# GMP Development Projects

Copyright 2000, 2001 Free Software Foundation, Inc.

This file is part of the GNU MP Library.

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This file current as of 25 Nov 2001. An up-to-date version is available at http://www.swox.com/gmp/projects.html. Please send comments about this page to bug-gmp@gnu.org.

This file lists projects suitable for volunteers. Please see the tasks file for smaller tasks.

If you want to work on any of the projects below, please let tege@swox.com know. If you want to help with a project that already somebody else is working on, please talk to that person too, tege@swox.com can put you in touch. (There are no email addresses of volunteers below, due to spamming problems.)

- Faster multiplication

  The current multiplication code uses Karatsuba, 3-way Toom-Cook, or Fermat FFT. Several new developments are desirable:

  1. Handle multiplication of operands with different digit count better than today. We now split the operands in a very inefficient way, see mpn/generic/mul.c.
  2. Consider N-way Toom-Cook. See Knuth's Seminumerical Algorithms for details on the method. Code implementing it exists. This is asymptotically inferior to FFTs, but is finer grained. A toom-4 might fit in between toom-3 and the FFTs (or it might not).
  3. It's possible CPU dependent effects like cache locality will have a greater impact on speed than algorithmic improvements.
  4. Add support for partial products, either a given number of low limbs or high limbs of the result. A high partial product can be used by `mpf_mul` now, a low half partial product might be of use in a future sub-quadratic REDC. On small sizes a partial product will be faster simply through fewer cross-products, similar to the way squaring is faster. But work by Thom Mulders shows that for Karatsuba and higher order algorithms the advantage is progressively lost, so for large sizes partial products turn out to be no faster.

  Another possibility would be an optimized cube. In the basecase that should definitely be able to save cross-products in a similar fashion to squaring, but some investigation might be needed for how best to adapt the higher-order algorithms. Not sure whether cubing or further small powers have any particularly important uses though.

- Assembly routines

  Write new and improve existing assembly routines. The tests/devel programs and the tune/speed.c and tune/many.pl programs are useful for testing and timing the routines you write. See the README files in those directories for more information.

  Please make sure your new routines are fast for these three situations:

  1. Operands that fit into the cache.
  2. Small operands of less than, say, 10 limbs.
  3. Huge operands that does not fit into the cache.

  The most important routines are mpn_addmul_1, mpn_mul_basecase and mpn_sqr_basecase. The latter two don't exist for all machines, while mpn_addmul_1 exists for almost all machines.

  Standard techniques for these routines are unrolling, software pipelining, and specialization for common operand values. For machines with poor integer multiplication, it is often possible to improve the performance using floating-point operations, or SIMD operations such as MMX or Sun's VIS.

  Using floating-point operations is interesting but somewhat tricky. Since IEEE double has 53 bit of mantissa, one has to split the operands in small prices, so that no result is greater than 2^53. For 32-bit computers, splitting one operand into 16-bit pieces works. For 64-bit machines, one operand can be split into 21-bit pieces and the other into 32-bit pieces. (A 64-bit operand can be split into just three 21-bit pieces if one allows the split operands to be negative!)

- Faster extended GCD

  We currently have extended GCD based on Lehmer's method. But the binary method can quite easily be improved for bignums in a similar way Lehmer improved Euclid's algorithm. The result should beat Lehmer's method by about a factor of 2. Furthermore, the multiprecision step of Lehmer's method and the binary method will be identical, so the current code can mostly be reused. It should be possible to share code between GCD and GCDEXT, and probably with Jacobi symbols too.

  Paul Zimmerman has worked on sub-quadratic GCD and GCDEXT, but it seems that the most likely algorithm doesn't kick in until about 3000 limbs.

- Math functions for the mpf layer

  Implement the functions of math.h for the GMP mpf layer! Check the book "Pi and the AGM" by Borwein and Borwein for ideas how to do this. These functions are desirable: acos, acosh, asin, asinh, atan, atanh, atan2, cos, cosh, exp, log, log10, pow, sin, sinh, tan, tanh.

  These are implemented in mpfr, redoing them in mpf might not be worth bothering with, if the long term plan is to bring mpfr in as the new mpf.

- Faster sqrt

  The current code uses divisions, which are reasonably fast, but it'd be possible to use only multiplications by computing 1/sqrt(A) using this formula:

                                          2
                         x   = x  (3 - A x )/2.
                          i+1   i         i

  And the final result

                           sqrt(A) = A x .
                                        n

  That final multiply might be the full size of the input (though it might only need the high half of that), so there may or may not be any speedup overall.

- Nth root

  Implement, at the mpn level, a routine for computing the nth root of a number. The routine should use Newton's method, preferably without using division.

  If the routine becomes fast enough, perhaps square roots could be computed using this function.

- Quotient-Only Division

  Some work can be saved when only the quotient is required in a division, basically the necessary correction -0, -1 or -2 to the estimated quotient can almost always be determined from only a few limbs of multiply and subtract, rather than forming a complete remainder. The greatest savings are when the quotient is small compared to the dividend and divisor.

  Some code along these lines can be found in the current `mpn_tdiv_qr`, though perhaps calculating bigger chunks of remainder than might be strictly necessary. That function in its current form actually then always goes on to calculate a full remainder. Burnikel and Zeigler describe a similar approach for the divide and conquer case.

- Sub-Quadratic REDC and Exact Division

  `mpn_bdivmod` and the `redc` in `mpz_powm` should use some sort of divide and conquer algorithm. This would benefit `mpz_divexact`, and `mpn_gcd` on large unequal size operands. See "Exact Division with Karatsuba Complexity" by Jebelean for a (brief) description.

  Failing that, some sort of `DIVEXACT_THRESHOLD` could be added to control whether `mpz_divexact` uses `mpn_bdivmod` or `mpn_tdiv_qr`, since the latter is faster on large divisors.

  For the REDC, basically all that's needed is Montgomery's algorithm done in multi-limb integers. R is multiple limbs, and the inverse and operands are multi-precision.

  For exact division the time to calculate a multi-limb inverse is not amortized across many modular operations, but instead will probably create a threshold below which the current style `mpn_bdivmod` is best. There's also Krandick and Jebelean, "Bidirectional Exact Integer Division" to basically use a low to high exact division for the low half quotient, and a quotient-only division for the high half.

  It will be noted that low-half and high-half multiplies, and a low-half square, can be used. These ought to each take as little as half the time of a full multiply, or square, though work by Thom Mulders shows the advantage is progressively lost as Karatsuba and higher algorithms are applied.

- Test Suite

  Add a test suite for old bugs. These tests wouldn't loop or use random numbers, but just test for specific bugs found in the past.

  More importantly, improve the random number controls for test collections:

  1. Use the new random number framework.
  2. Have every test accept a seed parameter.
  3. Allow `make check' to set the seed parameter.
  4. Add more tests for important, now untested functions.

  With this in place, it should be possible to rid GMP of practically all bugs by having some dedicated GMP test machines running "make check" with ever increasing seeds (and periodically updating to the latest GMP).

  If a few more ASSERTs were sprinkled through the code, running some tests with --enable-assert might be useful, though not a substitute for tests on the normal build.

  An important feature of any random tests will be to record the seeds used, and perhaps to snapshot operands before performing each test, so any problem exposed can be reproduced.