# Notes on Ferveo & SCRAPE papers



Doing now:


Try calculating this "Lagrange polynomial" in `prepare_combine_simple`

Extra notes on "lambda"

Simple variant of Ferveo uses SCRAPE/Pi protocol calculations, straight up:


Notice how the "fast" decryption variant is using different kind of lambda

There is no exponentiation! And also, the thing is indexed by omega. What does it mean?
Looking at Lagrange interpolation from Ferveo docs: https://nikkolasg.github.io/ferveo/lagrange.html


They are using some "subproduct domain" technique to calculate "lambda" (?). Notice the use of omega, which is the domain.

We've already established that Lagrange coefficients are "partiotioned" by domain, which has to do with how multiple entities have multiple shares.

So apparently this is what `prepare_combine` is doing

Is this how the Li is calculated?

    let s = SubproductDomain::<E::Fr>::new(domain);
    let mut lagrange = s.inverse_lagrange_coefficients(); // L
    ark_ff::batch_inversion_and_mul(&mut lagrange, &n_0);
    
    // ... 
    
    // L_i * [b] Z_{i,omega_i}
    blinded_key_share.mul(*lambda)

Earlier in the `setup` function:

blinded_key_shares.multiply_by_omega_inv(domain_inv);

So from this:

We have to reverse engineer how to reverse engineer L_{i}(0)

Ideas for next steps:
- Read how the original, "improper", L_i is calculated
- Actually read the implementation of the whole things, line-by-line and underneath, the function impls. etc
- Figure out how to rewrite it
- Or maybe replace it with different evaluations method from arkworks
- Also take note of the setup function and any additional pre-processing you may find there

Use this `LagrangeInterpolate` construction?
https://github.com/arkworks-rs/r1cs-std/blob/6d64f379a27011b3629cf4c9cb38b7b7b695d5a0/src/poly/evaluations/univariate/lagrange_interpolator.rs#L111-L143

https://github.com/arkworks-rs/algebra/blob/402e7f9603fca7a68b86baf296b6feaf904939f5/poly/src/domain/general.rs#L100

https://github.com/arkworks-rs/r1cs-std/blob/6d64f379a27011b3629cf4c9cb38b7b7b695d5a0/src/poly/evaluations/univariate/mod.rs#L90

https://github.com/arkworks-rs/algebra/blob/402e7f9603fca7a68b86baf296b6feaf904939f5/poly/src/domain/mixed_radix.rs#L441

https://github.com/arkworks-rs/algebra/blob/3ef9864dd3afd44ff5be432c8bfdafa201c58749/poly/src/domain/mod.rs#L149

https://github.com/arkworks-rs/algebra/blob/d3e888d9a7489a89bdc278618899c4604da26dbd/poly/README.md#evaluations