diff options
Diffstat (limited to 'vendor/gmp-6.3.0/doc/projects.html')
| -rw-r--r-- | vendor/gmp-6.3.0/doc/projects.html | 476 |
1 files changed, 476 insertions, 0 deletions
diff --git a/vendor/gmp-6.3.0/doc/projects.html b/vendor/gmp-6.3.0/doc/projects.html new file mode 100644 index 0000000..34bbb52 --- /dev/null +++ b/vendor/gmp-6.3.0/doc/projects.html @@ -0,0 +1,476 @@ +<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> +<html> +<head> + <title>GMP Development Projects</title> + <link rel="shortcut icon" href="favicon.ico"> + <link rel="stylesheet" href="gmp.css"> + <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> +</head> + +<center> + <h1> + GMP Development Projects + </h1> +</center> + +<font size=-1> +<pre> +Copyright 2000-2006, 2008-2011 Free Software Foundation, Inc. + +This file is part of the GNU MP Library. + +The GNU MP Library is free software; you can redistribute it and/or modify +it under the terms of either: + + * the GNU Lesser General Public License as published by the Free + Software Foundation; either version 3 of the License, or (at your + option) any later version. + +or + + * the GNU General Public License as published by the Free Software + Foundation; either version 2 of the License, or (at your option) any + later version. + +or both in parallel, as here. + +The GNU MP Library is distributed in the hope that it will be useful, but +WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY +or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License +for more details. + +You should have received copies of the GNU General Public License and the +GNU Lesser General Public License along with the GNU MP Library. If not, +see https://www.gnu.org/licenses/. +</pre> +</font> + +<hr> +<!-- NB. timestamp updated automatically by emacs --> + This file current as of 29 Jan 2014. An up-to-date version is available at + <a href="https://gmplib.org/projects.html">https://gmplib.org/projects.html</a>. + Please send comments about this page to gmp-devel<font>@</font>gmplib.org. + +<p> This file lists projects suitable for volunteers. Please see the + <a href="tasks.html">tasks file</a> for smaller tasks. + +<p> If you want to work on any of the projects below, please let + gmp-devel<font>@</font>gmplib.org know. If you want to help with a project + that already somebody else is working on, you will get in touch through + gmp-devel<font>@</font>gmplib.org. (There are no email addresses of + volunteers below, due to spamming problems.) + +<ul> +<li> <strong>Faster multiplication</strong> + + <ol> + + <li> Work on the algorithm selection code for unbalanced multiplication. + + <li> 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. <br><br> + + <p> [We now have two implementations of this algorithm, one by Tommy + Färnqvist and one by Niels Möller.] + + <li> Work on short products. Our mullo and mulmid are probably K, but we + lack mulhi. + + </ol> + + <p> 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. + + +<li> <strong>Assembly routines</strong> + + <p> 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. + + <p> Please make sure your new routines are fast for these three situations: + <ol> + <li> Small operands of less than, say, 10 limbs. + <li> Medium size operands, that fit into the cache. + <li> Huge operands that does not fit into the cache. + </ol> + + <p> 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. + + <p> 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). + + <p> 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!) + + +<li> <strong>Faster sqrt</strong> + + <p> 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: + <pre> + 2 + x = x (3 − A x )/2 + i+1 i i </pre> + The square root can then be computed like this: + <pre> + sqrt(A) = A x + n </pre> + <p> 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. + + <p> 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. + + +<li> <strong>Nth root</strong> + + <p> 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 <i>average</i> complexity a function of the output. + + +<li> <strong>Fat binaries</strong> + + <p> Add more functions to the set of fat functions. + + <p> The speed of multiplication is today highly dependent on combination + functions like <code>addlsh1_n</code>. 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. + + <p> 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. + + +<li> <strong>Exceptions</strong> + + <p> 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 <code>gmp_errno</code> + is not actually documented, except for the old <code>gmp_randinit</code> + function. Being currently just a plain global means it's not + thread-safe. + + <p> 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 <code>longjmp</code> or + similar recovery from it may be difficult. A combination would be + possible, for instance by allowing the handler to return an error code. + + <p> 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. + + <p> 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. + + <p> A C++ <code>throw</code> might be a good optional extra exceptions + mechanism, perhaps under a build option. For + GCC <code>-fexceptions</code> will add the necessary frame information to + plain C code, or GMP could be compiled as C++. + + <p> Out-of-memory exceptions are expected to be handled by the + <code>mp_set_memory_functions</code> 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 <code>mpn_add_n</code> and + <code>mpn_addmul_1</code> 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. + + <p> A <code>longjmp</code> 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. + + <p> One scheme for out-of-memory would be to note that all GMP allocations go + through the <code>mp_set_memory_functions</code> routines. So if the + application has an intended <code>setjmp</code> 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 <code>malloc</code>. 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. + + <p> Applications desiring a fine-grained error control, for instance a + language interpreter, would very possibly not be well served by a scheme + requiring <code>longjmp</code>. Wrapping every GMP function call with a + <code>setjmp</code> would be very inconvenient. + + <p> Another option would be to let <code>mpz_t</code> 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 <code>mpn</code> 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 + <code>mpz_t</code> etc structure for them to indicate failure in. + + <p> 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 <code>alloca</code>, but + merely adjusts <code>%esp</code>. A big enough <code>alloca</code> 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 + <code>--enable-alloca=malloc-reentrant</code> (or + <code>notreentrant</code>). Arranging the default to be + <code>alloca</code> only on blocks up to a certain size and + <code>malloc</code> thereafter might be a better approach and would have + the advantage of not having calculations limited by available stack. + + <p> Actually recovering from stack overflow is of course another problem. It + might be possible to catch a <code>SIGSEGV</code> in the stack redzone + and do something in a <code>sigaltstack</code>, 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. + + <p> Operand overflow is another exception to be addressed. It's easy for + instance to ask <code>mpz_pow_ui</code> for a result bigger than an + <code>mpz_t</code> 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 + <code>mpz_mul</code> won't overflow. + + +<li> <strong>Performance Tool</strong> + + <p> 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, + + <ol> + <li> Checking speed variations between compilers. + <li> Checking relative performance between systems or CPUs. + </ol> + + <p> A combination of measuring some fundamental routines and some + representative application routines might satisfy these. + + <p> The tune/time.c routines would be the easiest way to get good accurate + measurements on lots of different systems. The high level + <code>speed_measure</code> may or may not suit, but the basic + <code>speed_starttime</code> and <code>speed_endtime</code> would cover + lots of portability and accuracy questions. + + +<li> <strong>Using <code>restrict</code></strong> + + <p> There might be some value in judicious use of C99 style + <code>restrict</code> on various pointers, but this would need some + careful thought about what it implies for the various operand overlaps + permitted in GMP. + + <p> Rumour has it some pre-C99 compilers had <code>restrict</code>, but + expressing tighter (or perhaps looser) requirements. Might be worth + investigating that before using <code>restrict</code> unconditionally. + + <p> 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 <code>_Pragma</code> gives an equivalent effect. + + <p> One thing to note is that Microsoft C headers (on ia64 at least) contain + <code>__declspec(restrict)</code>, so a <code>#define</code> of + <code>restrict</code> should be avoided. It might be wisest to setup a + <code>gmp_restrict</code>. + + +<li> <strong>Prime Testing</strong> + + <p> 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. + + <p> The present "repetitions" argument to <code>mpz_probab_prime_p</code> 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. + + <p> <code>mpz_probab_prime_p</code> always initializes a new + <code>gmp_randstate_t</code> 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. + + <p> <code>mpn_mod_34lsub1</code> 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 +<pre> +#define MP_LIMB_DIVISIBLE_7_P(n) \ + ((n) * MODLIMB_INVERSE_7 <= MP_LIMB_T_MAX/7) +</pre> + 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. + + <p> 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. <code>udiv_qrnnd_preinv</code> 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. + + <p> 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. + + <p> 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. + + <p> 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. + + <p> 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 <code>mpz_probab_prime_p</code> 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. + + <p> 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. + + <p> An <code>mpz_nthprime</code> might be cute, but is almost certainly + impractical for anything but small n. + + +<li> <strong>Intra-Library Calls</strong> + + <p> 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. + + <p> 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. + + <p> The <code>visibility</code> 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 <code>protected</code>, and purely + internal things not wanted by test programs or anything can be + <code>internal</code>. + + <p> Unfortunately, on i386 it seems <code>protected</code> 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). + + <p> 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. + + <p> 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). + + +<li> <strong>Math functions for the mpf layer</strong> + + <p> 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. + + <p> Note that the <a href="https://www.mpfr.org/">mpfr</a> functions already + provide these functions, and that we usually recommend new programs to use + mpfr instead of mpf. +</ul> +<hr> + +</body> +</html> + +<!-- +Local variables: +eval: (add-hook 'write-file-hooks 'time-stamp) +time-stamp-start: "This file current as of " +time-stamp-format: "%:d %3b %:y" +time-stamp-end: "\\." +time-stamp-line-limit: 50 +End: +--> |