84 lines
2.9 KiB
Plaintext
84 lines
2.9 KiB
Plaintext
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Bignum support (Fast Number Theoretic Transform or FNT):
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========================================================
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Bignum arithmetic in libmpdec uses the scheme for fast convolution
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of integer sequences from:
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J. M. Pollard: The fast Fourier transform in a finite field
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http://www.ams.org/journals/mcom/1971-25-114/S0025-5718-1971-0301966-0/home.html
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The transform in a finite field can be used for convolution in the same
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way as the Fourier Transform. The main advantages of the Number Theoretic
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Transform are that it is both exact and very memory efficient.
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Convolution in pseudo-code:
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~~~~~~~~~~~~~~~~~~~~~~~~~~~
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fnt_convolute(a, b):
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x = fnt(a) # forward transform of a
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y = fnt(b) # forward transform of b
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z = pairwise multiply x[i] and y[i]
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result = inv_fnt(z) # backward transform of z.
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Extending the maximum transform length (Chinese Remainder Theorem):
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-------------------------------------------------------------------
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The maximum transform length is quite limited when using a single
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prime field. However, it is possible to use multiple primes and
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recover the result using the Chinese Remainder Theorem.
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Multiplication in pseudo-code:
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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_mpd_fntmul(u, v):
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c1 = fnt_convolute(u, v, P1) # convolute modulo prime1
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c2 = fnt_convolute(u, v, P2) # convolute modulo prime2
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c3 = fnt_convolute(u, v, P3) # convolute modulo prime3
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result = crt3(c1, c2, c3) # Chinese Remainder Theorem
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Optimized transform functions:
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------------------------------
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There are three different fnt() functions:
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std_fnt: "standard" decimation in frequency transform for array lengths
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of 2**n. Performs well up to 1024 words.
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sixstep: Cache-friendly algorithm for array lengths of 2**n. Outperforms
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std_fnt for large arrays.
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fourstep: Algorithm for array lengths of 3 * 2**n. Also cache friendly
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in large parts.
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List of bignum-only files:
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--------------------------
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Functions from these files are only used in _mpd_fntmul().
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umodarith.h -> fast low level routines for unsigned modular arithmetic
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numbertheory.c -> routines for setting up the FNT
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difradix2.c -> decimation in frequency transform, used as the
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"base case" by the following three files:
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fnt.c -> standard transform for smaller arrays
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sixstep.c -> transform large arrays of length 2**n
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fourstep.c -> transform arrays of length 3 * 2**n
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convolute.c -> do the actual fast convolution, using one of
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the three transform functions.
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transpose.c -> transpositions needed for the sixstep algorithm.
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crt.c -> Chinese Remainder Theorem: use information from three
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transforms modulo three different primes to get the
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final result.
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