Issue 936813: fast modular exponentiation (original) (raw)
Created on 2004-04-17 08:16 by trevp, last changed 2022-04-11 14:56 by admin. This issue is now closed.
Messages (27)
Author: Trevor Perrin (trevp)
Date: 2004-04-17 08:16
For crypto-sized numbers, Python mod-exp is several times slower than GMP or OpenSSL (6x or more). Those libraries do crazy special-case stuff, + assembly, platform-specific tuning, and so on.
However, there's some low-hanging fruit: this patch has a few basic optimizations giving a 2-3x speedup for numbers in the 1000-8000 bit range (that's what I've mostly tested; but the patch should improve, or at least not hurt, everything else):
x_mul() is special-cased for squaring, which is almost twice as fast as general multiplication.
x_mul() uses pointers instead of indices for iteration, giving ~10% speedup (under VC6).
the right-to-left square-and-multiply exponentiation algorithm is replaced with a left-to-right square-and-multiply, which takes advantage of small bases.
when the exponent is above a certain size, "k-ary" exponentiation is used to reduce the number of multiplications via precalculation.
when the modulus is odd, Montgomery reduction is used.
the Karatsuba cutoff seems too low. For multiplicands in the range of 500-5000 bits, Karatsuba slows multiplication by around ~25% (VC6sp4, Intel P4M 1.7 Ghz). For larger numbers, the benefits of Karatsuba are less than they could be.
Currently, the cutoff is 35 digits (525 bits). I've tried 70, 140, 280, and 560. 70, 140, and 280 are roughly the same: they don't slow down small values, and they have good speedup on large ones. 560 is not quite as good for large values, but at least it doesn't hurt small ones.
I know this is platform-dependent, but I think we should err on the side of making the cutoff too high and losing some optimization, instead of putting it too low and slowing things down. I suggest 70.
A couple misc. things:
- Negative exponents with a modulus no longer give an error, when the base is coprime with the modulus. Instead, it calculates the multiplicative inverse of the base with respect to the modulus, using the extended euclidean algorithm, and exponentiates that.
Libraries like GMP and LibTomMath work the same way. Being able to take inverses mod a number is useful for cryptography (e.g. RSA, DSA, and Elgamal).
The diff includes patch 923643, which supports converting longs to byte-strings. Ignore the last few diff entries, if you don't want this.
I haven't looked into harmonizing with int_pow(). Something may have to be done.
Author: Trevor Perrin (trevp)
Date: 2004-07-13 08:04
Logged In: YES user_id=973611
Uploading 2nd version of longobject.diff - the only change is that patch 923643 is removed from this diff. That was a diff for converting longs to byte-strings, which I unnecessarily left in.
Author: Tim Peters (tim.peters) * 
Date: 2004-07-17 03:06
Logged In: YES user_id=31435
Notes after a brief eyeball scan:
Note that the expression
a & 1 == 1
groups as
a & (1 == 1)
in C -- comparisons have higher precedence in C than bit- fiddling operators. Stuff like that is usually best resolved by explicitly parenthesizing any "impure" expression fiddling with bits. In this case, in a boolean expression plain
a & 1
has the hoped-for effect. and is clearer anyway.
Would be better to use "**" than "^" in comments when exponentiation is intended, since "^" means xor in both Python and C.
Doc changes are needed, because you're changing visible semantics in some cases.
Tests are needed, especially for new semantics.
l_invmod can return NULL for more than one reason, but one of its callers ignores this, assuming that all NULL returns are due to lack of coprimality. It's unreasonable to, e.g., replace a MemoryError with a complaint about coprimality; this needs reworking. l_invmod should probably set an exception in the "not coprime" case. As is, it's a weird function, sometimes setting an exception when it returns NULL, but not setting one when coprimality doesn't obtain. That makes life difficult for callers (which isn't apparent in the patch, because its callers are currently ignoring this issue).
The Montgomery reduction gimmicks grossly complicate this patch -- they're long-winded and hard to follow. That may come with the territory, but it's the only part of the patch that made me want to vomit . I'd be happier if it weren't there, for aesthetic, clarity, and maintainability reasons. How much of a speedup does it actually buy?
You're right that int pow must deliver the same results as long pow, so code is needed for that too. "short int" versus "unbounded int" is increasingly meant to be an invisible internal implementation detail in Python. I'm also in favor of giving this meaning to modular negative exponents, btw, so no problem with that. An easy way would be to change int pow to delegate to long pow when this is needed.
Pragmatics: there's a better chance of making 2.4 if the patch were done in bite-size stages. For example, no doc changes are needed to switch to 5-ary left-to-right exponentation, and that has no effect on the int implementation either, etc. A patch that did just that much probably would have gone in a long time ago.
Author: Trevor Perrin (trevp)
Date: 2004-07-19 11:00
Logged In: YES user_id=973611
Tim, thanks for the feedback. I'm uploading a new patch against CVS latest that fixes those issues, and adds docs and tests. Also, I cleaned up the code quite a bit, and got it properly handling (I hope) all the varied combinations of ints/longs, positives/negatives/zeros etc..
Unfortunately, Montgomery is the bulk of the speedup: http://trevp.net/long_pow/
But I could split out the negative exponent handling into a separate patch, if that would be easier.
Anyways, I'd like to add more tests for the exponentiation stuff. Aside from that, I think the patch is complete. And more robust than previously, though I still wouldn't trust it until another person or two gives it a serious looking-over....
Author: Tim Peters (tim.peters) *
Date: 2004-07-21 19:29
Logged In: YES user_id=31435
Pragmatics are a real problem here, Trevor. I don't foresee being able to make a solid block of sufficient hours to give to reviewing this before Python 2.4 is history (which is why I've left this patch unassigned, BTW -- I just can't promise to make enough time). So if nobody else can volunteer to review it, that alone is likely to leave the patch sitting here unapplied.
But there are several independent changes in this patch, and it could be broken into several smaller patches. I tossed that bait out before, but you didn't bite. You should .
Author: Trevor Perrin (trevp)
Date: 2004-07-22 08:39
Logged In: YES user_id=973611
Pragmatics isn't my strong suit... but I get your drift :-). I split it into 3 diffs:
- x_mul optimizations: (pointers instead of indices, special-case squaring, changing Karatsuba cutoff)
- rewriting long_pow() for left-to-right 5-ary
- Montgomery reduction. This also includes l_invmod(), since it's necessary for Montgomery.
I've left out the code which exposes l_invmod() to the user (and associated docs, tests, and intobject changes). We could slap that on afterwards or not...
Anyways, these are applied sequentially: longobject.c + longobject1.diff = longobject1.c longobject1.c + longobject2.diff = longobject2.c longobject2.c + longobject2.diff = longobject3.c
Should I open new tracker items for them?
Author: Tim Peters (tim.peters) *
Date: 2004-08-29 22:21
Logged In: YES user_id=31435
Checked in the first part of the patch, with major format changes (Python's C coding standard is hard 8-column tabs), and minor code changes:
Include/longintrepr.h 2.15 Misc/ACKS 1.280 Misc/NEWS 1.1119 Objects/longobject.c 1.162
I don't know whether it's possible for me to get to part 2 of the patch before 2.4a3, but I would like to. It seems plainly impossible that I'll be able to get to part 3 before 2.4a3.
Author: Tim Peters (tim.peters) *
Date: 2004-08-30 02:47
Logged In: YES user_id=31435
Same deal with the 2nd part of the patch (major format changes, minor code changes). Incidentally fixed an old leak bug in long_pow() during the review. Added code to raise a compile-time error (C) if SHIFT isn't divisible by 5, and removed long_pow's new hardcoded assumption that SHIFT is exactly 15.
Include/longintrepr.h 2.16 Misc/NEWS 1.1120 Objects/longobject.c 1.163
This is cool stuff (& thank you!), but I'm sorry to say I can't foresee making time for the 3rd part of the patch for weeks.
Author: Trevor Perrin (trevp)
Date: 2004-09-13 08:20
Logged In: YES user_id=973611
Here's the 3rd part of the patch (long_mont.diff; Montgomery Reduction), diff'd against 2.4a3 and cleaned up a bit.
Note that this doesn't include negative exponent handling. If this patch is accepted, I'll make a new tracker item for that, since it's not an optimization, just an "opportunistic feature" (it builds on one of the helper functions needed for Montgomery).
Author: Trevor Perrin (trevp)
Date: 2004-10-04 05:43
Logged In: YES user_id=973611
I did more code review, testing, and timing. The only change in this new patch (long_mont2.diff) is a couple "int"s were changed to "digits"s, and it's against CVS head.
As far as testing, I used the random module and GMPY to check it on ~3 million random input values. That's about an hour of testing. I'll leave the tests running for a few days and see if anything crops up.
As far as timing, I updated the benchmarks with a new machine (OpenBSD): http://trevp.net/long_pow/ On 3 different machines, Montgomery gives a speedup of 2x, 3x, and 4x. That dwarfs what we've done so far, so I'm crossing my fingers for 2.4. Let me know if I can explain or improve the code, or anything..
(The below crypto library comes with a "book" which has an explanation of Montgomery I found helpful): http://math.libtomcrypt.org/download.html
Author: Trevor Perrin (trevp)
Date: 2004-10-04 07:48
Logged In: YES user_id=973611
oops. Good thing for random testing, carry propagation was buggy. Submitting long_mont3.diff.
Author: Trevor Perrin (trevp)
Date: 2004-10-05 07:25
Logged In: YES user_id=973611
Montgomery has a fixed cost, so it slows down small exponents. For example modular squaring is slowed ~5x. I added a MONTGOMERY_CUTOFF to take care of this. Submitting long_mont4.diff.
Author: Trevor Perrin (trevp)
Date: 2005-09-29 06:29
Logged In: YES user_id=973611
I updated this patch to CVS head, but didn't change it otherwise. It's still a bit hairy. However, it's also still a big speedup (see benchmarks from 2004-10-03).
If I can do anything to help this make it in 2.5, let me know.
Author: Christian Heimes (christian.heimes) *
Date: 2008-01-12 04:31
Re-targeting for 2.6 We should discuss it at the bug day.
Author: Christian Heimes (christian.heimes) *
Date: 2008-01-12 17:05
Mark, as the second math guru in our team, you are probably interested in these patches. Trevor has written an interesting patch to optimize longs for cryptographic problems. In fact it might be two patches now, one for the Montgomery Reduction and one containing other optimizations. The 2005 patch applies almost cleanly.
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-02-14 14:18
I'll see if I can find time to look at this; I'm currently looking at various ways to improve long integer arithmetic in 2.7/3.1.
Author: Trevor Perrin (trevp)
Date: 2009-02-19 20:39
Hi Mark,
Let me know if I can give you any help with this. The original patch was split into 3 parts. The only part remaining unapplied is the Montgomery Reduction.
It appeared to be a significant speedup when I was last testing, and is frequently used in other modular exponentiation libraries, but it does, admittedly, complicate the code.
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-02-20 16:16
Thanks, Trevor. I'm currently working on the 15-bit -> 30-bit digit change (issue 4258), since it seems sensible to get that in before considering other optimizations, but I certainly hope to find time to look at this before the 3.1 release cycle gets underway.
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-02-20 16:25
By the way, I'd be interested to know if you (Trevor) have any thoughts on the multiplication optimizations that are in the patch
30bit_longdigit13+optimizations.patch
in the issue 4258 discussion. These have been giving me some quite spectacular speedups (4 or 5 times faster) for multiplication on 64-bit machines; smaller speedups on 32-bit. Currently, those speedups render your earlier special-case-squaring patch obsolete; I wonder whether there's a way to get the same speedup for squaring.
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-12-29 21:07
This patch still(!) applies almost perfectly cleanly to trunk. On a 64- bit machine, I'm getting a failure in test_auto_overflow, coming from:
pow(0L, 0, 9223372036854775807) 28051505152L
I haven't looked hard to figure out where this is coming from, but my guess is that the 15-bitness of digits is hard-coded in the patch somewhere.
My general feeling is that three-argument pow is such a little-used operation in Python that it's not worth the extra code to speed it up.
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-12-30 13:55
Looks like the test failure is a result of a misplaced (twodigits) cast: replacing the line
carry += (twodigits) ( (*aptr) + (u * (*mptr++)) );
in function mont_reduce with
carry += *aptr + (twodigits)u * *mptr++;
fixes this.
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-12-30 19:02
Here's a slightly modified version of Trevor's patch:
- update to apply cleanly to trunk
- fix misplaced twodigits cast described above
- add 'PyLong_' prefix to BASE, MASK and SHIFT
- no need for _PyLong_Copy in l_invmod: just copy and INCREF the pointers
- don't calculate d - q*c by hand in l_invmod, since l_divmod computes the remainder already
- simplify reference counting in l_invmod
- add a '* 1U' to a digit-by-digit multiplication, to avoid possible (in theory; unlikely in practice) undefined behaviour resulting from signed integer overflow.
The rest of the patch looks fine to me, modulo some minor style issues.
Two points:
it seems that l_invmod is only ever used to compute the inverse of a single-digit long modulo PyLong_BASE. Mightn't it be more efficient to simply do a twodigit/digit-based calculation here instead, to reduce the overhead of setting up the Montgomery reductions?
the current algorithm for three-argument pow involves many allocations and deallocations of integers; it seems to me that these could almost all be eliminated: for pow(a, b, c) with c an n-digit PyLong, just allocate 2n (or possibly 2n+1) digits of workspace to begin with and use that to store intermediate products; the Montgomery reduction could then be done more-or-less in place. This doesn't work with the non- Montgomery method, though, since l_divmod doesn't operate in place.
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-12-31 15:20
Here's a second revision of Trevor's patch:
- factor out the code for creating Montgomery representatives; this simplifies the changes to the main long_pow function
- get rid of l_invmod and use a simple function for computing the negation of an inverse of a single digit modulo PyLong_BASE instead
- Montgomery reduction wasn't being used for multidigit exponent b if the last digit of the exponent is small. Fix this.
- add 'static' qualifier to mont_reduce function
- use type Py_ssize_t instead of int for digit indices in mont_reduce
- various other unimportant style fixes
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-12-31 16:32
Some timings on my machine (OS X 10.6, 64-bit nondebug build, trunk r77157). These are just doing an RSA-like powmod pow(c, d, n), with n the product of two similarly-sized primes, d the inverse of 7 modulo eulerPhi(n), and c of similar magnitude to n.
Without the patch:
Mark-Dickinsons-MacBook-Pro:trunk dickinsm$ ./python.exe ../time_powmod.py 64-bit modulus: 0.000031 253-bit modulus: 0.000274 1023-bit modulus: 0.008032
With the patch:
Mark-Dickinsons-MacBook-Pro:trunk-issue936813 dickinsm$ ./python.exe ../time_powmod.py 64-bit modulus: 0.000025 253-bit modulus: 0.000209 1023-bit modulus: 0.006717
So I'm seeing a speedup of 20-30%.
I've attached the (rather primtive) timing script. Anyone else want to contribute timings?
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-12-31 16:39
Hmm. For smaller inputs, I'm actually getting significant slowdowns:
Unpatched:
timeit('pow(123, 123456789, 123456789L)') 7.355183839797974
Patched:
timeit('pow(123, 123456789, 123456789L)') 8.873976945877075
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-12-31 16:59
One more lot of timings, from Trevor's pow_benchmark.txt:
Unpatched
1024 bits: 0.008256 2048 bits: 0.052324 3072 bits: 0.159689 4096 bits: 0.357264
Patched (percent speedup)
1024 bits: 0.006576 (+25.5%) 2048 bits: 0.045878 (+14.1%) 3072 bits: 0.135740 (+17.6%) 4096 bits: 0.310756 (+15.0%)
I'm not quite sure why I'm not seeing the same level of speedup that Trevor originally
reported. Perhaps this a result of some of the other optimizations that have been
applied to the long implementation (30-bit digits, a reworked division algorithm), or
perhaps the same level of speedup isn't available on 64-bit machines for some reason.
I hope I haven't somehow negated the effects of the patch in my refactoring.
Author: Mark Dickinson (mark.dickinson) *
Date: 2009-12-31 19:39
Okay, I retested the original patch without any of my refactoring (besides fixing the twodigits cast), and got pretty much the same numbers.
On a 32-bit non-debug trunk build (still on OS X 10.6), I get:
Unpatched
Mark-Dickinsons-MacBook-Pro:trunk dickinsm$ ./python.exe ../pow_benchmark.py 1024 bits: 0.033691 2048 bits: 0.224796 3072 bits: 0.712510 4096 bits: 1.691484 Mark-Dickinsons-MacBook-Pro:trunk dickinsm$ ./python.exe ../time_powmod.py 64-bit modulus: 0.000054 253-bit modulus: 0.000981 1023-bit modulus: 0.034314
Patched
Mark-Dickinsons-MacBook-Pro:trunk-issue936813 dickinsm$ ./python.exe ../pow_benchmark.py 1024 bits: 0.027317 (+23.3%) 2048 bits: 0.181053 (+24.2%) 3072 bits: 0.571688 (+24.6%) 4096 bits: 1.251051 (+35.2%) Mark-Dickinsons-MacBook-Pro:trunk-issue936813 dickinsm$ ./python.exe ../time_powmod.py 64-bit modulus: 0.000045 (+20.0%) 253-bit modulus: 0.000773 (+26.9%) 1023-bit modulus: 0.026983 (+27.2%)
I must admit I was hoping for a bit more than this. IMO, these speedups aren't big enough to justify the extra complexity for what's really a bit of a niche function, so I'm going to reject this 3rd part and close this issue (but marking it as 'accepted' because most of the original 2004 patch went in).
History
Date
User
Action
Args
2022-04-11 14:56:03
admin
set
github: 40160
2009-12-31 19:39:35
mark.dickinson
set
status: open -> closed
resolution: accepted
messages: +
stage: patch review -> resolved
2009-12-31 16:59:39
mark.dickinson
set
messages: +
2009-12-31 16:39:22
mark.dickinson
set
messages: +
2009-12-31 16:32:25
mark.dickinson
set
files: + time_powmod.py
messages: +
2009-12-31 15:20:26
mark.dickinson
set
files: + long_pow_2009_12_31.diff
messages: +
2009-12-30 19:02:41
mark.dickinson
set
files: + long_pow_2009_12_30.diff
messages: +
2009-12-30 13:55:37
mark.dickinson
set
messages: +
2009-12-29 21:07:11
mark.dickinson
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messages: +
2009-02-20 16:25:58
mark.dickinson
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messages: +
2009-02-20 16:16:53
mark.dickinson
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messages: +
2009-02-19 21:02:04
gregory.p.smith
set
nosy: + gregory.p.smith
2009-02-19 20:39:49
trevp
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messages: +
2009-02-14 14🔞07
mark.dickinson
set
assignee: tim.peters -> mark.dickinson
messages: +
2009-02-14 13:53:47
ajaksu2
set
stage: patch review
versions: + Python 3.1, Python 2.7, - Python 2.6, Python 3.0
2008-01-12 17:05:02
christian.heimes
set
nosy: + mark.dickinson
messages: +
versions: + Python 3.0
2008-01-12 04:31:15
christian.heimes
set
nosy: + christian.heimes
type: enhancement
messages: +
versions: + Python 2.6, - Python 2.5
2004-04-17 08:16:31
trevp
create