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Two kinds of division polynomials for twisted Edwards curves

Abstract

This paper presents two kinds of division polynomials for twisted Edwards curves. Their chief property is that they characterise the n-torsion points of a given twisted Edwards curve. We present recursions for the division polynomials, which differ in their flavour. We prove a uniqueness form for elements of the function field of an Edwards curve. We also present results concerning the coefficients of these polynomials, which may aid computation.

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References

  1. 1

    Abel, N.H.: Recherches sur les fonctions elliptiques. J. Reine Angew. Math. 2, 3, 101–181, 160–190, 1827, 1828

  2. 2

    Bernstein, D., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards curves. In: Progress in Cryptology—AFRICACRYPT 2008, vol. 5023 of Lecture Notes in Computer Science, pp. 389–405. Springer, Berlin (2008)

  3. 3

    Bernstein, D., Lange, T.: Faster addition and doubling on elliptic curves. In: Advances in Cryptology—ASIACRYPT 2007, vol. 4833 of Lecture Notes in Computer Science, pp. 29–50. Springer, Berlin (2007)

  4. 4

    Bernstein D., Lange T.: A complete set of addition laws for incomplete Edwards curves. J. Num. Th. 131(1), 858–872 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5

    Castryck, W., Galbraith, S., Rezaeian Farashahi, R.: Efficient arithmetic on elliptic curves using a mixed Edwards-Montgomery representation. Cryptology ePrint Archive, (2008). http://www.eprint.iacr.org/2008/218

  6. 6

    Cox D.: Galois Theory. Wiley-Interscience, New york (2004)

    Book  MATH  Google Scholar 

  7. 7

    Dewaghe L.: Remarks on the Schoof-Elkies-Atkin algorithm. Math. Comp. 67(223), 1247–1252 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8

    Edwards H.: A normal form for elliptic curves. Bull. Am. Math. Soc. (N.S.) 44(3), 393–422 (2007)

    Article  MATH  Google Scholar 

  9. 9

    Eisenstein G.: Über die Irreductibilität und einige andere Eigenschaften der Gleichung, von welcher die Theilung der ganzen Lemniscate abhängt. J. Reine Angew. Math. 39, 160–179 (1850)

    Article  MATH  Google Scholar 

  10. 10

    Eisenstein G.: Über einige allgemeine Eigenschaften der Gleichung, von welcher die Theilung der ganzen Lemniscate abhängt nebst Anwendungen derselben auf die Zahlentheorie. J. Reine Angew. Math. 39, 224–274 (1850)

    Article  MATH  Google Scholar 

  11. 11

    Gauss, C.: Werke, Band III. Göttinger Digitalierungszentrum, p. 405 (1863). http://www.resolver.sub.uni-goettingen.de/purl?PPN235999628

  12. 12

    Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, No. 52, Springer, Berlin (1977)

  13. 13

    Justus, B., Loebenberger, D.: Differential addition in generalized Edwards coordinates. In: Advances in Information and Computer Security, Lecture Notes in Computer Science, pp. 316–325. Springer, Berlin (2010)

  14. 14

    Lang S.: Elliptic Curves: Diophantine Analysis. Springer, Berlin (1970)

    Google Scholar 

  15. 15

    Lidl R., Niederreiter H.: Finite Fields. 2nd edn. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  16. 16

    Montgomery P.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comp. 48(177), 243–264 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17

    Okeya, K., Kurumatani, H., Sakurai, K.: Elliptic curves with the Montgomery-form and their cryptographic applications. In: Public Key Cryptography (Melbourne, 2000), vol. 1751 of Lecture Notes in Computer Science, pp. 238–257. Springer, Berlin (2000)

  18. 18

    Schoof R.: Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp. 44(170), 483–494 (1985)

    MATH  MathSciNet  Google Scholar 

  19. 19

    Silverman J.: The Arithmetic of Elliptic Curves. 2nd edn. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  20. 20

    Ward M.: Memoir on elliptic divisibility sequences. Am. J. Math. 70, 31–74 (1948)

    Article  MATH  Google Scholar 

  21. 21

    Washington L.: Elliptic Curves. 2nd edn. Chapman & Hall/CRC, London (2008)

    Book  MATH  Google Scholar 

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Correspondence to Gary McGuire.

Additional information

Research supported by Claude Shannon Institute, Science Foundation Ireland Grant 06/MI/006, and Grant 07/RFP/MATF846, and the Irish Research Council for Science, Engineering and Technology.

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Moloney, R., McGuire, G. Two kinds of division polynomials for twisted Edwards curves. AAECC 22, 321–345 (2011). https://doi.org/10.1007/s00200-011-0153-5

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Keywords

  • Elliptic curve
  • Division polynomial
  • Edwards