Người báo cáo: Tuan-Thuong Dang
Thời gian: 14h, Thứ 5, ngày 7 tháng 1, năm 2016
Địa điểm: Phòng 303, Nhà A5, Viện Toán học, 18 Hoàng Quốc Việt Cầu Giấy Hà Nội
Tóm tắt: The aim of the seminar focuses on isogenies between elliptic curves over finite fields, whose characteristic is $
e 2,3$. Let $E_1, E_2$ be two elliptic curves defined over $mathbb{F}_q$, the theorem of J. Tate implies that, there exists an isogeny defined over $mathbb{F}_q$ between them iff $#E_1(mathbb{F}_q)=#E_2(mathbb{F}_q)$. The explicit construction of these isogenies plays an important role in Cryptography because it allows us to transform the discrete logarithm problem (DLP) from a curve to another one. Furthermore, it is one of the current trends in post-quantum cryptography, with the design of hash functions, and public-key exchange protocols, based on the isogenies graph. This graph has vertices as (classes of isomorphic) elliptic curves over $mathbb{F}_q$, and edges as isogenies defined over $mathbb{F}_q$ between them. And we are concerned with some questions:
begin{enumerate}
item Given two elliptic curves $E_1,E_2$ such that they have the same number of points over $mathbb{F}_q$. How can we construct an isogeny between them?
item Given an elliptic curve $E/mathbb{F}_q$, how many curves are there in its isogeny class? That means, how many elliptic curves defined over $mathbb{F}_q$, that have the same number of points with $E$?
item From the theorem of Hasse-Weil, $(sqrt{q}-1)^2le#E(mathbb{F}_q)le(sqrt{q}+1)^2$. Now, given an integer $N$, that lies between $[(sqrt{q}-1)^2,(sqrt{q}+1)^2]$. Under which assumptions can we construct an elliptic curve $E(mathbb{F}_q)$ such that $#E(mathbb{F}_q)=N$? And is there any method to do this?
end{enumerate}
In the first two weeks, I will give two talks. The first talk will focus on some aspects of the DLP over finite fields, quadratic curves, together with some of their arithmetic aspects, including the classification of quadratic forms over finite fields. The second talk will be devoted for the theory of elliptic curves. It will be mainly focused on the cryptographic and arithmetic aspects of Weil's pairing over finite fields. If time permits, we will discuss something about the Weil's conjecture for these curves, and the $l$-adic method (R. Schoof's algorithm) in counting the number of points on a given elliptic curve over finite fields.
The book of L. C. Washington ``Elliptic Curves: Number Theory and Cryptography" will be used for the seminars, and we will focus on Chapter XII about isogenies.
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