Helpers, mullow and modular inverse for radix integers

[fredrik-johansson]

We add more essential functionality for radix integers:

• Setting/getting/shifting/counting limbs
• Reduction mod $B^n$, both to $[0, B^n)$ and to $(\pm B^n / 2)$
• Multiplication and inverse mod $B^n$.

This is now starting to get usable for p-adics.

Check out these timings @thofma @fieker.

Multiplication mod $B^n$ where $B = 7^{22}$ (padic_mul, fmpz_mod_mul, mpn_mod_mul, radix_integer_mullow_limbs):

       n      padic   fmpz_mod    mpn_mod   radix_integer    speedup/padic  speedup/fmpz_mod
       1   6.99e-08   5.45e-09          -   9.40e-09         7.44x          0.58x
       2   9.26e-08   9.27e-09   8.02e-09   1.54e-08         6.01x          0.60x
       4   1.33e-07   7.00e-08   3.43e-08   3.37e-08         3.95x          2.08x
       8   2.11e-07   1.34e-07   7.03e-08   7.02e-08         3.01x          1.91x
      16   4.42e-07   3.37e-07   2.19e-07   2.35e-07         1.88x          1.43x
      32   1.16e-06   8.27e-07          -   6.03e-07         1.92x          1.37x
      64   3.43e-06   2.55e-06          -   1.87e-06         1.83x          1.36x
     128   1.05e-05   7.77e-06          -   5.88e-06         1.79x          1.32x
     256   3.27e-05   2.73e-05          -   9.73e-06         3.36x          2.81x
     512   9.30e-05   6.92e-05          -   2.03e-05         4.58x          3.41x
    1024   2.54e-04   1.46e-04          -   4.32e-05         5.88x          3.38x
    2048   5.96e-04   2.87e-04          -   8.93e-05         6.67x          3.21x
    4096   1.33e-03   6.73e-04          -   1.87e-04         7.11x          3.60x
    8192   2.75e-03   1.31e-03          -   3.99e-04         6.89x          3.28x
   16384   5.70e-03   2.74e-03          -   8.30e-04         6.87x          3.30x
   32768   1.21e-02   5.62e-03          -   1.72e-03         7.03x          3.27x
   65536   2.48e-02   1.16e-02          -   3.58e-03         6.93x          3.24x
  131072   5.28e-02   2.55e-02          -   7.62e-03         6.93x          3.35x
  262144   1.29e-01   5.53e-02          -   1.67e-02         7.72x          3.31x
  524288   2.71e-01   1.21e-01          -   3.77e-02         7.19x          3.21x
 1048576   5.79e-01   2.64e-01          -   8.43e-02         6.87x          3.13x
 2097152   1.26e+00   5.88e-01          -   1.82e-01         6.91x          3.23x
 4194304   2.77e+00   1.24e+00          -   3.93e-01         7.05x          3.16x
 8388608   6.15e+00   2.67e+00          -   8.19e-01         7.51x          3.26x

There will be a little less overhead for small $n$ if one uses the mpn-like interface instead of the mpz-like radix_integer.

Inversion of a unit mod $B^n$ where $B = 7^{22}$:

       n      padic   fmpz_mod    mpn_mod   radix_integer    speedup/padic  speedup/fmpz_mod
       1   2.10e-07   4.79e-08          -   3.09e-08         6.80x          1.55x
       2   3.57e-07   3.16e-07   2.59e-07   5.19e-08         6.88x          6.09x
       4   6.01e-07   6.35e-07   5.76e-07   1.12e-07         5.37x          5.67x
       8   8.83e-07   1.40e-06   1.33e-06   2.36e-07         3.74x          5.93x
      16   1.45e-06   2.89e-06   2.84e-06   5.75e-07         2.52x          5.03x
      32   3.00e-06   6.35e-06          -   1.56e-06         1.92x          4.07x
      64   7.70e-06   1.59e-05          -   4.26e-06         1.81x          3.73x
     128   2.23e-05   4.64e-05          -   1.28e-05         1.74x          3.63x
     256   6.61e-05   1.40e-04          -   2.84e-05         2.33x          4.93x
     512   1.92e-04   4.67e-04          -   5.97e-05         3.22x          7.82x
    1024   5.41e-04   1.39e-03          -   1.25e-04         4.33x          11.12x
    2048   1.47e-03   4.06e-03          -   2.66e-04         5.53x          15.26x
    4096   3.41e-03   1.13e-02          -   5.61e-04         6.08x          20.14x
    8192   7.21e-03   3.09e-02          -   1.17e-03         6.16x          26.41x
   16384   1.48e-02   8.85e-02          -   2.46e-03         6.02x          35.98x
   32768   3.02e-02   2.09e-01          -   5.08e-03         5.94x          41.14x
   65536   6.19e-02   5.10e-01          -   1.04e-02         5.95x          49.04x
  131072   1.38e-01   1.23e+00          -   2.21e-02         6.24x          55.43x
  262144   2.86e-01   2.90e+00          -   4.69e-02         6.10x          61.77x
  524288   6.04e-01   6.84e+00          -   1.02e-01         5.92x          67.03x
 1048576   1.28e+00   1.59e+01          -   2.23e-01         5.73x          71.22x
 2097152   2.73e+00   3.74e+01          -   4.80e-01         5.69x          77.89x
 4194304   5.85e+00   8.72e+01          -   1.06e+00         5.52x          82.20x
 8388608   1.29e+01   2.01e+02          -   2.22e+00         5.79x          90.59x

It remains to implement functions that accept a digit count rather than a limb count. These are more fiddly and will inevitably have some more overhead. In practice, the most efficient way to implement $\mathbb{Z} / p^e \mathbb{Z}$ for specific $e$ (or $\mathbb{Q}_p$ with digit-granular precision) will probably be to actually use whole limbs internally and reduce modulo a fractional limb only when strictly needed, e.g. when doing a comparison.