~~My last blog was a bit too esoteric~~

This blog assumes that you can do a basic convolution (e.g. FFT). After I saw that many people did not know how to do Chirp Z-Transform with arbitrary exponents fast from my blog 9 weeks ago I decided to ~~shitpost~~ write a short tutorial on it.

Let's say we have a polynomial $$$P(x)$$$, such that

Now, let's say we have $$$c$$$ and $$$m$$$, and we want to find $$$P(c^0),P(c^1),P(c^2),\dots,P(c^m)$$$. Let $$$b_j=P(c^j)$$$. Consider the implications of having $$$x$$$ as such a form:

Tinkering around with different expressions for $$$ij$$$, one finds that $$$ij=\frac{(i+j)^2}{2}-\frac{i^2}{2}-\frac{j^2}{2}$$$. This means that

Hence we can find $$$b_j$$$ from the difference-convolution of $$$a_ic^{-\frac{i^2}{2}}$$$ and $$$c^{\frac{i^2}{2}}$$$. However, in many cases — especially when working under a modulus — we can't find the $$$c^{\frac{i^2}{2}}$$$ as $$$i$$$ may be odd. We use a workaround: $$$ij=\binom{i+j}{2}-\binom{i}{2}-\binom{j}{2}$$$. Proof:

Modifying our formula a bit, we get

As for implmentation details, notice that

Define $$$C_i=a_{n-i}c^{-\binom{n-i}2}$$$; $$$D_i=c^{\binom i2}$$$. We hence have

(the second through definition of convolution)

You can test your implementations here, mod 998244353 and here, mod 10^9+7, although note that the second one is both intense in precision and constant factor.

**My code for the former**

This method can be used to cheese 1184A3 - Heidi Learns Hashing (Hard) and 1054H - Epic Convolution, and is also a core point in 901E - Cyclic Cipher.