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Journal articles on the topic 'Supersingular elliptic curves'

Author: Grafiati

Published: 14 June 2021

Last updated: 29 July 2025

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1

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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2

Sutherland, Andrew V. "Identifying supersingular elliptic curves." LMS Journal of Computation and Mathematics 15 (September 1, 2012): 317–25. http://dx.doi.org/10.1112/s1461157012001106.

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AbstractGiven an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits structural differences between ordinary and supersingular isogeny graphs. This yields a simple algorithm that, given E and a suitable non-residue in 𝔽p2, determines the supersingularity of E in O(n3log 2n) time and O(n) space, where n=O(log p) . Both these complexity bounds are significant improvements over existing methods, as we demons

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3

Martinez-Diaz, Ismel, Rashad Ali, and Muhammad Kamran Jamil. "On the Search for Supersingular Elliptic Curves and Their Applications." Mathematics 13, no. 2 (2025): 188. https://doi.org/10.3390/math13020188.

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Elliptic curves with the special quality known as supersingularity have gained much popularity in the rapidly developing field of cryptography. The conventional method of employing random search is quite ineffective in finding these curves. This paper analyzes the search of supersingular elliptic curves in the space of curves over Fp2. We show that naive random search is unsuitable to easily find any supersingular elliptic curves when the space size is greater than 1013. We improve the random search using a necessary condition for supersingularity. As our main result, we define for the first t

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4

Skuratovskii, Ruslan, and Volodymyr Osadchyy. "Elliptic and Edwards Curves Order Counting Method." International Journal of Mathematical Models and Methods in Applied Sciences 15 (April 5, 2021): 52–62. http://dx.doi.org/10.46300/9101.2021.15.8.

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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. In this paper, we extend our previous research into those Edwards algebraic curves over a finite field. We propose a novel effective method of point counting for both Edwards and elliptic curves. In addition to finding a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, we also find a general formula by which one can determine whether or not a

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5

Onuki, Hiroshi. "On oriented supersingular elliptic curves." Finite Fields and Their Applications 69 (January 2021): 101777. http://dx.doi.org/10.1016/j.ffa.2020.101777.

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6

Xiao, Guanju, Lixia Luo, and Yingpu Deng. "Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves." Journal of Mathematical Cryptology 15, no. 1 (2021): 454–64. http://dx.doi.org/10.1515/jmc-2020-0029.

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Abstract Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve E defined over 𝔽 p 2 , if an imaginary quadratic order O can be embedded in End(E) and a prime L splits into two principal ideals in O, we construct loops or cycles in the supersingular L-isogeny graph at the vertices which are next to j(E) in the supersingular ℓ-isogeny graph where ℓ is a prime different from L. Next, we discuss the lengths of these cycles especially for j(E) = 1728 and 0. Finally, we also determine an

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7

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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8

Colò, Leonardo, and David Kohel. "Orienting supersingular isogeny graphs." Journal of Mathematical Cryptology 14, no. 1 (2020): 414–37. http://dx.doi.org/10.1515/jmc-2019-0034.

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AbstractWe introduce a category of 𝓞-oriented supersingular elliptic curves and derive properties of the associated oriented and nonoriented ℓ-isogeny supersingular isogeny graphs. As an application we introduce an oriented supersingular isogeny Diffie-Hellman protocol (OSIDH), analogous to the supersingular isogeny Diffie-Hellman (SIDH) protocol and generalizing the commutative supersingular isogeny Diffie-Hellman (CSIDH) protocol.

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9

Luc, Nhu-Quynh, Quang-Trung Do, and Manh-Hung Le. "Implementation of Boneh - Lynn - Shacham short digital signature scheme using Weil bilinear pairing based on supersingular elliptic curves." Ministry of Science and Technology, Vietnam 64, no. 12 (2022): 3–9. http://dx.doi.org/10.31276/vjste.64(4).03-09.

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One option for a digital signature solution for devices with low memory and low bandwidth transmission over channels uses a short digital signature scheme based on Weil bilinear pairing aimed at short processing times, fast computation, and convenient deployment on applications. The computational technique of non-degenerate bilinear pairings uses supersingular elliptic curves over a finite field Fpl (where p is a sufficiently large prime number) and has the advantage of being able to avoid Weil-descent, Menezes-Okamoto-Vanstone (MOV) attacks, and attacks by the Number Field Sieve algorithm. Co

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10

Kane, Ben. "CM liftings of supersingular elliptic curves." Journal de Théorie des Nombres de Bordeaux 21, no. 3 (2009): 635–63. http://dx.doi.org/10.5802/jtnb.692.

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11

Bashmakov, M. I., and A. S. Kurochkin. "Selmer groups of supersingular elliptic curves." Journal of Soviet Mathematics 37, no. 2 (1987): 924–28. http://dx.doi.org/10.1007/bf01089082.

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12

Yang, Tonghai. "Minimal CM Liftings of Supersingular Elliptic Curves." Pure and Applied Mathematics Quarterly 4, no. 4 (2008): 1317–26. http://dx.doi.org/10.4310/pamq.2008.v4.n4.a14.

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13

Monks, Keenan. "On supersingular elliptic curves and hypergeometric functions." Involve, a Journal of Mathematics 5, no. 1 (2012): 99–113. http://dx.doi.org/10.2140/involve.2012.5.99.

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14

Pitman, Sarah. "3F2-hypergeometric functions and supersingular elliptic curves." Involve, a Journal of Mathematics 8, no. 3 (2015): 481–90. http://dx.doi.org/10.2140/involve.2015.8.481.

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15

Hatley, Jeffrey. "Rank parity for congruent supersingular elliptic curves." Proceedings of the American Mathematical Society 145, no. 9 (2017): 3775–86. http://dx.doi.org/10.1090/proc/13545.

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16

Murty, M. Ram. "On the supersingular reduction of elliptic curves." Proceedings of the Indian Academy of Sciences - Section A 97, no. 1-3 (1987): 247–50. http://dx.doi.org/10.1007/bf02837827.

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17

Takahashi, Yasushi, Momonari Kudo, Ryoya Fukasaku, Yasuhiko Ikematsu, Masaya Yasuda, and Kazuhiro Yokoyama. "Algebraic approaches for solving isogeny problems of prime power degrees." Journal of Mathematical Cryptology 15, no. 1 (2020): 31–44. http://dx.doi.org/10.1515/jmc-2020-0072.

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AbstractRecently, supersingular isogeny cryptosystems have received attention as a candidate of post-quantum cryptography (PQC). Their security relies on the hardness of solving isogeny problems over supersingular elliptic curves. The meet-in-the-middle approach seems the most practical to solve isogeny problems with classical computers. In this paper, we propose two algebraic approaches for isogeny problems of prime power degrees. Our strategy is to reduce isogeny problems to a system of algebraic equations, and to solve it by Gröbner basis computation. The first one uses modular polynomials,

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18

Granger, R., D. Page, and M. Stam. "On Small Characteristic Algebraic Tori in Pairing-Based Cryptography." LMS Journal of Computation and Mathematics 9 (2006): 64–85. http://dx.doi.org/10.1112/s1461157000001194.

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The value ot the late pairing on an elliptic curve over a finite field may be viewed as an element of an algebraic torus. Using this simple observation, we transfer techniques recently developed for torus-based cryptography to pairing-based cryptography, resulting in more efficient computations, and lower bandwidth requirements. To illustrate the efficacy of this approach, we apply the method to pairings on supersingular elliptic curves in characteristic three.

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19

LEI, ANTONIO. "NON-COMMUTATIVE p-ADIC L-FUNCTIONS FOR SUPERSINGULAR PRIMES." International Journal of Number Theory 08, no. 08 (2012): 1813–30. http://dx.doi.org/10.1142/s1793042112501047.

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Let E/ℚ be an elliptic curve with good supersingular reduction at p with ap(E) = 0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the ℤp-cyclotomic extension of a finite Galois extension of ℚ where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.

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20

Li, Songsong, Yi Ouyang, and Zheng Xu. "Endomorphism rings of supersingular elliptic curves over Fp." Finite Fields and Their Applications 62 (February 2020): 101619. http://dx.doi.org/10.1016/j.ffa.2019.101619.

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21

Kobayashi, Shin-ichi. "Iwasawa theory for elliptic curves at supersingular primes." Inventiones Mathematicae 152, no. 1 (2003): 1–36. http://dx.doi.org/10.1007/s00222-002-0265-4.

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22

Du (B. D.) Kim, Byoung. "Signed-Selmer Groups over the ℤ2p-extension of an Imaginary Quadratic Field". Canadian Journal of Mathematics 66, № 4 (2014): 826–43. http://dx.doi.org/10.4153/cjm-2013-043-2.

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AbstractLet E be an elliptic curve over ℚ that has good supersingular reduction at p > 3. We construct what we call the ±/±-Selmer groups of E over the ℤ2p-extension of an imaginary quadratic field K when the prime p splits completely over K/ℚ, and prove that they enjoy a property analogous to Mazur's control theorem.Furthermore, we propose a conjectural connection between the±/±-Selmer groups and Loeffler's two-variable ±/±-p-adic L-functions of elliptic curves.

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23

El-Guindy, Ahmad. "Legendre Drinfeld modules and universal supersingular polynomials." International Journal of Number Theory 10, no. 05 (2014): 1277–89. http://dx.doi.org/10.1142/s1793042114500262.

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We introduce a certain family of Drinfeld modules that we propose as analogues of the Legendre normal form elliptic curves. We exhibit explicit formulas for a certain period of such Drinfeld modules as well as formulas for the supersingular locus in that family, establishing a connection between these two kinds of formulas. Lastly, we also provide a closed formula for the supersingular polynomial in the j-invariant for generic Drinfeld modules.

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24

KAWAHARA, Yuto, Tetsutaro KOBAYASHI, Gen TAKAHASHI, and Tsuyoshi TAKAGI. "Faster MapToPoint on Supersingular Elliptic Curves in Characteristic 3." IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E94-A, no. 1 (2011): 150–55. http://dx.doi.org/10.1587/transfun.e94.a.150.

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25

Tran, Ying-Ying. "Generalization of Atkin’s orthogonal polynomials and supersingular elliptic curves." Proceedings of the American Mathematical Society 141, no. 4 (2012): 1135–41. http://dx.doi.org/10.1090/s0002-9939-2012-11433-9.

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26

Brown, M. L. "Note On Supersingular Primes of Elliptic Curves Over Q." Bulletin of the London Mathematical Society 20, no. 4 (1988): 293–96. http://dx.doi.org/10.1112/blms/20.4.293.

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27

Saikia, Anupam. "On the Iwasawa $\mu $-invariants of supersingular elliptic curves." Acta Arithmetica 194, no. 2 (2020): 179–86. http://dx.doi.org/10.4064/aa190213-28-8.

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28

Chevyrev, Ilya, and Steven D. Galbraith. "Constructing supersingular elliptic curves with a given endomorphism ring." LMS Journal of Computation and Mathematics 17, A (2014): 71–91. http://dx.doi.org/10.1112/s1461157014000254.

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AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{O}$ be a maximal order in the quaternion algebra $B_p$ over $\mathbb{Q}$ ramified at $p$ and $\infty $. The paper is about the computational problem: construct a supersingular elliptic curve $E$ over $\mathbb{F}_p$ such that ${\rm End}(E) \cong \mathcal{O}$. We present an algorithm that solves this problem by taking gcds of the reductions modulo $p$ of Hilbert cl

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29

SAITO, T. "Candidate One-Way Functions on Non-Supersingular Elliptic Curves." IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E89-A, no. 1 (2006): 144–50. http://dx.doi.org/10.1093/ietfec/e89-a.1.144.

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30

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 (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 dist

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31

Pollack, Robert, and Karl Rubin. "The main conjecture for CM elliptic curves at supersingular primes." Annals of Mathematics 159, no. 1 (2004): 447–64. http://dx.doi.org/10.4007/annals.2004.159.447.

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32

YOSHIDA, Reo, and Katsuyuki TAKASHIMA. "Computing a Sequence of 2-Isogenies on Supersingular Elliptic Curves." IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E96.A, no. 1 (2013): 158–65. http://dx.doi.org/10.1587/transfun.e96.a.158.

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33

Kim, Byoung Du (B D. ). "The parity conjecture for elliptic curves at supersingular reduction primes." Compositio Mathematica 143, no. 01 (2007): 47–72. http://dx.doi.org/10.1112/s0010437x06002569.

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34

Xiao, Guanju, Zijian Zhou, and Longjiang Qu. "Oriented supersingular elliptic curves and Eichler orders of prime level." Finite Fields and Their Applications 100 (December 2024): 102501. http://dx.doi.org/10.1016/j.ffa.2024.102501.

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35

Emerton, Matthew. "Supersingular elliptic curves, theta series and weight two modular forms." Journal of the American Mathematical Society 15, no. 3 (2002): 671–714. http://dx.doi.org/10.1090/s0894-0347-02-00390-9.

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36

Chen, BingLong, and Chang-An Zhao. "Self-pairings on supersingular elliptic curves with embedding degree three." Finite Fields and Their Applications 28 (July 2014): 79–93. http://dx.doi.org/10.1016/j.ffa.2014.01.013.

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37

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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38

Lozano-Robledo, Álvaro. "Formal groups of elliptic curves with potential good supersingular reduction." Pacific Journal of Mathematics 261, no. 1 (2013): 145–64. http://dx.doi.org/10.2140/pjm.2013.261.145.

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39

Mazzoli, M. "Non-commutative digit expansions for arithmetic on supersingular elliptic curves." Acta Mathematica Hungarica 149, no. 1 (2016): 149–59. http://dx.doi.org/10.1007/s10474-016-0596-z.

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40

Finotti, Luís R. A. "An elementary proof for the number of supersingular elliptic curves." São Paulo Journal of Mathematical Sciences 14, no. 2 (2020): 531–38. http://dx.doi.org/10.1007/s40863-020-00170-8.

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41

DOBI, DORIS, NICHOLAS WAGE, and IRENA WANG. "SUPERSINGULAR RANK TWO DRINFEL'D MODULES AND ANALOGS OF ATKIN'S ORTHOGONAL POLYNOMIALS." International Journal of Number Theory 05, no. 05 (2009): 885–95. http://dx.doi.org/10.1142/s1793042109002444.

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The theory of elliptic curves and modular forms provides a precise relationship between the supersingular j-invariants and the congruences between modular forms. Kaneko and Zagier discuss a surprising generalization of these results in their paper on Atkin orthogonal polynomials. In this paper, we define an analog of the Atkin orthogonal polynomials for rank two Drinfel'd modules.

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42

Koshelev, Dmitrii. "The most efficient indifferentiable hashing to elliptic curves of j-invariant 1728." Journal of Mathematical Cryptology 16, no. 1 (2022): 298–309. http://dx.doi.org/10.1515/jmc-2021-0051.

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Abstract This article makes an important contribution to solving the long-standing problem of whether all elliptic curves can be equipped with a hash function (indifferentiable from a random oracle) whose running time amounts to one exponentiation in the basic finite field F q {{\mathbb{F}}}_{q} . More precisely, we construct a new indifferentiable hash function to any ordinary elliptic F q {{\mathbb{F}}}_{q} -curve E a {E}_{a} of j-invariant 1728 with the cost of extracting one quartic root in F q {{\mathbb{F}}}_{q} . As is known, the latter operation is equivalent to one exponentiation in fi

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43

Sakai, Yuichi. "The Atkin orthogonal polynomials for the Fricke groups of levels 5 and 7." International Journal of Number Theory 10, no. 08 (2014): 2243–55. http://dx.doi.org/10.1142/s1793042114500766.

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We define Atkin's inner product for the Fricke groups of levels 5 and 7, and show that under certain condition the Atkin orthogonal polynomials in these cases exist. We also show the connection between extremal quasimodular forms of depth 1 and the Atkin orthogonal polynomials for these groups. Finally we propose a conjecture that relates our Atkin orthogonal polynomials to supersingular elliptic curves.

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44

Tomita, Takumi, and Tsuyoshi Takagi. "Efficient system parameters for Identity-Based Encryption using supersingular elliptic curves." JSIAM Letters 6 (2014): 13–16. http://dx.doi.org/10.14495/jsiaml.6.13.

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45

Samaa Fuad Ibraheem. "Constructing Supersingular Elliptic Curves Depending on the Coefficients of Weierstrass Equation." Journal of the College of Basic Education 16, no. 65 (2022): 159–74. http://dx.doi.org/10.35950/cbej.vi.7647.

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في استخدام المنحنيات الإهليلجية في تطبيقاتها (خاصة في علم التشفير) ، غالبًا ما يحتاج المرء إلى بناء منحنيات بيضاوية بنوع معروف (عدد نقاط) على حقل محدد Fp ، حيث تكون بعض الأنواع أكثر أمانًا من الآخرين. لذلك نقدم طريقة بسيطة لبناء منحنى إهليلجي فائق الشكل عن طريق حساب معامل xp-1 في f (x) (p-1) / 2 حيث و (س) = x3 + فأس + ب. ونعطي خوارزمية تحسب هذا المعامل و بناء المنحنى المطلوب.

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46

Ouyang, Yi, and Zheng Xu. "Loops of isogeny graphs of supersingular elliptic curves at j = 0." Finite Fields and Their Applications 58 (July 2019): 174–76. http://dx.doi.org/10.1016/j.ffa.2019.04.002.

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47

Lei, Antonio. "Iwasawa theory for modular forms at supersingular primes." Compositio Mathematica 147, no. 3 (2011): 803–38. http://dx.doi.org/10.1112/s0010437x10005130.

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AbstractWe generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.

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48

Lei, Antonio, David Loeffler, and Zerbes Sarah Livia. "On the Asymptotic Growth of Bloch-Kato-Shafarevich-Tate Groups of Modular Forms Over Cyclotomic Extensions." Canadian Journal of Mathematics 69, no. 4 (2017): 826–50. http://dx.doi.org/10.4153/cjm-2016-034-x.

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AbstractWe study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic ℤp-extension of ℚ under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara, and Sprung for supersingular elliptic curves.

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49

NOGAMI, Yasuyuki, Hiroto KAGOTANI, Kengo IOKIBE, Hiroyuki MIYATAKE, and Takashi NARITA. "FPGA Implementation of Various Elliptic Curve Pairings over Odd Characteristic Field with Non Supersingular Curves." IEICE Transactions on Information and Systems E99.D, no. 4 (2016): 805–15. http://dx.doi.org/10.1587/transinf.2015icp0018.

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50

Hamidi, Parham, та Jishnu Ray. "Conjecture A and μ-invariant for Selmer groups of supersingular elliptic curves". Journal de Théorie des Nombres de Bordeaux 33, № 3.1 (2022): 853–86. http://dx.doi.org/10.5802/jtnb.1181.

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