# GMP Development Projects

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This file current as of 21 Apr 2006. An up-to-date version is available at http://swox.com/gmp/projects.html. Please send comments about this page to gmp-devel@swox.com.

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 gmp-devel@swox.com know. If you want to help with a project that already somebody else is working on, you will get in touch through gmp-devel@swox.com. (There are no email addresses of volunteers below, due to spamming problems.)

- Faster multiplication

  The current multiplication code uses Karatsuba, 3-way Toom, and 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. The best operands splitting strategy depends on the underlying multiplication algorithm to be used.
  2. Implement an FFT variant computing the coefficients mod m different limb size primes of the form l*2^k+1. i.e., compute m separate FFTs. The wanted coefficients will at the end be found by lifting with CRT (Chinese Remainder Theorem). If we let m = 3, i.e., use 3 primes, we can split the operands into coefficients at limb boundaries, and if our machine uses b-bit limbs, we can multiply numbers with close to 2^b limbs without coefficient overflow. For smaller multiplication, we might perhaps let m = 1, and instead of splitting our operands at limb boundaries, split them in much smaller pieces. We might also use 4 or more primes, and split operands into bigger than b-bit chunks. By using more primes, the gain in shorter transform length, but lose in having to do more FFTs, but that is a slight total save. We then lose in more expensive CRT.
     An nearly complete implementation has been done by Tommy Färnqvist.
  3. Perhaps consider N-way Toom, N > 3. 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).
  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` and by Newton approximations, 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 GCD

  Work on Schönhage GCD and GCDEXT for large numbers is in progress. Contact Niels Möller if you want to help.

- 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.

- 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

  The square root can then be computed like this:

                           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

  Improve mpn_rootrem. The current code is really naive, using full precision from the first iteration. Also, calling mpn_pow_1 isn't very clever, as only 1/n of the result bits will be used; truncation after each multiplication would be better. Avoiding division might also be possible. Work mostly done, see http://swox.com/gmp/development.html.

- 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

  See also http://swox.com/gmp/development.html for some new code for divexact.

  `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.

- Exceptions

  Some sort of scheme for exceptions handling would be desirable. Presently the only thing documented is that divide by zero in GMP functions provokes a deliberate machine divide by zero (on those systems where such a thing exists at least). The global `gmp_errno` is not actually documented, except for the old `gmp_randinit` function. Being currently just a plain global means it's not thread-safe.

  The basic choices for exceptions are returning an error code or having a handler function to be called. The disadvantage of error returns is they have to be checked, leading to tedious and rarely executed code, and strictly speaking such a scheme wouldn't be source or binary compatible. The disadvantage of a handler function is that a `longjmp` or similar recovery from it may be difficult. A combination would be possible, for instance by allowing the handler to return an error code.

  Divide-by-zero, sqrt-of-negative, and similar operand range errors can normally be detected at the start of functions, so exception handling would have a clean state. What's worth considering though is that the GMP function detecting the exception may have been called via some third party library or self contained application module, and hence have various bits of state to be cleaned up above it. It'd be highly desirable for an exceptions scheme to allow for such cleanups.

  The C++ destructor mechanism could help with cleanups both internally and externally, but being a plain C library we don't want to depend on that.

  A C++ `throw` might be a good optional extra exceptions mechanism, perhaps under a build option. For GCC `-fexceptions` will add the necessary frame information to plain C code, or GMP could be compiled as C++.

  Out-of-memory exceptions are expected to be handled by the `mp_set_memory_functions` routines, rather than being a prospective part of divide-by-zero etc. Some similar considerations apply but what differs is that out-of-memory can arise deep within GMP internals. Even fundamental routines like `mpn_add_n` and `mpn_addmul_1` can use temporary memory (for instance on Cray vector systems). Allowing for an error code return would require an awful lot of checking internally. Perhaps it'd still be worthwhile, but it'd be a lot of changes and the extra code would probably be rather rarely executed in normal usages.

  A `longjmp` recovery for out-of-memory will currently, in general, lead to memory leaks and may leave GMP variables operated on in inconsistent states. Maybe it'd be possible to record recovery information for use by the relevant allocate or reallocate function, but that too would be a lot of changes.

  One scheme for out-of-memory would be to note that all GMP allocations go through the `mp_set_memory_functions` routines. So if the application has an intended `setjmp` recovery point it can record memory activity by GMP and abandon space allocated and variables initialized after that point. This might be as simple as directing the allocation functions to a separate pool, but in general would have the disadvantage of needing application-level bookkeeping on top of the normal system `malloc`. An advantage however is that it needs nothing from GMP itself and on that basis doesn't burden applications not needing recovery. Note that there's probably some details to be worked out here about reallocs of existing variables, and perhaps about copying or swapping between "permanent" and "temporary" variables.

  Applications desiring a fine-grained error control, for instance a language interpreter, would very possibly not be well served by a scheme requiring `longjmp`. Wrapping every GMP function call with a `setjmp` would be very inconvenient.

  Another option would be to let `mpz_t` etc hold a sort of NaN, a special value indicating an out-of-memory or other failure. This would be similar to NaNs in mpfr. Unfortunately such a scheme could only be used by programs prepared to handle such special values, since for instance a program waiting for some condition to be satisfied could become an infinite loop if it wasn't also watching for NaNs. The work to implement this would be significant too, lots of checking of inputs and intermediate results. And if `mpn` routines were to participate in this (which they would have to internally) a lot of new return values would need to be added, since of course there's no `mpz_t` etc structure for them to indicate failure in.

  Stack overflow is another possible exception, but perhaps not one that can be easily detected in general. On i386 GNU/Linux for instance GCC normally doesn't generate stack probes for an `alloca`, but merely adjusts `%esp`. A big enough `alloca` can miss the stack redzone and hit arbitrary data. GMP stack usage is normally a function of operand size, which might be enough for some applications to know they'll be safe. Otherwise a fixed maximum usage can probably be obtained by building with `--enable-alloca=malloc-reentrant` (or `notreentrant`). Arranging the default to be `alloca` only on blocks up to a certain size and `malloc` thereafter might be a better approach and would have the advantage of not having calculations limited by available stack.

  Actually recovering from stack overflow is of course another problem. It might be possible to catch a `SIGSEGV` in the stack redzone and do something in a `sigaltstack`, on systems which have that, but recovery might otherwise not be possible. This is worth bearing in mind because there's no point worrying about tight and careful out-of-memory recovery if an out-of-stack is fatal.

  Operand overflow is another exception to be addressed. It's easy for instance to ask `mpz_pow_ui` for a result bigger than an `mpz_t` can possibly represent. Currently overflows in limb or byte count calculations will go undetected. Often they'll still end up asking the memory functions for blocks bigger than available memory, but that's by no means certain and results are unpredictable in general. It'd be desirable to tighten up such size calculations. Probably only selected routines would need checks, if it's assumed say that no input will be more than half of all memory and hence size additions like say `mpz_mul` won't overflow.

- Performance Tool

  It'd be nice to have some sort of tool for getting an overview of performance. Clearly a great many things could be done, but some primary uses would be,

  1. Checking speed variations between compilers.
  2. Checking relative performance between systems or CPUs.

  A combination of measuring some fundamental routines and some representative application routines might satisfy these.

  The tune/time.c routines would be the easiest way to get good accurate measurements on lots of different systems. The high level `speed_measure` may or may not suit, but the basic `speed_starttime` and `speed_endtime` would cover lots of portability and accuracy questions.

- Using `restrict`

  There might be some value in judicious use of C99 style `restrict` on various pointers, but this would need some careful thought about what it implies for the various operand overlaps permitted in GMP.

  Rumour has it some pre-C99 compilers had `restrict`, but expressing tighter (or perhaps looser) requirements. Might be worth investigating that before using `restrict` unconditionally.

  Loops are presumably where the greatest benefit would be had, by allowing the compiler to advance reads ahead of writes, perhaps as part of loop unrolling. However critical loops are generally coded in assembler, so there might not be very much to gain. And on Cray systems the explicit use of `_Pragma` gives an equivalent effect.

  One thing to note is that Microsoft C headers (on ia64 at least) contain `__declspec(restrict)`, so a `#define` of `restrict` should be avoided. It might be wisest to setup a `gmp_restrict`.

- Nx1 Division

  The limb-by-limb dependencies in the existing Nx1 division (and remainder) code means that chips with multiple execution units or pipelined multipliers are not fully utilized.

  One possibility is to follow the current preinv method but taking two limbs at a time. That means a 2x2->4 and a 2x1->2 multiply for each two limbs processed, and because the 2x2 and 2x1 can each be done in parallel the latency will be not much more than 2 multiplies for two limbs, whereas the single limb method has a 2 multiply latency for just one limb. A version of `mpn_divrem_1` doing this has been written in C, but not yet tested on likely chips. Clearly this scheme would extend to 3x3->9 and 3x1->3 etc, though with diminishing returns.

  For `mpn_mod_1`, Peter L. Montgomery proposes the following scheme. For a limb R=2^`bits_per_mp_limb`, pre-calculate values R mod N, R^2 mod N, R^3 mod N, R^4 mod N. Then take dividend limbs and multiply them by those values, thereby reducing them (moving them down) by the corresponding factor. The products can be added to produce an intermediate remainder of 2 or 3 limbs to be similarly included in the next step. The point is that such multiplies can be done in parallel, meaning as little as 1 multiply worth of latency for 4 limbs. If the modulus N is less than R/4 (or is it R/5?) the summed products will fit in 2 limbs, otherwise 3 will be required, but with the high only being small. Clearly this extends to as many factors of R as a chip can efficiently apply.

  The logical conclusion for powers R^i is a whole array "p[i] = R^i mod N" for i up to k, the size of the dividend. This could then be applied at multiplier throughput speed like an inner product. If the powers took roughly k divide steps to calculate then there'd be an advantage any time the same N was used three or more times. Suggested by Victor Shoup in connection with chinese-remainder style decompositions, but perhaps with other uses.

  `mpn_modexact_1_odd` calculates an x in the range 0<=x<d satisfying a = q*d + x*b^n, where b=2^bits_per_limb. The factor b^n needed to get the true remainder r could be calculated by a powering algorithm, allowing `mpn_modexact_1_odd` to be pressed into service for an `mpn_mod_1`. `modexact_1` is simpler and on some chips can run noticeably faster than plain `mod_1`, on Athlon for instance 11 cycles/limb instead of 17. Such a difference could soon overcome the time to calculate b^n. The requirement for an odd divisor in `modexact` can be handled by some shifting on-the-fly, or perhaps by an extra partial-limb step at the end.

- Factorial

  The removal of twos in the current code could be extended to factors of 3 or 5. Taking this to its logical conclusion would be a complete decomposition into powers of primes. The power for a prime p is of course floor(n/p)+floor(n/p^2)+... Conrad Curry found this is quite fast (using simultaneous powering as per Handbook of Applied Cryptography algorithm 14.88).

  A difficulty with using all primes is that quite large n can be calculated on a system with enough memory, larger than we'd probably want for a table of primes, so some sort of sieving would be wanted. Perhaps just taking out the factors of 3 and 5 would give most of the speedup that a prime decomposition can offer.

- Binomial Coefficients

  An obvious improvement to the current code would be to strip factors of 2 from each multiplier and divisor and count them separately, to be applied with a bit shift at the end. Factors of 3 and perhaps 5 could even be handled similarly.

  Conrad Curry reports a big speedup for binomial coefficients using a prime powering scheme, at least for k near n/2. Of course this is only practical for moderate size n since again it requires primes up to n.

  When k is small the current (n-k+1)...n/1...k will be fastest. Some sort of rule would be needed for when to use this or when to use prime powering. Such a rule will be a function of both n and k. Some investigation is needed to see what sort of shape the crossover line will have, the usual parameter tuning can of course find machine dependent constants to fill in where necessary.

  An easier possibility also reported by Conrad Curry is that it may be faster not to divide out the denominator (1...k) one-limb at a time, but do one big division at the end. Is this because a big divisor in `mpn_bdivmod` trades the latency of `mpn_divexact_1` for the throughput of `mpn_submul_1`? Overheads must hurt though.

  Another reason a big divisor might help is that `mpn_divexact_1` won't be getting a full limb in `mpz_bin_uiui`. It's called when the n accumulator is full but the k may be far from full. Perhaps the two could be decoupled so k is applied when full. It'd be necessary to delay consideration of k terms until the corresponding n terms had been applied though, since otherwise the division won't be exact.

- Perfect Power Testing

  `mpz_perfect_power_p` could be improved in a number of ways, for instance p-adic arithmetic to find possible roots.

  Non-powers can be quickly identified by checking for Nth power residues modulo small primes, like `mpn_perfect_square_p` does for squares. The residues to each power N for a given remainder could be grouped into a bit mask, the masks for the remainders to each divisor would then be "and"ed together to hopefully leave only a few candidate powers. Need to think about how wide to make such masks, ie. how many powers to examine in this way.

  Any zero remainders found in residue testing reveal factors which can be divided out, with the multiplicity restricting the powers that need to be considered, as per the current code. Further prime dividing should be grouped into limbs like `PP`. Need to think about how much dividing to do like that, probably more for bigger inputs, less for smaller inputs.

  `mpn_gcd_1` would probably be better than the current private GCD routine. The use it's put to isn't time-critical, and it might help ensure correctness to just use the main GCD routine.

- Prime Testing

  GMP is not really a number theory library and probably shouldn't have large amounts of code dedicated to sophisticated prime testing algorithms, but basic things well-implemented would suit. Tests offering certainty are probably all too big or too slow (or both!) to justify inclusion in the main library. Demo programs showing some possibilities would be good though.

  The present "repetitions" argument to `mpz_probab_prime_p` is rather specific to the Miller-Rabin tests of the current implementation. Better would be some sort of parameter asking perhaps for a maximum chance 1/2^x of a probable prime in fact being composite. If applications follow the advice that the present reps gives 1/4^reps chance then perhaps such a change is unnecessary, but an explicitly described 1/2^x would allow for changes in the implementation or even for new proofs about the theory.

  `mpz_probab_prime_p` always initializes a new `gmp_randstate_t` for randomized tests, which unfortunately means it's not really very random and in particular always runs the same tests for a given input. Perhaps a new interface could accept an rstate to use, so successive tests could increase confidence in the result.

  `mpn_mod_34lsub1` is an obvious and easy improvement to the trial divisions. And since the various prime factors are constants, the remainder can be tested with something like

      #define MP_LIMB_DIVISIBLE_7_P(n) \
        ((n) * MODLIMB_INVERSE_7 <= MP_LIMB_T_MAX/7)

  Which would help compilers that don't know how to optimize divisions by constants, and is even an improvement on current gcc 3.2 code. This technique works for any modulus, see Granlund and Montgomery "Division by Invariant Integers" section 9.

  The trial divisions are done with primes generated and grouped at runtime. This could instead be a table of data, with pre-calculated inverses too. Storing deltas, ie. amounts to add, rather than actual primes would save space. `udiv_qrnnd_preinv` style inverses can be made to exist by adding dummy factors of 2 if necessary. Some thought needs to be given as to how big such a table should be, based on how much dividing would be profitable for what sort of size inputs. The data could be shared by the perfect power testing.

  Jason Moxham points out that if a sqrt(-1) mod N exists then any factor of N must be == 1 mod 4, saving half the work in trial dividing. (If x^2==-1 mod N then for a prime factor p we have x^2==-1 mod p and so the jacobi symbol (-1/p)=1. But also (-1/p)=(-1)^((p-1)/2), hence must have p==1 mod 4.) But knowing whether sqrt(-1) mod N exists is not too easy. A strong pseudoprime test can reveal one, so perhaps such a test could be inserted part way though the dividing.

  Jon Grantham "Frobenius Pseudoprimes" (www.pseudoprime.com) describes a quadratic pseudoprime test taking about 3x longer than a plain test, but with only a 1/7710 chance of error (whereas 3 plain Miller-Rabin tests would offer only (1/4)^3 == 1/64). Such a test needs completely random parameters to satisfy the theory, though single-limb values would run faster. It's probably best to do at least one plain Miller-Rabin before any quadratic tests, since that can identify composites in less total time.

  Some thought needs to be given to the structure of which tests (trial division, Miller-Rabin, quadratic) and how many are done, based on what sort of inputs we expect, with a view to minimizing average time.

  It might be a good idea to break out subroutines for the various tests, so that an application can combine them in ways it prefers, if sensible defaults in `mpz_probab_prime_p` don't suit. In particular this would let applications skip tests it knew would be unprofitable, like trial dividing when an input is already known to have no small factors.

  For small inputs, combinations of theory and explicit search make it relatively easy to offer certainty. For instance numbers up to 2^32 could be handled with a strong pseudoprime test and table lookup. But it's rather doubtful whether a smallnum prime test belongs in a bignum library. Perhaps if it had other internal uses.

  An `mpz_nthprime` might be cute, but is almost certainly impractical for anything but small n.

- Intra-Library Calls

  On various systems, calls within libgmp still go through the PLT, TOC or other mechanism, which makes the code bigger and slower than it needs to be.

  The theory would be to have all GMP intra-library calls resolved directly to the routines in the library. An application wouldn't be able to replace a routine, the way it can normally, but there seems no good reason to do that, in normal circumstances.

  The `visibility` attribute in recent gcc is good for this, because it lets gcc omit unnecessary GOT pointer setups or whatever if it finds all calls are local and there's no global data references. Documented entrypoints would be `protected`, and purely internal things not wanted by test programs or anything can be `internal`.

  Unfortunately, on i386 it seems `protected` ends up causing text segment relocations within libgmp.so, meaning the library code can't be shared between processes, defeating the purpose of a shared library. Perhaps this is just a gremlin in binutils (debian packaged 2.13.90.0.16-1).

  The linker can be told directly (with a link script, or options) to do the same sort of thing. This doesn't change the code emitted by gcc of course, but it does mean calls are resolved directly to their targets, avoiding a PLT entry.

  Keeping symbols private to libgmp.so is probably a good thing in general too, to stop anyone even attempting to access them. But some undocumented things will need or want to be kept visible, for use by mpfr, or the test and tune programs. Libtool has a standard option for selecting public symbols (used now for libmp).