Constructions of elliptic curves endomorphisms
Matematičeskie voprosy kriptografii, Tome 5 (2014) no. 2, pp. 99-102
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Let $\mathbb K$ be an imaginary quadratic field. Consider an elliptic curve $E(\mathbb F_p)$ defined over prime field $\mathbb F_p$ with given ring of endomorphisms $o_\mathbb K$, where $o_\mathbb K$ is an order in a ring of integers $\mathbb Z_\mathbb K$. An algorithm permitting to construct endomorphism of the curve $E(\mathbb F_p)$ corresponding to the complex number $\tau\in o_\mathbb K$ is presented. The endomorphism is represented as a pair of rational functions with coefficients in $\mathbb F_p$. To construct these functions we use continued fraction expansion for values of Weierstrass function. After that we reduce the rational functions modulo prime ideal in finite extension of $\mathbb K$. One can use such endomorphism for elliptic curve point exponentiation.
@article{MVK_2014_5_2_a10,
author = {A. Yu. Nesterenko},
title = {Constructions of elliptic curves endomorphisms},
journal = {Matemati\v{c}eskie voprosy kriptografii},
pages = {99--102},
year = {2014},
volume = {5},
number = {2},
language = {en},
url = {http://geodesic.mathdoc.fr/item/MVK_2014_5_2_a10/}
}
A. Yu. Nesterenko. Constructions of elliptic curves endomorphisms. Matematičeskie voprosy kriptografii, Tome 5 (2014) no. 2, pp. 99-102. http://geodesic.mathdoc.fr/item/MVK_2014_5_2_a10/
[1] Borevich Z. I., Shafarevich I. R., Number Theory, Academic Press, 1966, 436 pp. | MR | Zbl
[2] Cox D., Primes of the form $x^2+ny^2$: Fermat, Class Field Theory and Complex Multiplication, J. Wiles and Sons, 1989, 363 pp. | MR | Zbl
[3] Galant R., Lambert R., Vanstone S., “Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms”, CRYPTO 01, Proceedings of the 21st Annual International Conference on Advances Of Cryptology, 2001, 190–200 | MR