Polynomial ring operations

[mattsuffern]

Module Description

This module provides the implementation of various operations in the polynomial ring:

$$ R = \mathbb{Z}_q[X] / (X^d + 1) $$

Based on the "modular arithmetic primitive using wrapping operations" module by @pycckuu (unchanged)

Implemented Functions:

• Polynomial Addition: add()
• Polynomial Multiplication: mul()
• Inner Product / Dot Product: inner_product()
• Polynomial Subtraction: sub()
• Polynomial Negation: neg()
• Scalar Multiplication: scalar_mul()
• Division by Monomials: div_by_monomial()
• Polynomial Evaluation: eval()
• Zero Check: is_zero()
• Polynomial Equality Check: is_equal()

(closes #5)

[pycckuu]

Great Work Matteo! 🚀 See my suggestions 😄

Additionally, I suggest:

1. Rename the struct to Rq to align with cryptographical terms and rename poly_rings.rs to rq.rs

2. Implement operator traits for natural arithmetic syntax. This will enable idiomatic usage like let sum = &rq1 + &rq2; and if rq1 == rq2 while maintaining ring arithmetic correctness.

3. I noticed you went beyond, but the original task only required:

   • Addition
   • Multiplication
   • Modular Reduction

Not sure if we need the additional operations. Maybe @NiDimi can confirm.

[maksimryndin]

Very good job Mateo! 👍 You're doing a great progress with Rust! 🔥

Igor has made a thorough review 👍 I've only added some minors. Will review again later.

[NiDimi]

Not sure if we need the additional operations. Maybe @NiDimi can confirm.

@pycckuu My reasoning for adding it was that we need it for all the summation operations on the paper. For example for Ajtai:

The prover has to compute the commitment vector t as:

$t = As \in R$

The vector is computed using matrix-vector multiplication:

$t_i = \sum_j{A_{i,j} \cdot s_j}$

That requires adding the results of the polynomial multiplication.

Of course this is a suggestion if you think it can be done better feel free to suggest how to do it.

[NiDimi]

1. Left some comments. 😄
2. You need to consider what happens when the polynomials do not have the same number of coefficient. Right now If I am not mistaken the code will just fail.
3. In rust commonly we use // for line comment and /// for doc comments. What that means is that we should use /// when explaining what the function does.

Overall a very good starting step 🔥

[mattsuffern]

Thank you very much, @pycckuu, @maksimryndin, @NiDimi, for all the help and comments! I think all the issues are now resolved, but if there's any other change I should make, please let me know

[pycckuu]

@mattsuffern Code looks great! See my very minor suggestions!

[maksimryndin]

Awesome job! 🥇
let's finish some minor things and move further! 👍

[maksimryndin]

Awesome, @mattsuffern ! 👍 great progress!

[maksimryndin]

Not sure if we need the additional operations. Maybe @NiDimi can confirm.

@pycckuu My reasoning for adding it was that we need it for all the summation operations on the paper. For example for Ajtai:

The prover has to compute the commitment vector t as:

t = A s ∈ R

The vector is computed using matrix-vector multiplication:

t i = ∑ j A i , j ⋅ s j

That requires adding the results of the polynomial multiplication.

Of course this is a suggestion if you think it can be done better feel free to suggest how to do it.

@NiDimi I believe this is already resolved. Could you please take a final look? :)