Cryptography-Digest Digest #72, Volume #11 Tue, 8 Feb 00 14:13:02 EST
Contents:
Re: permission to do crypto research (Bill Unruh)
Re: Strip Security ("karl malbrain")
Re: Hill Climbing ("Michael Darling")
Re: NSA opens up to US News (wtshaw)
Re: Seeking Information on FRACTAL CRYPTOGRAPHY (Tim Tyler)
Re: Elliptic and Rivest ([EMAIL PROTECTED])
Re: Hill Climbing (Mark VandeWettering)
Re: Hill Climbing (Mark VandeWettering)
Re: Anti-crack (Glenn Larsson)
Re: Strip Security ("Brian Keener")
Re: Hill Climbing (Mark VandeWettering)
Re: Hill Climbing (Jim Gillogly)
Re: DVD crypt Q (Troed)
Re: question about PKI... (Mike Rosing)
Re: Strip Security ("karl malbrain")
Re: Latin Squares (was Re: Reversibly combining two bytes?) ("r.e.s.")
----------------------------------------------------------------------------
From: [EMAIL PROTECTED] (Bill Unruh)
Crossposted-To: talk.politics.crypto,misc.int-property,misc.legal.computing
Subject: Re: permission to do crypto research
Date: 8 Feb 2000 17:15:11 GMT
In <87l12q$96o$[EMAIL PROTECTED]> [EMAIL PROTECTED] (Paul Rubin) writes:
>In article <87k55g$i5o$[EMAIL PROTECTED]>,
>Roger Schlafly <[EMAIL PROTECTED]> wrote:
>>According to US copyright law on "circumvention of copyright
>>protection systems" (17 US 1201), certain encryption research
>>is permissable only if "the person made a good faith effort to
>>obtain authorization before the circumvention".
>Is the US Copyright Patrol supposed to have jurisdiction in Norway?
Apparently they do. People have been arrested and charged.
This law has always been totally stupid and flies in the face of
centuries of copyright legal theory. It is completely a case of certain
big businesses attempting to impose of Soviet model of the economy on
the USA-- State granted monopolies, with the full force of the law to
punish those who question.
------------------------------
Reply-To: "karl malbrain" <[EMAIL PROTECTED]>
From: "karl malbrain" <[EMAIL PROTECTED]>
Crossposted-To: comp.sys.palmtops.pilot,alt.comp.sys.palmtops.pilot,comp.sys.handhelds
Subject: Re: Strip Security
Date: Tue, 8 Feb 2000 09:16:43 -0800
Steven G. Tyler <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]...
> karl malbrain wrote:
>
> > Highdesertman <[EMAIL PROTECTED]> wrote...
>
> > > I am wondering how you arrived at 10,000 possible combinations with a
> > > four digit pin. I don't doubt it is correct, . . .
>
> > The formula for 10 things (the digits 0 - 9) taken 6 at a time with
> > replacement is: 10*10*10*10*10*10.
>
> Formulae aside, remember also that the number of "possible combinations"
> must be calculated on the basis of the largest allowable PIN, rather
> than the size of the actual PIN. In other words, you might have actually
> used a 4-digit PIN, but if the encryption software will allow up to,
> say, 12 digits, a would-be cracker can't just assume the smaller number
> but has to deal with the (much) larger number of combinations generated
> by the larger number.
What on earth does this mean? If you are placing a 4-digit PIN ahead of a
12-digit encryption algorithm, there are still only 10,000 inputs. Perhaps
you mean that not all 10,000 PINs are actually used? Karl M
------------------------------
From: "Michael Darling" <[EMAIL PROTECTED]>
Subject: Re: Hill Climbing
Date: Tue, 8 Feb 2000 17:14:12 -0000
Thanks for this. I'll definitely add these to my mutation possibilities.
How about for a certain number of the top percentages you did all of these
things - would
that be a good idea?
regards,
Mike.
> I also found this and asked Mark directly about his approach. What's
really
> needed is just a good spread of "mutations" (it seems). What can happen in
a
> Playfair grid?
> No mutation
> Single letter swaps
> Row/row or column/column swaps
> Row/column swap
> Row/row or column/column inversions
> Totally new grid
> etc.
>
> The difficult part is deciding what ratios to assign each of these
mutations
> to the children. The highest percentage in mine left the grid well alone,
> and the others were held around 5%. The hill-climbing algorithm was also
> treated as a type of mutation which happened about 0.01% of the time
(rather
> more often that Mark suggested). It slows down the mutation cycle
> *enormously*, but also provides the quickest convergence.
>
> To be honest I think of my program as primarily a hill-climber, which the
GA
> simply providing a good random starting point selection. Once I finally
had
> it working in the way I had anticipated it found the "almost solution"
after
> 16 generations. A small amount of hand tweaking of the grid then solved it
> properly. Only a few letters were bad in the plaintext anyway, so it could
> be read without a problem.
>
> Stan
------------------------------
From: [EMAIL PROTECTED] (wtshaw)
Subject: Re: NSA opens up to US News
Date: Tue, 08 Feb 2000 10:34:31 -0600
In article <[EMAIL PROTECTED]>, Terje Mathisen
<[EMAIL PROTECTED]> wrote:
> By inventing a Base-72 encoding scheme, email attachments could
> theoretically be transferred a little bit more efficiently, but the
> gains are probably too small to matter. :-(
>
Consider that normal text uses few UC letters. Select a rare character
such as accent grave and simply place that ahead of a letter to be
capitalized. As a by product, you get a bunch of other ready codes beyond
the alphabet which are free for assignment, including a double ag to stand
for one of them, then there are no missing common keyboard characters.
The gains in text handling are worth it. instead of 94+ characters, since
47 keys less 26 is 21, you can handle something like 94+20, including
characters which can be dedicated to format stuff. Space should be coded
too, taking another base key since spaces are so common. I find these
things quite fun, especially when they are made to work.
--
Life is full of upturns and downturns, with varying periods of
stabilty mixed in. It is a fool's errand to assume that what is
happening any one day predicts the same as a constant future.
------------------------------
From: Tim Tyler <[EMAIL PROTECTED]>
Subject: Re: Seeking Information on FRACTAL CRYPTOGRAPHY
Reply-To: [EMAIL PROTECTED]
Date: Tue, 8 Feb 2000 17:11:50 GMT
M. Hackett <[EMAIL PROTECTED]> wrote:
: I am seeking information on FRACTAL CRYPTOGRAPHY -- patents, programs and
: / or otherwise available on the Internet.
: Send me any links or information that you may find, as I am having some
: trouble assimilating this info.
http://chilidog.lascruces.nmhs.edu/~wbaker/tech2.html explicitly relates
fractals to cryptography. It appears to be from a high school.
Since - to most people's ears - "fractal cryptography" immediately sounds
like snake oil, it may be that most people who use fractal-like
iterated nonlinear functions to generate structures such as chaotic
avalanches as part of their cryptosystems give the term a wide berth.
--
__________
|im |yler The Mandala Centre http://www.mandala.co.uk/ [EMAIL PROTECTED]
There's no fuel like an old fuel.
------------------------------
From: [EMAIL PROTECTED]
Subject: Re: Elliptic and Rivest
Date: Tue, 08 Feb 2000 17:24:44 GMT
In article <
31ov9s49j0ho69n7tsoooo0chv07ulm8gq@4ax.
com>,
Paris Cristiano <[EMAIL PROTECTED]> wrote:
> Hi !
>
> Do you know where I can get the paper "Expose the Eavesdropper" by
> Rivest in which he shows the Interlock protocol (or any article that
> talks about it) ?
>
The Rivest paper is in Communications of the
ACM, vol. 27, pages 393-395, April 1984.
> Besides, I'm looking for a good introduction on Elliptic Curve
> Cryptography...
>
> Please help, thanks !
>
You might try the survey article "Elliptic
Curve Cryptosystems" by Robshaw & Yin which
you can probably get a copy of from
rsasecurity.com. For books consider "Elliptic
Curves in Cryptography" by Seroussi, et al. or
"Implementing Elliptic Curve Cryptography" by
Rosing. There is an intoductory book by
Andreas Enge which might be very good but it
costs $115. If you order one of these books
from, say amazon.com, and don't like it then
you can get a refund.
P.S. If I go to jail for pretending to be the NSA
then you can return the favor by sending me an
encrypted cake with frosting and a saw on the
inside.
- da NSA
Sent via Deja.com http://www.deja.com/
Before you buy.
------------------------------
From: Mark VandeWettering <[EMAIL PROTECTED]>
Subject: Re: Hill Climbing
Date: Tue, 08 Feb 2000 09:07:57 -0800
Michael Darling wrote:
>
> After reading the Mark VandeWettering post on the CypherChallenge I've built
> myself a Trigraph Frequency Dictionary
> and I am going to attempt his 5000 squares method.
>
> He doesn't really mention how to mutate his chosen squares so I'm just going
> to do random swapping of letters to
> begin with, and see how that goes.
Golly, it's nice that so many people are goofing around with this, I
thought I
might pop my head in and give a few comments...
My mutation operators were:
do nothing...
swap a row...
swap a column...
swap two letters...
replace the square with a random new one...
and "optimize".
Optimize is a procedure which iteratively selects the best pair of
letters in a
square and swaps them until no further improvement can be made. This
procedure
was invoked very seldom (1/1000), but without it the method converges
rather slowly.
Doing it more often may cause it to converge in fewer generations, but
it is
expensive enough that it doesn't help get the result in any less wall
clock time.
My GA was entirely mutation based, never could figure out how to make
crossover work.
You can experiment with percentages on which mutation operation to
perform, I basically
tried to make drastic moves (optimize, replace with random) very low
probability, and less
drastic ones more often. The chance of doing nothing was approximately
50%.
Hope this helps....
Mark
--
Mark T. VandeWettering Telescope Information (and more)
Email: <[EMAIL PROTECTED]> http://raytracer.org
------------------------------
From: Mark VandeWettering <[EMAIL PROTECTED]>
Subject: Re: Hill Climbing
Date: Tue, 08 Feb 2000 09:13:14 -0800
G Winstanley wrote:
>
> On Tue, 8 Feb 2000 09:14:56 -0000, the cup of "Michael Darling"
> <[EMAIL PROTECTED]> overfloweth with the following:
> I also found this and asked Mark directly about his approach. What's really
> needed is just a good spread of "mutations" (it seems). What can happen in a
> Playfair grid?
> No mutation
> Single letter swaps
> Row/row or column/column swaps
> Row/column swap
> Row/row or column/column inversions
Later versions of my software used row and column reversals, I don't
swap rows with columns.
I solved the original code without any row/column reversals.
> Totally new grid
> etc.
>
> The difficult part is deciding what ratios to assign each of these mutations
> to the children. The highest percentage in mine left the grid well alone,
> and the others were held around 5%. The hill-climbing algorithm was also
> treated as a type of mutation which happened about 0.01% of the time (rather
> more often that Mark suggested). It slows down the mutation cycle
> *enormously*, but also provides the quickest convergence.
Yep, that sounds similar to mine.
> To be honest I think of my program as primarily a hill-climber, which the GA
> simply providing a good random starting point selection. Once I finally had
> it working in the way I had anticipated it found the "almost solution" after
> 16 generations. A small amount of hand tweaking of the grid then solved it
> properly. Only a few letters were bad in the plaintext anyway, so it could
> be read without a problem.
I think some interesting work can be done to try to optimize the weights
and costs
of various strategies to reduce the time it takes to solve ciphers.
Some time
after I break the remaining 4 stages, :-) perhaps I'll take time to do
so.
Mark
>
> Stan
--
Mark T. VandeWettering Telescope Information (and more)
Email: <[EMAIL PROTECTED]> http://raytracer.org
------------------------------
From: Glenn Larsson <[EMAIL PROTECTED]>
Reply-To: [EMAIL PROTECTED]
Subject: Re: Anti-crack
Date: Tue, 08 Feb 2000 18:39:08 +0100
> That may be right, but if it takes the cracker a sufficient amount
> of time more than without the protection, it might already pay - the
> product cycles today are so quick, the "brake" effect might
> be enough to stop them dead.
So you're suggesting a product cycle lifespan of 24 hours to a week?
Wake up.
Glenn
Ex cracker
Sweden
_________________________________________________
Spammers will be reported to their government and
Internet Service Provider along with possible legal
reprocussions of violating the Swedish "Personal
Information Act" of 1998. (PUL 1998:204)
This is punishable by a fine or 6 month to 2 years
imprisonment (Paragraph 49)
------------------------------
From: "Brian Keener" <[EMAIL PROTECTED]>
Crossposted-To: comp.sys.palmtops.pilot,alt.comp.sys.palmtops.pilot,comp.sys.handhelds
Subject: Re: Strip Security
Date: Tue, 8 Feb 2000 12:53:49 -0500
"karl malbrain" <[EMAIL PROTECTED]> wrote in message
news:1bYn4.2217$[EMAIL PROTECTED]...
>
> Steven G. Tyler <[EMAIL PROTECTED]> wrote in message
> news:[EMAIL PROTECTED]...
> > karl malbrain wrote:
> >
> > > Highdesertman <[EMAIL PROTECTED]> wrote...
> >
> > > > I am wondering how you arrived at 10,000 possible combinations with
a
> > > > four digit pin. I don't doubt it is correct, . . .
> >
> > > The formula for 10 things (the digits 0 - 9) taken 6 at a time with
> > > replacement is: 10*10*10*10*10*10.
> >
> > Formulae aside, remember also that the number of "possible combinations"
> > must be calculated on the basis of the largest allowable PIN, rather
> > than the size of the actual PIN. In other words, you might have actually
> > used a 4-digit PIN, but if the encryption software will allow up to,
> > say, 12 digits, a would-be cracker can't just assume the smaller number
> > but has to deal with the (much) larger number of combinations generated
> > by the larger number.
>
> What on earth does this mean? If you are placing a 4-digit PIN ahead of a
> 12-digit encryption algorithm, there are still only 10,000 inputs.
Perhaps
> you mean that not all 10,000 PINs are actually used? Karl M
>
>
If max password size is 12 characters, then you've got more possible
combinations. However, you might only have a password that's 4 characters
long, out of a possible maximum of 12. But, anyone trying to guess your
password won't know if you used all twelve digits or not.
Brian K
------------------------------
From: Mark VandeWettering <[EMAIL PROTECTED]>
Subject: Re: Hill Climbing
Date: Tue, 08 Feb 2000 09:51:35 -0800
"Douglas A. Gwyn" wrote:
>
> Michael Darling wrote:
> > He doesn't really mention how to mutate his chosen squares so I'm
> > just going to do random swapping of letters to begin with, ...
>
> Of course, you don't want to *just* swap at random -- the swapping
> should be directed toward the goal (improving the overall score),
> so mainly you want to swap only if it improves the score. The use
> for randomness is to keep from getting stuck on a local maximum;
> by frequently injecting a random swap, you keep alive the chance
> of migrating to another (taller) hill in the domain. This has a
> connection to both "simulated annealing" and "genetic" methods.
Actually, I'm not so sure that's a great idea.
My program does just do random mutations, without regard as to whether
they
are better or worse. It relies on the selection process to weed out bad
mutations. By only allowing mutations which improve, you are
effectively
implementing parallel hill climbing, no different than just sequentially
generating a random initial position, trying to improve it, and
eventually giving up and starting from other random place. The chance
of
finding the final solution is good if the final solutions space ramps
smoothly
to the final solution, but if it is "hilly", you have excellent chances
of
being caught in local minima.
My playfair cipher solver scored 90% of the letters correctly during one
run,
and required another 8 generations to improve to the final score. It
would be
interesting to see if _any_ of the local mutation operations resulted in
an
improvement, or whether you needed to go back to go forward.
I'm no expert on genetic algorithms, but I did ask an expert friend
about this, and
my reasoning is (at least to him) fairly sound.
Anyway, excellent fun, and good things to experiment with. :-)
Mark
--
Mark T. VandeWettering Telescope Information (and more)
Email: <[EMAIL PROTECTED]> http://raytracer.org
------------------------------
From: Jim Gillogly <[EMAIL PROTECTED]>
Subject: Re: Hill Climbing
Date: Tue, 08 Feb 2000 18:12:04 +0000
G Winstanley wrote:
> Hence I used them. Interestingly, I couldn't get it to converge nearly as
> fast when using tetragraph frequencies. I understand that with digraph
> frequencies you will get false positives due to the digraph nature of the
> cipher; does this somehow carry over to tetragraphs?
My experience is that tegragraphs are significantly better for most
applications... but only if you have a good dense set. My highest
frequency tetragraph is ofth at 38,504. I counted them from a bunch
of on-line literature (Gutenberg, Oxford and others), and from some
fairly carefully Perled Usenet data.
I mostly work on ciphers without word divisions, but if I suspected
word divisions for something I was going to try to break a lot of,
I'd re-do the tetragraphs with word divisions included. I imagine
they're more sensitive than trigraphs to getting that part right,
since they'd overlap more words.
--
Jim Gillogly
Highday, 18 Solmath S.R. 2000, 18:08
12.19.6.16.18, 8 Edznab 6 Pax, Fifth Lord of Night
------------------------------
From: [EMAIL PROTECTED] (Troed)
Crossposted-To: rec.video.dvd.tech
Subject: Re: DVD crypt Q
Reply-To: [EMAIL PROTECTED]
Date: Tue, 08 Feb 2000 18:12:14 GMT
Stephen Lee - Post replies please <[EMAIL PROTECTED]> wrote:
>I have a question about the crypto used on DVD. I have read articles
>on the web but there are some points not clear to me. This is
>regarding a PC/Mac with a DVD-ROM connected.
Of the top of my head, this, I might make a few mistakes:
>1) I understand DVDs use the UDF filesystem. Is a Video DVD just a
>DVD-ROM with specific files on it?
Yes, the VIDEO_TS directory. A DVD-Audio disc will use AUDIO_TS for
its contents.
It's possible to make "Audio DVD" discs with 30 frames of black and
the rest used for high quality AC3 5.1 sound instead - they'll work on
all stand alone DVD players that exist, removing the need for new DVD
audio players ...
>2) Why is authentication neccessary? I remember (maybe wrong)
>someone mentioning that some part of a DVD is not readable until the
>user authenticate with the drive. Is this true?
Yes. That's CSS-Auth, something that was published (source code) long
before DeCSS ...
>3) Why would the DVD-ROM need decryption key when decryption is done
>by separate hardware/software? Maybe I am misunderstanding, but if
>decryption is done by the drive, then wouldn't something like DeCSS be
>unneccessary?
Well, yes, but I think we're talking about the manufacturer key here -
of which there are only ~400. MoRE ripped out Xing's from their
software decoder, and that's the type of key that's supposed to be in
the drive instead.
(As if it would be any harder to get info from chips instead of
software ... sorry, it's not)
>4) I also read that DVD-Video Disc has some special part that stores
>the decryption keys and this part is blanked on recordable DVDs.
>
>4a) Is this true?
Yes.
>4b) Why does the Disc need to have the key? I thought the player has
>1 key and the Disc has its content scrambled (encrypted?) by ~400 keys?
The video is scrambled with one "master" key, and that key is
scrambled with the ~400 different keys. Each manufacturer can
therefore decode the master key and thus the content of the disc.
>5) Where does regional code come into the picture? Where is the
>regional code stored on a disc and how is it checked? I heard newer
It's stored in the files in the VIDEO_TS folder, in the .ifo file
(might be mistaken about the exact file, can't check at the moment).
Macrovision is also stored there, I believe. Since Macrovision
seriously hurts consumers - I can't get the video out put from my DVD
from the living room into the bedroom no matter what I try - it would
be a Good Thing(tm) if producers realised there's no benefit in using
Macrovision and stopped enabling it ..
>DVD-ROMs has the check built in, does it mean that older drives
>hasn't and it is checked somewhere else?
Some check purely in software, some don't.
>That's all I can think of for now. I am very confused. Please help
>me clear this up.
Are you a journalist or a lawyer? ;)
More answers can be found on www.opendvd.org
___/
_/ - who hasn't got a dvd-rom, and has never used decss
------------------------------
From: Mike Rosing <[EMAIL PROTECTED]>
Subject: Re: question about PKI...
Date: Tue, 08 Feb 2000 12:54:27 -0600
Palmpalmpalm wrote:
> Is there any method to transfer a "private key" and a certificate on line?
> I know that the certificate can be done, but the private key is very sensitive
> information and thus any secure channel is necessary or off-line method.
> Am I right?
Don't transfer the private key, regenerate it. This is easy if you use
a pass phrase and a hash function. By using ECC PKI methods, the
private
key is the output of the hash, there's no further computations. So when
the user goes from one device to another, they type in their pass phrase
and the private key is created on the fly. Transfer the public key over
the
net, along with all the public parameters for the PKI. They are already
public so that's no security problem.
Patience, persistence, truth,
Dr. mike
------------------------------
Reply-To: "karl malbrain" <[EMAIL PROTECTED]>
From: "karl malbrain" <[EMAIL PROTECTED]>
Crossposted-To: comp.sys.palmtops.pilot,alt.comp.sys.palmtops.pilot,comp.sys.handhelds
Subject: Re: Strip Security
Date: Tue, 8 Feb 2000 11:05:01 -0800
Brian Keener <[EMAIL PROTECTED]> wrote in message
news:87plki$cqt$[EMAIL PROTECTED]...
>
> "karl malbrain" <[EMAIL PROTECTED]> wrote in message
> news:1bYn4.2217$[EMAIL PROTECTED]...
> >
> > Steven G. Tyler <[EMAIL PROTECTED]> wrote in message
> > news:[EMAIL PROTECTED]...
> > > karl malbrain wrote:
> > >
> > > > Highdesertman <[EMAIL PROTECTED]> wrote...
> > >
> > > > > I am wondering how you arrived at 10,000 possible combinations
with
> a
> > > > > four digit pin. I don't doubt it is correct, . . .
> > >
> > > > The formula for 10 things (the digits 0 - 9) taken 6 at a time with
> > > > replacement is: 10*10*10*10*10*10.
> > >
> > > Formulae aside, remember also that the number of "possible
combinations"
> > > must be calculated on the basis of the largest allowable PIN, rather
> > > than the size of the actual PIN. In other words, you might have
actually
> > > used a 4-digit PIN, but if the encryption software will allow up to,
> > > say, 12 digits, a would-be cracker can't just assume the smaller
number
> > > but has to deal with the (much) larger number of combinations
generated
> > > by the larger number.
> >
> > What on earth does this mean? If you are placing a 4-digit PIN ahead of
a
> > 12-digit encryption algorithm, there are still only 10,000 inputs.
> Perhaps
> > you mean that not all 10,000 PINs are actually used? Karl M
> >
> >
>
> If max password size is 12 characters, then you've got more possible
> combinations. However, you might only have a password that's 4 characters
> long, out of a possible maximum of 12. But, anyone trying to guess your
> password won't know if you used all twelve digits or not.
No. If you mean the PIN is now 12 digits long instead of 4, and all PINS are
issued, anyone trying to guess your PIN would have to try to guess out of
10^12 possibilities, leading ZEROES not-with-standing. Karl M
------------------------------
From: "r.e.s." <[EMAIL PROTECTED]>
Subject: Re: Latin Squares (was Re: Reversibly combining two bytes?)
Date: Tue, 8 Feb 2000 11:03:58 -0800
"Tim Tyler" <[EMAIL PROTECTED]> wrote ...
: r.e.s. <[EMAIL PROTECTED]> wrote:
: : "Tim Tyler" <[EMAIL PROTECTED]> wrote ...
:
: : : : Note that if the answer to the second question is yes (all Latin
: : : : Squares are in PRT-form), then we can reduce any Latin Square of
: : : : order N to three arrays of N entries, which means that the internal
: : : : state of a square of order N is bounded by 3N, not by N^2. (Right?)
: : :
: : : The information content is smaller than this perhaps might suggest -
since
: : : each array entry is itself constrained.
: : :
: : : N! x N! x N! - for the Latin Square - compared to N^(N^2) for the
totally
: : : random table. ^^^^^^^
:
: : But the table can't be totally random if it's to be a workable
: : combiner. [...]
:
: Indeed not. Only a tiny fraction of these would be valid Latin squares.
You missed my point.
Your number N^(N^2) was said to be for a "totally random table".
I was pointing out that it's not to a totally random table that
one would want to make a comparison.
: : So I think that that latter number should be N!^N instead of N^(N^2).
:
: That is /an/ upper bound.
: It may be possible to do much better, though.
: N!^(N-1) is a smaller bound - since if N-1 columns are known, the last
: column is uniquely determined.
:
: I have a figure of 161820 possible 5x5 latin squares. This doesn't
: conform to the N!^N formula very well at all (though it's the right side
: of it when taken as a bound).
N!^N is not supposed to be the number of Lsquares.
It's the number of possible NxN combiners, against which
one might want to *compare* the number of Lsquare combiners.
: The last I heard, the function yielding the number of Latin squares of
: size NxN was one of the unsolved problems of mathematics - since no
: simple expression of it has yet been discovered.
Right. I cite a reference for this elsewhere in the thread.
--
r.e.s.
[EMAIL PROTECTED]
------------------------------
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