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python-clinic / Modules / _decimal / libmpdec / literature / bignum.txt

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Bignum support (Fast Number Theoretic Transform or FNT):
========================================================

Bignum arithmetic in libmpdec uses the scheme for fast convolution
of integer sequences from:

J. M. Pollard: The fast Fourier transform in a finite field
http://www.ams.org/journals/mcom/1971-25-114/S0025-5718-1971-0301966-0/home.html


The transform in a finite field can be used for convolution in the same
way as the Fourier Transform. The main advantages of the Number Theoretic
Transform are that it is both exact and very memory efficient.


Convolution in pseudo-code:
~~~~~~~~~~~~~~~~~~~~~~~~~~~

  fnt_convolute(a, b):
    x = fnt(a)                          # forward transform of a
    y = fnt(b)                          # forward transform of b
    z = pairwise multiply x[i] and y[i]
    result = inv_fnt(z)                 # backward transform of z.


Extending the maximum transform length (Chinese Remainder Theorem):
-------------------------------------------------------------------

The maximum transform length is quite limited when using a single
prime field. However, it is possible to use multiple primes and
recover the result using the Chinese Remainder Theorem.


Multiplication in pseudo-code:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

  _mpd_fntmul(u, v):
    c1 = fnt_convolute(u, v, P1)  # convolute modulo prime1
    c2 = fnt_convolute(u, v, P2)  # convolute modulo prime2
    c3 = fnt_convolute(u, v, P3)  # convolute modulo prime3
    result = crt3(c1, c2, c3)     # Chinese Remainder Theorem


Optimized transform functions:
------------------------------

There are three different fnt() functions:

   std_fnt: "standard" decimation in frequency transform for array lengths
            of 2**n. Performs well up to 1024 words.

   sixstep: Cache-friendly algorithm for array lengths of 2**n. Outperforms
            std_fnt for large arrays.

   fourstep: Algorithm for array lengths of 3 * 2**n. Also cache friendly
             in large parts.


List of bignum-only files:
--------------------------

Functions from these files are only used in _mpd_fntmul().

  umodarith.h    -> fast low level routines for unsigned modular arithmetic
  numbertheory.c -> routines for setting up the FNT
  difradix2.c    -> decimation in frequency transform, used as the
                    "base case" by the following three files:

      fnt.c          -> standard transform for smaller arrays
      sixstep.c      -> transform large arrays of length 2**n
      fourstep.c     -> transform arrays of length 3 * 2**n

  convolute.c    -> do the actual fast convolution, using one of
                    the three transform functions.
  transpose.c    -> transpositions needed for the sixstep algorithm.
  crt.c          -> Chinese Remainder Theorem: use information from three
                    transforms modulo three different primes to get the
                    final result.