# Topics in elliptic curves over finite fields

First of all, sorry if I didn't put this question in the correct category.

This a paper aimed for undergraduate math majors. So I am writing a general paper explaining about elliptic curves over finite fields for my senior undergraduate project. After summarizing the main topics in elliptic curves over finite fields I want to focus on the cryptographic applications since I have an interest in cryptography. However this topic isn't easy for me to understand and so I want to make sure I am including all the important things relating to my topic. So here are some important things I think I should include:

• Weierstrass Equation
• Hasse's Theorem
• How to add points on Elliptic Curves
• Frobenius Endomorphism
• Schoof's Algorithm & SEA
• Weil Conjectures
• Riemann Hypothesis
• Zeta Function
• Sato-Tate Conjecture
• Elliptic Curve Cryptography (ECC) (as well as a short explanation of public key cryptography)
• Elliptic Curve Discrete Logarithm Problem (ECDLP)
• Elliptic Curve ElGamal example
• Supersingular Curves

Some things I am not sure about:

• Mentioning specifically curves of characteristic 2

Is there anything from my list that I shouldn't include in my paper or is there any important topic I am missing? Also if you know of any good reference material that would be greatly appreciated!

I have two of Silverman's books, the arithmetic of Elliptic Curves and Rational Points over Elliptic Curves as well as some books on elliptic curve cryptography and some other scholarly articles on various topics.

• I have no idea how the Riemann hypothesis bears upon cryptography. Will your paper explain that? ${}\qquad{}$ – Michael Hardy May 1 '15 at 17:49
• @MichaelHardy Well I don't know...but isn't it important in elliptic curves? this is not a paper about cryptography, but a paper about elliptic curves over finite fields, like a general summary. I just decided to include some cryptographic elements as well – Math Major May 1 '15 at 17:53
• These are really questions you should be asking your advisor. – Potato May 1 '15 at 18:13
• Depending on how much detail you are expected to give that looks like anything between 50 and 500 pages. Are you sure the target audience is average math majors? :-) – Jyrki Lahtonen May 4 '15 at 13:16
• Is it possible that you upload this paper and post a link? I am highly interested in these topics. – Marm Oct 26 '15 at 17:11

## 1 Answer

Depending on how cryptography heavy/math heavy the resulting paper is supposed to be you might want to include theory relating to attacks on curves. This would be weil-tate pairing and possibly the background needed for the weil-descent attack. Who is supposed to be the target audience of the paper? Is this a math paper for academic cryptographers? Or more a math paper for mathematicians? Given the topics you chose I'm assuming this is not a paper particularly aimed at a wider audience.

If it was to be more crypto heavy you could look at some of the newer results on implementation speeds and security of things like Edwards form.

In the end the topics you chose seem to probably be sufficient. If you are going for more a mathematical approach I would point out the way you get addition on elliptic curves by lifting addition form complex numbers.

Anyway just my thoughts.

• The target audience is supposed to be any undergraduate math major. Thank you so much for your input! – Math Major May 1 '15 at 17:35