feat: jl projection

[Junochiu]

Summary

This pr implemented the JL projection method that projects the given polynomial ring matrix to a polynomial ring matrix with smaller dimension.

Example

Given a data matrix D_{2, 256} by setting the target dimension of 128, the expected result would be a data matrix of D_{2, 128}

[FoodChain1028]

Probably using reference instead of cloning is better here. I am not very sure but we can discuss here

Suggested change

	let mut projected_data = Vec::with_capacity(data.len());
	for data_vec in data.clone() {
	let mut projected_vector: Vec<RingGoldilock256> = Vec::with_capacity(target_dim);
	for projection_row in projection_matrix.clone() {
	let projection_value = RingGoldilock256::dot_product(&data_vec, &projection_row);
	projected_vector.push(projection_value);
	}
	projected_data.push(projected_vector);
	}
	let mut projected_data = Vec::with_capacity(data.len());
	for data_vec in &data {
	let mut projected_vector: Vec<RingGoldilock256> = Vec::with_capacity(target_dim);
	for projection_row in &projection_matrix {
	let projection_value = RingGoldilock256::dot_product(&data_vec, &projection_row);
	projected_vector.push(projection_value);
	}
	projected_data.push(projected_vector);
	}

[FoodChain1028]

And I reckon the result of project should be a vector of pure numbers

for example,

we have $r$ = 2 (the number of $\vec{s}$ ($\vec{s}=[s_1, s_2, ...]$ ))
$\vec{S_1} = [1,2, 3,4]$
$\vec{S_2} = [3,4, 5,6]$

lets the randomly sampled projection vector be

$\vec{\pi_1}^{(1)} = [1, 0, 1, -1]$
$\vec{\pi_2}^{(1)} = [0, 1, 1, 0]$

$p_1$ is calculated as below:

$$ \begin{aligned} p_1 &= <\vec{\pi_1}^{(1)}, \vec{s_1}> + <\vec{\pi_2}^{(1)}, \vec{s_2}> &= 1 \times 1 + 0\times2 + 1\times3 + (-1)\times4 + 0\times3 + 1\times4 + 1\times5 + 0\times6 &= 9 \end{aligned} $$

I think in the end we will get a 256-length vector $\vec{p}$, which each element is a pure number, i.e., [9, 2, 13, ...]

[Junochiu]

I have different understanding here, since the given data would be a polynomial ring matrix, the result should be a polynomial ring matrix with lower dimension.

[luckyyang]

I think @FoodChain1028 is correct, refer to page 35 of https://eprint.iacr.org/2024/311.pdf#page=34.19 , you will find the result of JL projection is in $Z_q^{2\lambda}$, which $\lambda = 128$

[pinhaocrypto]

In the Greyhound paper (https://eprint.iacr.org/2024/1293.pdf#page=28.41), they made some changes to the details of the JL project. I think we should follow these changes. But I'm not sure what the probability distribution of {1, -1}, maybe 1/2?

[Junochiu]

Closing the pr for now as we will be focusing on the poc prover implementation and will likely use the implementation from math library