Impl arithmetic ops for Fields

[justinfrevert]

Tracking an issue I faced during the ethglobal nyc hackathon. During the hackathon, I wanted to do things like implement counters using field elements. Doing so looked something like the following:

let counters_list = [BulletproofsField::from(0); 100];
for (i, f) in counters_list.into_iter().enumerate() {
    if i == alices_index {
        counters_list[i] = f + BulletproofsField::from(BigInt::ONE);
    }
}

Currently, this is not possible because Add is not implemented for Field. I recommend Add be implemented, as well as all the other arithmetic methods. This would be a convenient change for devs, because the current state is that they would need to fork the repo to implement themselves.

[samtay]

This seems like an oversight on our part, thank you for pointing it out. We have arithmetic implementations for field elements F behind ProgramNodes so that you can do arithmetic within a #[zkp_program] function, and I guess we mostly considered the BulletproofsField type on its own to be used for providing static-ish arguments. But I can definitely see how it would be convenient to do some arithmetic on them outside of the ZKP program.

In the meantime, you could just do your arithmetic on a different integral type, and then cast to the BulletproofsField. Your example above would change to something like:

// Create the list out of u32s
let mut counters_list = [0u32; 100];
for (i, f) in counters_list.into_iter().enumerate() {
    if i == alices_index {
        counters_list[i] = f + 1;
    }
}
// Then cast them all to BulletproofsField
let counters_list = counters_list.map(BulletproofsField::from);

For this particular example, there's also a nice stdlib function you could use:

let counters_list: [BulletproofsField; 100] = core::array::from_fn(|i| {
    if i == alices_index {
        1.into()
    } else {
        0.into()
    }
});