Parameterize Poly and Yashe with a Config trait (vs const generic)

[naure]

Tasks

• [x] Replace MAX_POLY_DEGREE
• [x] Replace Coeff
• [x] Add a trait for the Yashe config

Background

One approach is to replace const generic with a trait like this:

trait PolyConfig {
    const MAX_DEGREE: usize;

    fn poly_modulus() -> Poly<Self>;

    // … maybe also Coeff type.
}

And FULL_RES_POLY_DEGREE is replaced with:

struct FullResConfig;

impl PolyConfig for FullResConfig {
    const MAX_DEGREE: usize = 2048;
    …
}

Poly<FullResConfig>

See https://github.com/Inversed-Tech/eyelid/pull/47#discussion_r1570509963

I think it would be better to use MAX_POLY_DEGREE = 2047 and create aliases for maximal number of coefficients, and the degree of the modulus.

_Originally posted by @emmorais in https://github.com/Inversed-Tech/eyelid/pull/47#discussion_r1572468457_

Polynomials with degree MAX_POLY_DEGREE - 1 have MAX_POLY_DEGREE coefficients, because the coefficients have degrees 0, 1, ... , MAX_POLY_DEGREE - 1.

Having an even number of coefficients is a requirement for efficient Karatsuba implementations. Otherwise we have to pad the polynomial with the odd number of coefficients to the next power of two when we split them.

So currently we:

• declare MAX_POLY_DEGREE = 2048
• have a modulus with degree MAX_POLY_DEGREE = 2048 and 2049 coefficients
• have polynomials with degree MAX_POLY_DEGREE - 1 = 2047 and 2048 coefficients (which is a performance requirement)

This is slightly confusing, and could benefit from some renames and calculated constants. For example: https://github.com/Inversed-Tech/eyelid/pull/47/files/a6821dcd3aa5a11af24e57ce60b872babd3eeadc#r1571498268

Under the alternative scheme, we would:

• declare MAX_POLY_DEGREE = 2047
• have a modulus with degree MAX_POLY_DEGREE + 1 = 2048 and 2049 coefficients
• have polynomials with degree MAX_POLY_DEGREE = 2047 and 2048 coefficients

The underlying sizes wouldn't change, but this way of coding them seems to be more confusing, and could cause more off-by-one errors.

See the comment thread https://github.com/Inversed-Tech/eyelid/pull/47#discussion_r1570513537