Allow multiplication of extension field elements directly by the BasePrimeField element

[mmagician]

Description

Note, there was already a method to multiply by the extension field's BaseField: but for a field tower, a BaseField might not necessarily be the BasePrimeField, but rather an extension field itself.

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[mmagician]

@suyash67

[Pratyush]

Suggested change

	fn mul_by_base_field_elem(&self, elem: &Self::BasePrimeField) -> Self;
	fn mul_by_base_prime_field(&self, elem: &Self::BasePrimeField) -> Self;

Also, should we move this into a Mul<Self::BasePrimeField, Output = Self> bound? (And maybe add for<'a> Mul<&'a Self::BasePrimeField, Output = Self>, MulAssign<Self::BasePrimeField>, and for<'a> MulAssign<&'a Self::BasePrimeField> bounds too?

[mmagician]

I tried this initially, but we already have Mul<Self> coming from a macro, and adding a trait bound as you suggest, and then implementing:

impl<P: QuadExtConfig> Mul<<Self as Field>::BasePrimeField> for QuadExtField<P> {
    type Output = Self;
    ...
}

causes a conflicting implementation error.
I believe this is what you meant, right?

[Pratyush]

Ouch that's annoying...

I guess Rust doesn't know that for QuadExtField, BasePrimeField is always different from Self...

But it should be able to figure that out: BasePrimeField: PrimeField, while Self doesn't implement that.

[Pratyush]

Where is the existing multiplication by base field?

[mmagician]

@Pratyush We have a method directly on QuadExtField:

algebra/ff/src/fields/models/quadratic_extension.rs

Line 170 in 9cb911f

pub fn mul_assign_by_basefield(&mut self, element: &P::BaseField) {

[mmagician]

Actually, we also have the counterpart for CubicExtField, although the names weren't consistent, notice the underscore in base_field:

algebra/ff/src/fields/models/cubic_extension.rs

Line 122 in 2369347

pub fn mul_assign_by_base_field(&mut self, value: &P::BaseField) {

I would propose to unify these two as part of this PR, at least when it comes to the name. We could pull that method out into a separate trait, but I'm not sure it's worth it (it is not used anywhere in the arkworks org).