Investigate using `BigInt`'s with `< 4` limbs for `FieldElement`'s

[michaeljklein]

Problem

bn254 FieldElement's are defined in acir_field

pub type FieldElement = field_element::FieldElement<ark_bn254::Fr>;

Where field_element::FieldElement wraps its argument and ensures it's a PrimeField:

pub struct FieldElement<F: PrimeField>(F);

ark_bn254::Fr is defined as:

pub type Fr = Fp256<MontBackend<FrConfig, 4>>;

Where Fp256 defines the contents (MontBackend.. defines the FpConfig):

pub type Fp256<P> = Fp<P, 4>;

And Fp:

pub struct Fp<P: FpConfig<N>, const N: usize>(pub BigInt<N>, _);

Is an N-limbed BigInt:

pub struct BigInt<const N: usize>(pub [u64; N]);

So our bn254 FieldElement's are always represented by [u64; 4], even when their values are much smaller than u64::MAX

Happy Case

Investigate client-size memory savings and performance overhead of using a flexibly-sized big integer type instead of BigInt<N>, e.g. this BigInt, which uses a Vec of limbs

Workaround

None

Workaround Description

No response

Additional Context

No response

Project Impact

Nice-to-have

Blocker Context

No response

Would you like to submit a PR for this Issue?

None

Support Needs

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