Introduce secp256k1 module with field and group classes to test framework #26222

[rajarshimaitra]

Session Details

• Date: 08-06-2023
• Time: IST 20:00 (UTC 14:30)
• PR: https://github.com/bitcoin/bitcoin/pull/26222
• Context: [Crypto]
• Difficulty: Beginer
• Language: [python]

Learning

• Elliptic Curve Cryptography.
• Fields and Groups.
• Public Keys, Private Keys, Signature, Verification.
• Python Implementation of basic crypto.
• Testing cryptographic primitives.

• Functional test framework.

  Note: Drop your PR and review related questions below

Summary

This PR redefines the cryptographic primitives used in the Bitcoin core's functional testing framework. And adds a new module in the python framework as secp256k1.py. This module includes all the cryptographic primitives. So the way to review this PR is to understand some fun basic crypto maths.

To start with, Cover the first 3 chapters of the Programming Bitcoin Book. If you haven't done programming Bitcoin already, skim through the concepts and skip the exercises.

A free PDF is available here

Cover the below concepts from the book

• Modulo arithmatic and finite fields
• Groups and group operations
• Elliptic Curve Operations

• Public Key and Private Key

  Theory

Bitcoin uses secp256k1 elliptic curve for all its cryptographic operations like key/address generation, digital signatures etc. See curve specifications for the full curve details.

Mathematically:

• Elliptic curves are groups.
• A Public key is denoted by a point (x, y coordinate) in the curve, and they are theoretically group elements.
• A point is constructed by two coordinates, x and y, and these are field elements.
• A Private Key is another field element from which a curve point on the elliptic curve is generated.
• A Generator Point is a predefined point (group element) on the elliptic curve, which is used to "generate" the public key, from the private key.

Questions

In the questions below, GE = group element and FE = field element

1. Did you review the PR? Concept ACK, approach ACK, tested ACK, or NACK? What was your review approach?
2. Write a test using class FE for basic field element operations - addition, multiplication.
3. Verify Fermat's little theorem using class FE.
4. class GE represents infinity explicitly. what is point at infinity in an elliptic curve? Why do we need this point? Where are regions in the code where we need to handle infinity scenarios properly?
5. What's a generator point in an elliptic curve?
6. Are there any downsides to rewriting the elliptic curve logic using fields/groups?