On division polynomial PIT and supersingularity

  • Javad DoliskaniEmail author
Original Paper


For an elliptic curve E over a finite field \(\mathbb {F}_q\), where q is a prime power, we propose new algorithms for testing the supersingularity of E. Our algorithms are based on the polynomial identity testing problem for the p-th division polynomial of E. In particular, an efficient algorithm using points of high order on E is given.


Division polynomials Polynomial identity testing Elliptic curves 

Mathematics Subject Classification

Primary 11Y16 14H52 Secondary 12Y05 



The author would like to thank Felipe Voloch for his valuable feedback on Sect. 4, and Luca De Feo for helpful comments. This work was partially supported by NSERC, CryptoWorks21, and Public Works and Government Services Canada.


  1. 1.
    Bröker, R.: Constructing supersingular elliptic curves. J. Comb. Number Theory 1(3), 269–273 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chang, M.-C., Kerr, B., Shparlinski, I.E., Zannier, U.: Elements of large order on varieties over prime finite fields. J. Théor. Nombres Bordx. 26(3), 579–593 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Charles, D.X., Lauter, K.E., Goren, E.Z.: Cryptographic hash functions from expander graphs. J. Cryptol. 22(1), 93–113 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Finch, S.R.: Mathematical constants, vol. 93. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  5. 5.
    Hooley, C.: On artins conjecture. J. Reine Angew. Math. 225(209–220), 248 (1967)MathSciNetGoogle Scholar
  6. 6.
    Husemöller, D.: Elliptic Curves, Volume 111 of Graduate Texts in Mathematics. Springer, Berlin (1987)Google Scholar
  7. 7.
    Jao, D., De Feo, L.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. In: International Workshop on Post-Quantum Cryptography. Springer, Berlin, pp. 19–34 (2011)CrossRefGoogle Scholar
  8. 8.
    Jao, D., Soukharev, V.: Isogeny-based quantum-resistant undeniable signatures. In: International Workshop on Post-Quantum Cryptography. Springer, Berlin, pp. 160–179 (2014)zbMATHGoogle Scholar
  9. 9.
    Kohel, D.: Endomorphism Rings of Elliptic Curves Over Finite Fields. PhD thesis, University of California at Berkeley (1996)Google Scholar
  10. 10.
    Matthews, K.R.: A generalisation of artin’s conjecture for primitive roots. Acta Arith. 29, 113–146 (1976)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Saxena, N.: Progress on polynomial identity testing. Bull. EATCS 99, 49–79 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Schoof, R.: Elliptic curves over finite fields and the computation of square roots mod \(p\). Math. Comput. 44(170), 483–494 (1985)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM: JACM 27(4), 701–717 (1980)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Shoup, V.: Fast construction of irreducible polynomials over finite fields. J. Symb. Comput. 17(5), 371–391 (1994)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Shoup, V., et al.: NTL: A library for doing number theory (2016).
  16. 16.
    Silverman, J.H.: The Arithmetic of Elliptic Curves, vol. 106. Springer, Berlin (2009)zbMATHGoogle Scholar
  17. 17.
    Stein, W., et al.: Sage: open source mathematical software. 7 December 2009 (2016).
  18. 18.
    Sutherland, A.V.: Identifying supersingular elliptic curves. LMS J. Comput. Math. 15, 317–325 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    The PARI Group, Bordeaux. PARI/GP, version 2.8.0 (2016)Google Scholar
  20. 20.
    Voloch, J.F.: On the order of points on curves over finite fields. Integers Electron. J. Comb. Number Theory 7(A49), 1 (2007)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Voloch, J.F.: Elements of high order on finite fields from elliptic curves. Bull. Aust. Math. Soc. 81(03), 425–429 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, New York (1999)zbMATHGoogle Scholar
  23. 23.
    Washington, L.C.: Elliptic Curves: Number Theory and Cryptography. CRC Press, Boca Raton (2008)CrossRefGoogle Scholar
  24. 24.
    Zippel, R.: Probabilistic Algorithms for Sparse Polynomials. Springer, Berlin (1979)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada

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