Finding working modulus for FFT over finite fields

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I would like to implement multiplication of polynomials using NTT. I followed Number-theoretic transform (integer DFT) and it seems to work.



Now I would like to implement multiplication of polynomials over finite fields $mathbbZ_p[x]$ where $p$ is arbitrary prime number.



Does it changes anything that the coefficients are now bounded by $p$, compared to the former unbounded case?



In particular, original NTT required to find prime number $N$ as the working modulus that is larger than $(magnitude of largest element of input vector)^2 times (length of input vector) + 1$ so that the result never overflows. If the result is going to be bounded by modulo that $p$ prime anyway, how small can the modulus be? Note that $p - 1$ does not have to be of form $(some positive integer) * (length of input vector)$.










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    up vote
    1
    down vote

    favorite












    I would like to implement multiplication of polynomials using NTT. I followed Number-theoretic transform (integer DFT) and it seems to work.



    Now I would like to implement multiplication of polynomials over finite fields $mathbbZ_p[x]$ where $p$ is arbitrary prime number.



    Does it changes anything that the coefficients are now bounded by $p$, compared to the former unbounded case?



    In particular, original NTT required to find prime number $N$ as the working modulus that is larger than $(magnitude of largest element of input vector)^2 times (length of input vector) + 1$ so that the result never overflows. If the result is going to be bounded by modulo that $p$ prime anyway, how small can the modulus be? Note that $p - 1$ does not have to be of form $(some positive integer) * (length of input vector)$.










    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I would like to implement multiplication of polynomials using NTT. I followed Number-theoretic transform (integer DFT) and it seems to work.



      Now I would like to implement multiplication of polynomials over finite fields $mathbbZ_p[x]$ where $p$ is arbitrary prime number.



      Does it changes anything that the coefficients are now bounded by $p$, compared to the former unbounded case?



      In particular, original NTT required to find prime number $N$ as the working modulus that is larger than $(magnitude of largest element of input vector)^2 times (length of input vector) + 1$ so that the result never overflows. If the result is going to be bounded by modulo that $p$ prime anyway, how small can the modulus be? Note that $p - 1$ does not have to be of form $(some positive integer) * (length of input vector)$.










      share|cite|improve this question













      I would like to implement multiplication of polynomials using NTT. I followed Number-theoretic transform (integer DFT) and it seems to work.



      Now I would like to implement multiplication of polynomials over finite fields $mathbbZ_p[x]$ where $p$ is arbitrary prime number.



      Does it changes anything that the coefficients are now bounded by $p$, compared to the former unbounded case?



      In particular, original NTT required to find prime number $N$ as the working modulus that is larger than $(magnitude of largest element of input vector)^2 times (length of input vector) + 1$ so that the result never overflows. If the result is going to be bounded by modulo that $p$ prime anyway, how small can the modulus be? Note that $p - 1$ does not have to be of form $(some positive integer) * (length of input vector)$.







      number-theory finite-fields convolution fast-fourier-transform






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      asked Sep 10 at 18:24









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