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The thing I like best about this article is that it touches on elliptic curves over finite fields in more than a cursory way. But I would love to see an introdu
by andrewla 7y ago
The thing I like best about this article is that it touches on elliptic curves over finite fields in more than a cursory way. But I would love to see an introduction to elliptic curves that completely ignores the behavior in the real numbers, and instead focuses on driving the motivations from the finite field perspective.
There are elements that seem particularly interesting to me around this, especially since even elliptic curves with no rational points on them still have points in finite fields, and how to think about what it means for points to be on a line, and derive things like point addition and point doubling from that perspective.
I'm familiar enough with EC math to hand wave around these issues, but it's still something that I struggle with.
- empath75 7y ago> even elliptic curves with no rational points on them still have points in finite fields That’s interesting. How is that possible?
- andrewla 7y agoTake the elliptic curve y^2 = x^3 - 5 that has no rational solutions [1]. But it's easy to see (from brute force) that, for example, 3^2 - 5 = 22 = 27^2 (mod 101). [1] https://kconrad.math.uconn.edu/blurbs/gradnumthy/mordelleqn1.pdf https://kconrad.math.uconn.edu/blurbs/gradnumthy/mordelleqn1...
- frutiger 7y agoI can recommend https://www.amazon.com/Elliptic-Tales-Curves-Counting-Number/dp/0691151199 https://www.amazon.com/Elliptic-Tales-Curves-Counting-Number.... It goes in-depth into the motivating use cases for elliptic curves (as well as their group law) and explains some very elegant theorems (Bézout's theorem) as well as open problems in Maths (the Birch-Swinnerton-Dyer conjecture).