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Using Fft to Efficiently Multiply Multiple Polynomials Modulo N

David Miller • • 2 min read
Using Fft to Efficiently Multiply Multiple Polynomials Modulo N
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I have a set of polynomials $P_k$ defined as:

$$ P_k = \sum_{n = 0}^{n = N} p_{kn} x^n $$

$N$ is large ($2^{16}$)

I would like to perform multiplication on the polynomials:

$$ Q = \prod_{k=0}^{k=K} P_k $$

$K = 16$

But I only want to keep the coefficients from $0$ to $N$ i.e.

$$ Q = \sum_{n = 0}^{n = N} q_nx^n $$

I know the fastest way to multiply two large polynomials is using the FFT to perform linear convolution by zero padding the two polynomials by N before taking the forward transform, point wise multiplying and then taking the inverse transform. However, for $K$ polynomials I would have to pad $K*N$. Is there an efficient method where I can pad by N, do the forward transform on all the polynomials, and just take one inverse transform at the end?

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David Miller
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David Miller

David Miller brings 15 years of experience in global economics, personal finance strategy, and market dynamics. He specializes in turning complex economic trends into actionable insights for everyday readers.