ISBN: 978-981-11-3671-9 DOI: 10.18178/wcse.2017.06.163

## Constructing Weak Curves ECDLP-Equivalent to Ordinary Curves via Elliptic Surfaces

*Yuhong Zhang, Bo Peng, Maozhi Xu*

*Abstract*— The elliptic curve discrete logarithm problem(ECDLP) is the basis for the security of elliptic
curve cryptography. Up to now, there is no subexponential algorithm for ECDLP. Only certain classes of
weak curves exist whose standard ECDLP can be reduced to sample problems, like supersingular curves,
anomalous curves etc. Elliptic surfaces are algebraic surfaces containing a pencil of elliptic curves (also
means fibers). In order to reduce ordinary curves to weak curves using connections among these fibers, we
define the ECDLP-equivalence of specialized-reduced points in the sense of the same section to elliptic
surfaces (hereafter shortened as ECDLP-equivalence). We make a discovery that the ECDLP-equivalence is
only related to the order of specialized-reduced points, and present an algorithm for constructing
supersingular curves ECDLP-equivalent to ordinary curves via elliptic surfaces. In the end, we illustrate a
256-bit example.

*Index Terms*— elliptic surfaces, ECDLP-Equivalence, supersingular curves, rank one lifting problem.

Yuhong Zhang, Bo Peng, Maozhi Xu

LMAM, School of Mathematical Sciences, Peking University, No.5 Yiheyuan Rd., Haidian District, CHINA

Cite: Yuhong Zhang, Bo Peng, Maozhi Xu, "Constructing Weak Curves ECDLP-Equivalent to Ordinary Curves via Elliptic Surfaces," *Proceedings of 2017 the 7th International Workshop on Computer Science and Engineering, pp. 940-944, Beijing, 25-27 June, 2017.*