Elliptic Curve Cryptography in Rust

Explore finite groups with concrete examples: addition modulo four forms an abelian group, while multiplication modulo four fails due to missing inverses for some elements.

Explore cyclic groups, their order, and generators using modulo 11 to show how a generator yields all elements and how a non-generator differs, foundational for elliptic curve cryptography.

Visualize elliptic curve operations such as point addition and doubling on a well-known curve with an Ubuntu program. Observe line reflections, inverses, and abelian group properties like commutativity and associativity.

Implement the top layer of elliptic curve cryptography in Rust, coding point addition and multiplication, with coordinates or identity and on-curve checks via y^2 = x^3 + 8x mod p.

Demonstrate adding opposite points on the elliptic curve yields the identity when x1 = x2 and y1 + y2 = 0, with a test using 16 and 5.

Identify and address the overlooked case where a point has an x coordinate and the other coordinate is zero, involving point doubling, the tangent line, and the identity.

We reorganize the elliptic curve cryptography library into a standalone crate named generic, exporting elliptic curve, finite field, and point structures for flexible use in digital signatures, version 0.1.9.

Generate a private key as a random positive number within a bound, then derive the public key by multiplying the elliptic curve generator by the private key in Rust.

Explore ecdsa secp256k1 implementation in Rust, creating private and public keys, signing and verifying messages with sha-256, using a crate elliptic curve library, and testing tamper scenarios.