https://github.com/sec-bit/learning-zkp/tree/master/plonk-intro-cn

Roots of Unity

https://github.com/sec-bit/learning-zkp/blob/master/plonk-intro-cn/2-plonk-lagrange-basis.md

Use Roots of Unity in Vanishing polynomial can reduce verifier cost from

$$ O(n) \to O(\log{n}) $$

Compute $x^N - 1$ only needs $\log{n}$

Permutation Check

learning-zkp/plonk-intro-cn/3-plonk-permutation.md at master · sec-bit/learning-zkp (github.com)