feat: algebra

[lonerapier]

It changes the following:

• FiniteGroup trait
• MultiplicativePrimeGroup struct
• CurveGroup trait
• refactor ECDSA to generic group

[lonerapier]

@Autoparallel Thanks for the review.

• refactored as algebra module with separate submodules for group, field, ring (in future).
• Tried adding Group, Field trait separately, but we need function overriding in derived traits because operation in Group and FiniteGroup can be done in different ways.
  • Is there any other way that we can define Group trait's operation and FiniteGroup trait's operation separately?

[Autoparallel]

@lonerapier let me take a look.

One thing we could also consider is making the ring/field + op be a AbelianGroup. Food for thought.

Give me a sec to go look through this.

[lonerapier]

One thing we could also consider is making the ring/field + op be a AbelianGroup. Food for thought.

Regarding this suggestion, I can't find a way to add this without complicating the traits and structs more

Like it wouldn't be easy for a beginner to implement 5 traits just to have a PrimeField implementation

[Autoparallel]

Like it wouldn't be easy for a beginner to implement 5 traits just to have a PrimeField implementation

That's true... it would introduce having some wrapper types and what not. The only benefit it would give is that if you did something like:

let element: impl FiniteField = ...;
let (element as impl AbelianGroup) + ... 

but that's probably not worth it.

Another option is something like

trait FiniteField {
    type Monoid: ...
    type AbelianGroup: ...
    ...
}

but then I think we are hitting abstraction for the sake of abstraction.

You're right, let's not make the implementation needs abusive to beginners.

[lonerapier]

That's true... it would introduce having some wrapper types and what not.

Maybe we can implement this as an example demonstration with Ring implementation. Like showing a ring with inverse and commutativity forms a field, and then comparing it with the actual Field implementation.