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

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Coinmonks

2 min readApr 23, 2022

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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

Image 1: Elliptic curve over real numbers

Image 2: Elliptic curve over a finite field

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

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Written by Daniel Olshansky