Hi Codeforces!

I have recently come up with a really neat and simple recursive algorithm for multiplying polynomials in $$$O(n \log n)$$$ time. It is so neat and simple that I think it might possibly revolutionize the way that fast polynomial multiplication is taught and implemented.

Prerequisite: Polynomial quotient and remainder: Wiki article, Stackexchange example

Task:

Given two polynomials $$$P$$$ and $$$Q$$$ and an integer $$$n$$$, where degree $$$P < n$$$ and degree $$$Q < n$$$. Your task is to calculate the polynomial $$$P(x) \, Q(x) \% (x^n - c)$$$. You may assume that $$$n$$$ is a power of two.

Solution:

We can create a divide and conquer algorithm for $$$P(x) \, Q(x) \% (x^n - c)$$$ based on the difference of squares formula. Assuming $$$n$$$ is even, then $$$(x^n - c) = (x^{n/2} - \sqrt{c}) (x^{n/2} + \sqrt{c})$$$. The idea behind the algorithm is to calculate $$$P(x) \, Q(x) \% (x^{n/2} - \sqrt{c})$$$ and $$$P(x) \, Q(x) \% (x^{n/2} + \sqrt{c})$$$ using 2 recursive calls, and then use CRT (Chinese remainder theorem) to calculate $$$P(x) \, Q(x) \% (x^n - c)$$$.

So how do we actually calculate $$$P(x) \, Q(x) \% (x^n - c)$$$ using $$$P(x) \, Q(x) \% (x^{n/2} - \sqrt{c})$$$ and $$$P(x) \, Q(x) \% (x^{n/2} + \sqrt{c})$$$?

Well we use the following formula: