Can the inner-most loop in NTT be parallelized?

[Sword-Smith]

This is the performance critical code. It would be cool if the inner-most loop here could be parallelized:

for _ in 0..log_2_of_n {
        let w_m = omega.mod_pow_u32(n / (2 * m));
        let mut k = 0;
        while k < n {
            let mut w = BFieldElement::one();
            for j in 0..m {
                let u = x[(k + j) as usize];
                let mut v = x[(k + j + m) as usize];
                v *= w;
                x[(k + j) as usize] = u + v;
                x[(k + j + m) as usize] = u - v;
                w *= w_m;
            }

            k += 2 * m;
        }

        m *= 2;
    }

[jan-ferdinand]

Maybe this inner loop can be vectorized using AVX instructions?