2
votes
1answer
64 views

Primitive $2^k$-th roots of unity in $GF(p)$

While debugging an NTT implementation I've noticed something. Looks like if I have a primitive $(n = 2^k)$-th root of unity $\omega$ in a $GF(p)$, then $\omega ^0 = -\omega ^{n/2} + p,$ $\omega ^1 = ...
2
votes
1answer
134 views

The digit base and the NTT convolution

Suppose I'm using a number theoretic transform (NTT) in an integer field $GF(p)$. I assume that $2n$-th root of unity exists for such a $p$, and I want to compute a convolution of two $n$-length ...
3
votes
1answer
394 views

Long integer multiplication using FFT in integer rings

I would like to perform long integer (~= polynomial) multiplication using the FFT or its direct analogue, but never leave integer rings. Please excuse in advance all my mistakes in formulation and ...
8
votes
1answer
241 views

Expectation of a Random Subset of the Roots of Unity.

Let $p$ be a prime. If $1_A(x)$ denotes the indicator function of the set $A\subset\mathbb{Z}/p\mathbb{Z}$ and $$\hat{1}_A(t):=\frac{1}{p}\sum_{n=1}^p 1_A(n)e^{2\pi i \frac{nt}{p}}$$ denotes the ...