Abstract
Elliptic curves and hyperelliptic curves over finite fields are of great interest in public key cryptography. Using much smaller field for same security makes the genus 2 curves more competitive than elliptic curves. However, point counting algorithms for the Jacobians of genus 2 curves are not as efficient as what we have for elliptic curves. We give a method to generate genus 2 curves for which the point counting problems can be easily solved with efficient algorithms for elliptic curves. As an application, an example of a hyperelliptic curve whose order is a 256-bit prime is given. The method relies on the construction of a cover map from a hyperelliptic curve to an elliptic curve. Another important application of the construction is to generate the cover for the cover-decomposition attack on the discrete logarithm problems in elliptic curves.
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Acknowledgement
We thank the anonymous reviewers for their helpful comments. This work was supported by the National Natural Science Foundation of China (No. 61802401, No. 61772515 and No. 61872442).
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Fan, X., Tian, S., Li, B., Li, W. (2019). Constructing Hyperelliptic Covers for Elliptic Curves over Quadratic Extension Fields. In: Jang-Jaccard, J., Guo, F. (eds) Information Security and Privacy. ACISP 2019. Lecture Notes in Computer Science(), vol 11547. Springer, Cham. https://doi.org/10.1007/978-3-030-21548-4_35
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