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On division polynomial PIT and supersingularity

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

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.

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Notes

  1. The dual of an isogeny \(\phi : E_1 \rightarrow E_2\) of degree m is a unique isogeny \(\hat{\phi }: E_2 \rightarrow E_1\) such that \(\phi \circ \hat{\phi } = [m]\), see [16, III.6].

  2. See [20].

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Acknowledgements

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.

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  1. Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada

    Javad Doliskani

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  1. Javad Doliskani
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Correspondence to Javad Doliskani.

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Doliskani, J. On division polynomial PIT and supersingularity. AAECC 29, 393–407 (2018). https://doi.org/10.1007/s00200-018-0349-z

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  • DOI: https://doi.org/10.1007/s00200-018-0349-z

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