[Python-bugs-list] [ python-Bugs-457066 ] pow(a,b,c) should accept b<0

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Sat, 01 Sep 2001 13:04:48 -0700


Bugs item #457066, was opened at 2001-08-30 18:17
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Category: Python Library
Group: Feature Request
Status: Open
Resolution: Rejected
Priority: 5
Submitted By: Nobody/Anonymous (nobody)
Assigned to: Tim Peters (tim_one)
Summary: pow(a,b,c) should accept b<0

Initial Comment:
You should be able to raise integers to negative powers
mod n.  If b<0, pow(a,b,c)==pow(pow(a,-1,c),-b,c)
where pow(a,-1,c) is a's multiplicative inverse mod c,
computed with the extended Euclid algorithm.  This
would be in Python's spirit of math completeness and
would save people from the trouble of having to figure
out the algorithm over and over.  

I can come up with a patch for this if it's agreed on
as desirable.


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>Comment By: Tim Peters (tim_one)
Date: 2001-09-01 13:04

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Changed Resolution to None since this was opened again.

I still don't like this.  It's a wart no matter how you cut 
it:  implement the egcd meaning, and it's still a wart, 
because the "multiplicative inverse" meaning doesn't always 
make sense.  For example, pow(4, -1, 6) -- 4 has no 
multiplicative inverse mod 6.  The best it can return is 2, 
i.e. the best pow(i, -1, k) can return is an integer x s.t. 
i*x is congruent to gcd(i, k) mod k.  But Python provides 
no way to get the gcd, so there's no way (short of 
computing gcd separately) to know what the result of pow
(i, -1, k) really means (and it doesn't mean "inverse" 
unless the gcd is 1; OTOH, raise an exception if the gcd 
isn't one, and then you've artificially ruled out 
legitimate uses of egcd apparently not related to Paul's 
particular interest today).

The natural way to spell egcd as a library routine would 
return the gcd too; e.g.,

def egcd(aorig, borig):
.    """Return (g, i) s.t. g=gcd(aorig, borig) and i*aorig 
% borig = g."""
.    a, b = aorig, borig
.    a1, a2 = 1, 0
.    while b:
.        q, r = divmod(a, b)
.        a1, a2 = a2, a1-q*a2
.        a, b = b, r
.    if __debug__:
.        b1, r = divmod(a - a1*aorig, borig)
.        assert r == 0
.        assert a1*aorig + b1*borig == a
.    return a, a1

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Comment By: Guido van Rossum (gvanrossum)
Date: 2001-09-01 11:19

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Hm. There's something to say for making 3-arg pow() only
work for ints (and longs), and then the egcd would make
sense. But that would mean removing the 3-arg pow() for
floats. Why would anyone use 3-arg pow() with floats? I
don't know, but that doesn't mean it doesn't happen. *If*
there are no community objections against making 3-arg pow()
raise a TypeError on float or complex arguments, I'm OK with
the egcd interpretation. But this is PEP material -- that's
the only way to find out. phr, would you mind writing a PEP?
It can be short and sweet.

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Comment By: paul rubin (phr)
Date: 2001-08-31 23:38

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Argh.  I meant to say I'm NOT enthused about changing /.
This item is jinxed :-).

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Comment By: Nobody/Anonymous (nobody)
Date: 2001-08-31 23:34

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Making a 3-arg integer pow return a tiny floating point
number seems weird to me.  I don't see any situation
where I'd want that.  If I call pow with b<0 without
expecting it to use the egcd to get the integer answer mod
c, I've almost certainly made a mistake.  So my first
preference is still egcd, but second preference is to stay
with the old behavior of throwing an exception rather than
have my integer calculation suddenly turn into a floating
point calculation without my realizing it.  

I'm also enthused about changing / to turn 2/3 into a
float, but at least I understand the argument that 2/3=0
confuses newbies.  But newbies won't be using 3-arg pow(),
so we needn't worry about confusing them.  IMHO anyone using
3-arg pow on integers will expect an integer result.

By the way (off topic), 3-arg pow with ~150 decimal digits
is about 5x slower in Python than in carefully optimized
asm on an x86, which is pretty good.  But on a StrongARM
it appears to be about 30x slower :-(.  This can't really
be fixed without adding a lot more code.  Sigh.

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Comment By: Tim Peters (tim_one)
Date: 2001-08-30 22:23

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The desire is understandable but this isn't the right way 
to do it, so I'm rejecting this.  While 2.2a2 changed the 
specifics, the general rule remains that

pow(a, b, c) == a**b % c

except that the LHS may be very much faster for large 
integer arguments.

"The right way" to do modular arithmetic is to define a 
class for it, and do the full job (+ - * /, not just 
modular pow).  egcd is easy to code in Python, and because 
it's an obscure speciality need (it gets asked about maybe 
twice per year on c.l.py) doesn't really belong in the core 
even if it weren't.  I'm not even sure how 3-argument pow 
got in, but am grateful it did and don't want to press our 
luck <wink>.

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Comment By: Guido van Rossum (gvanrossum)
Date: 2001-08-30 18:49

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Of course I meant (1.0/(a**-b))%c. Sorry!

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Comment By: Guido van Rossum (gvanrossum)
Date: 2001-08-30 18:46

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Hm.

In 2.2a2, currently, pow(a, b, c) for ints a, b, c where b <
0 is defined as pow(float(a), float(b), float(c)), IOW
(1.0/(a**b))%c. This doesn't make a lot of sense for the
three-argument version though, because the result tends to
be between 0.0 and 1.0...  But it is consistent with the
(future) rule that operations on integers and floats should
give results with the same value only a different type.

Assigning to Tim, whose mathematical sensibilities are more
refined than mine...

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