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[We now have two implementations of this algorithm, one by Tommy Färnqvist and one by Niels Möller.]
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.
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:
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 sometimes possible to remedy the situation using floating-point operations or SIMD operations such as MMX (x86) (x86), SSE (x86), VMX (PowerPC), VIS (Sparc).
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 pieces, so that no intermediates are 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!)
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 iteration:
2 x = x (3 − A x )/2 i+1 i iThe 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.
We should probably allow a special exponent-like parameter, to speed computations of a precise square root of a small number in mpf and mpfr.
Improve mpn_rootrem. The current code is not too bad, but its time complexity is a function of the input, while it is possible to make the average complexity a function of the output.
Add more functions to the set of fat functions.
The speed of multiplication is today highly dependent on combination
functions like addlsh1_n. A fat binary will never use any such
functions, since they are classified as optional. Ideally, we should use
them, but making the current compile-time selections of optional functions
become run-time selections for fat binaries.
If we make fat binaries work really well, we should move away frm tehe current configure scheme (at least by default) and instead include all code always.
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.
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,
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.
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.
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.
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).
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.
Note that the mpfr functions already provide these functions, and that we usually recommend new programs to use mpfr instead of mpf.