COGS 2294. [Hzoi 2015] release (FFT mod any prime) __fft

Topic Description Transmission Door

Title: To two times bounded by N polynomial, the product of the two polynomials, output of the first x of the 0 items to the n-1 of the coefficient mod 23333333

The NTT can only find the value under the FFT modulus. For any modulus problem, we can select three FFT modules to do the NTT, and finally the Chinese remainder theorem to merge.
After the first convolution, each number can reach 1023 (N∗MOD2) 10^{23} (n*mod^2), so we need to select the FFT modulus of three product greater than 1023 10^{23}. When merging 1023>264 10^{23}>2^{64}, we can not use the Chinese remainder theorem directly, need a little skill.
ANS≡A1 (mod M1) ans\equiv A1 \space (Mod\space M1)
ANS≡A2 (mod m2) ans\equiv a2 \space (mod \space m2)
ANS≡A3 (mod m3) ans\equiv a3\space (mod \space m3)
The Ans≡a (mod m) ans\equiv A \space (mod \space m) is obtained by merging the A1,A2 A1,A2 with the Chinese remainder theorem.
of which a=a1∗m2∗m2−1+a2∗m1∗m1−1,m=m1∗m2 a=a1*m2*m2^{-1}+a2*m1*m1^{-1},m=m1*m2
Ans=k∗m+a=x∗m3+a3 ANS=K*M+A=X*M3+A3
K∗m≡a3−a (mod m3) k*m\equiv a3-a \space (mod \space m3)
K≡ (a3−a) ∗m−1 (mod m3)