017/s0305004116000347.

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AbstractLet E/Q be an elliptic curve, p a prime and K∞/K the anticyclotomic Zp-extension of a quadratic imaginary field K satisfying the Heegner hypothesis. In this paper we give a new proof to a theorem of Bertolini which determines the value of the Λ-corank of Selp∞(E/K∞) in the case where E has ordinary reduction at p. In the case where E has supersingular reduction at p we make a conjecture about the structure of the module of Heegner points mod p. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines the value of the Λ-corank of Selp∞(E/K∞) in the case wh

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