# 5P;1R — Bitcoin’s Elliptic Curve Cryptography

Published in

2 min readApr 23, 2022

This is the first of a potential series of articles I want to call “*5 points & 1 resource*” (think tl;dr but 5p;1r) where I summarize a list of bullet points that would have helped me start learning a new topic. It is intentionally far from a complete source of data.

- This is an
**Elliptic Curve**:`y^2 = x^3 + ax + b`

; see the continuous function Image 1. **Elliptic Curve Cryptography**is defined over a**Finite Field**(very large prime)**p**along with a**Generator Point**((x,y) coordinate)**G like so**:`y^2 ≡ (x^3 + ax + b) mod p`

; see the scatter plot Image 2.- Bitcoin uses a
**secp256k1**, which is an Elliptic Curve with carefully selected parameters to achieve certain security guarantees:

•`a=0`

•`b = 7`

•`p = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1`

•`G = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8`

- A public key
**P**(a point on the curve) is just a private key**pk**(a random integer) multiplied by the**Generator Point (**the pre-selected point on the curve):`P = pk * G.`

- The
**Elliptic Curve Discrete Logarim Problem**makes it hard to find the integer**pk**when you know (x,y) coordinates of**P**and**G**fitting the equation in (2) using the parameters in (3) or some other elliptic curve.

If I only had to recommend one resource out of all the references I looked through, it would be: https://cryptobook.nakov.com/asymmetric-key-ciphers/elliptic-curve-cryptography-ecc#the-generator-point-in-ecc

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