Cryptography-Digest Digest #516, Volume #14       Mon, 4 Jun 01 17:13:01 EDT

Contents:
  Re: Knapsack security??? Ah....huh (John Bailey)
  Re: Diffusion limits in block ciphers (Tim Tyler)
  Re: Diffusion limits in block ciphers ("Tom St Denis")
  Re: Def'n of bijection (John Savard)
  Re: WEB PAGES (SCOTT19U.ZIP_GUY)
  Re: Welcoming another Anti-Evidence Eliminator stooge to USENET  (P.  Dulles / AKA 
Loki) (Dave Howe)
  Re: Welcoming another Anti-Evidence Eliminator stooge to USENET  (P.  Dulles / AKA 
Loki) ("Tom St Denis")
  Re: Def'n of bijection (SCOTT19U.ZIP_GUY)
  Re: Best, Strongest Algorithm (gone from any reasonable topic) (SCOTT19U.ZIP_GUY)
  Re: Dynamic Transposition Revisited Again (long) ([EMAIL PROTECTED])
  Re: Help with Comparison Of Complexity of Discrete Logs, Knapsack, and Large Primes 
("Joseph Ashwood")
  Re: Def'n of bijection ([EMAIL PROTECTED])
  Re: National Security Nightmare? (SCOTT19U.ZIP_GUY)
  Re: WEB PAGES (SCOTT19U.ZIP_GUY)
  Re: Welcoming another Anti-Evidence Eliminator stooge to USENET  (Kyle Paskewitz)

----------------------------------------------------------------------------

From: [EMAIL PROTECTED] (John Bailey)
Subject: Re: Knapsack security??? Ah....huh
Date: Mon, 04 Jun 2001 18:43:20 GMT

On 3 Jun 2001 21:32:07 -0700, [EMAIL PROTECTED] (Merc42)
wrote:

>I was wondering if there are any knapsack systems that are still
>secure.  Any that don't use modular arithmatic to change the keys are
>of special interest to me.  Furthermore, if anybody does know of any,
>could you please tell me some reference where i could learn more about
>them.  As always, any help is appreciated...

 I wonder too.  This post in January on this news group did not get
any responses.

(quoting myself)
I was amazed to note that the NTRU Public Key method:
(reference)
Public key cryptosystem method and apparatus
http://www.delphion.com/details?pn=US06081597__
is formally equivalent to a knapsack system.
Referencing the last section of the NTRU tutorial
http://www.ntru.com/technology/tutorials/pkcstutorial.htm
e = r*h + m, where e is the encrypted message, h is a public key, r is
randomly chosen and m is the message.
The message m is recovered by  finding f*e mod q mod p = m.
In the NTRU case,  e, r, h, m, and f are truncated polynomials
whereas, in the cases mentioned at the beginning of this post, they
would simply be large numbers.  In either case, essentially the same
modulo algebra applies, showing the recoverability of encrypted
plaintext using private keys.

Other knapsaci systems:
A Comsat patent:
Simple and effective public-key cryptosystem 
http://www.delphion.com/details?&pn=US04306111__
and
Diophantine encryption for public key encoding
http://www.frontiernet.net/~jmb184/interests/sci.crypt/numerical_encryption.html
In this last one, an encrypted message e = r*h + m*k where e is the
encrypted message, r is a random number, h and k are public key
values, and m is the encrypted message (number)  m is recovered by
computing m = e*g mod p mod q where g, p, and q are calculated from
the private keys.

Simplified knapsack systems are easily implemented  and would provide
a nice means for simple, numeric only  public key tasks such as
symmetric key exchange except they have a history of being ultimately
breakable.

Quoting A. M. Odlyzko of Bell Labs:
The rise and fall of knapsack cryptosystems, 
http://www.research.att.com/~amo/doc/crypto.html
Abstract:
Cryptosystems based on the knapsack problem were among the first
public key systems to be invented, and for a while were considered to
be among the most promising. However, essentially all of the knapsack
cryptosystems that have been proposed so far have been broken. These
notes outline the basic constructions of these cryptosystems and
attacks that have been developed on them. (end quote)

Assuming the NTRU system has a new twist, are there also other
unexploited avenues in which the formalism for simple modulo knapsacks
might lead to interesting public key systems?  An example of such a
system might use digital Fourier Transforms (FFT) to form knapsacks,
along lines paralleling NTRU's use of truncated polynomials.

John Bailey

------------------------------

From: Tim Tyler <[EMAIL PROTECTED]>
Subject: Re: Diffusion limits in block ciphers
Reply-To: [EMAIL PROTECTED]
Date: Mon, 4 Jun 2001 19:02:06 GMT

Scott Fluhrer <[EMAIL PROTECTED]> wrote:

: You are correct, everything being equal, larger blocks are better than
: smaller ones.  However, no matter what you diffuse the information over, the
: diffusion really must be complete -- any partial diffusion [1] can give the
: attacker clues about what the last few rounds look like, and that's a Very
: Bad Thing.  And, as the size of the block grows, it tends to get
: increasingly difficult to maintain the amount of diffusion (without also
: increasing the amount of time spent per bit encrypting), and hence in
: practice, we tend to arrive at a compromise, where the block size is big
: enough that we don't get problems from it being too small, and not much
: bigger than that.

You can do diffusion by preprocessing, whitening, or compression - then
it doesn't much matter if it's patchy, irregular, or incomplete.

For example D. Scott has advocated compressing the file, reversing it
and compressing it again before finally encrypting.  This is normally 
quite effective at making every byte depend on every other one - but if
there is partial diffusion, it doesn't much matter since it's covered by
subsequent encryption.
-- 
__________
 |im |yler  [EMAIL PROTECTED]  Home page: http://alife.co.uk/tim/

------------------------------

From: "Tom St Denis" <[EMAIL PROTECTED]>
Subject: Re: Diffusion limits in block ciphers
Date: Mon, 04 Jun 2001 19:18:51 GMT


"Tim Tyler" <[EMAIL PROTECTED]> wrote in message news:[EMAIL PROTECTED]...
> Scott Fluhrer <[EMAIL PROTECTED]> wrote:
>
> : You are correct, everything being equal, larger blocks are better than
> : smaller ones.  However, no matter what you diffuse the information over,
the
> : diffusion really must be complete -- any partial diffusion [1] can give
the
> : attacker clues about what the last few rounds look like, and that's a
Very
> : Bad Thing.  And, as the size of the block grows, it tends to get
> : increasingly difficult to maintain the amount of diffusion (without also
> : increasing the amount of time spent per bit encrypting), and hence in
> : practice, we tend to arrive at a compromise, where the block size is big
> : enough that we don't get problems from it being too small, and not much
> : bigger than that.
>
> You can do diffusion by preprocessing, whitening, or compression - then
> it doesn't much matter if it's patchy, irregular, or incomplete.
>
> For example D. Scott has advocated compressing the file, reversing it
> and compressing it again before finally encrypting.  This is normally
> quite effective at making every byte depend on every other one - but if
> there is partial diffusion, it doesn't much matter since it's covered by
> subsequent encryption.

This is a sign of amateur cryptology.  Anyone can make complete inefficient
transforms.

Generally ideal crypto would consist of each part being as good as possible.
I am not an advocate (for example) of fast weak round functions and adding
tons of rounds.  Quite obvious a compromise must be made.  If we look at DES
for example one round is not terribly strong but 16 are.  I would be in
favour of stronger rounds.  For example, a simple 64-bit block cipher could
be made up of a 4x4 MDS over GF(2^8) with good sboxes.  If we had sboxes
with a DPmax of say 4/256 this function would have a maximal DP probability
of 4/256 compared to the 14/64 in DES. (that is 4/256 vs 56/256).  We could
get away with probably eight rounds in this case.

In this case the 4x4 MDS could be implemented just as easily (if not faster)
than the DES round function and would be more secure from itterative
attacks.

Also in the case of the MDS o SBOXES style round functions they become
complete after three rounds which is very ideal.  Unfortunately MDS o SBOXES
style round functions are vulnerable to saturation style attacks (i.e square
attack).  Since the # of active boxes follows the binomial distribution it
can be exploited.  (I think that's how it goes).

Tom



------------------------------

From: [EMAIL PROTECTED] (John Savard)
Subject: Re: Def'n of bijection
Date: Mon, 04 Jun 2001 19:41:22 GMT

On Mon, 4 Jun 2001 16:58:23 GMT, "Douglas A. Gwyn" <[EMAIL PROTECTED]>
wrote, in part:

>Note, I was agreeing with D.Scott and disagreeing with Tom St. Denis.
>Otherwise, what you say is correct and relevant.

I suspected that might be the case, but since your post didn't
indicate directly who you were agreeing or disagreeing with, I thought
it made sense to note where the points were applicable.

Given Mr. Scott's usual reputation, when one is saying that a fallacy
has been committed in a thread that concerns his ideas, one kind of
has to spell it out when the fallacy is not his. I didn't intend to
claim you had things wrong, but I was concerned that your post as it
stood, although unintentionally, might confuse some casual readers.

John Savard
http://home.ecn.ab.ca/~jsavard/frhome.htm

------------------------------

From: [EMAIL PROTECTED] (SCOTT19U.ZIP_GUY)
Crossposted-To: comp.compression
Subject: Re: WEB PAGES
Date: 4 Jun 2001 19:51:38 GMT

[EMAIL PROTECTED] (Mikael Lundqvist) wrote in
<[EMAIL PROTECTED]>: 

>
>
>"SCOTT19U.ZIP_GUY" wrote:
>
>>   Dear Anyone
>> I have been having a terible time even accessing my site
>> at http://members.nbci.com/ecil/index.htm
>> it seems problems are only getting worse. Does
>> anyone have recomendations as to where a alternative
>> free webpage hosting occurs.
>> Thank You.
>>
>
>Hi David.
>You should talk with them first. Your homepage seems to have been
>removed.
>But if you're tired of all the trouble, is here a list:
>http://directory.google.com/Top/Computers/Internet/Web_Design_and_Develop
>ment/Hosting/Free/Personal/ 
>
>I don't know what could be the best choice. It's up to you.
>
>Regards,
>--

  I have written them several times about the only response
is they are sorry for the email trouble. Its like they
don't eve bother to anwser why it down or not. I don't they
they care since its free. I don't care about the email account.
I don't even ask about the email.


David A. Scott
-- 
SCOTT19U.ZIP NOW AVAILABLE WORLD WIDE "OLD VERSIOM"
        http://www.jim.com/jamesd/Kong/scott19u.zip
My website http://members.nbci.com/ecil/index.htm
My crypto code http://radiusnet.net/crypto/archive/scott/
MY Compression Page http://members.nbci.com/ecil/compress.htm
**NOTE FOR EMAIL drop the roman "five" ***
Disclaimer:I am in no way responsible for any of the statements
 made in the above text. For all I know I might be drugged or
 something..
 No I'm not paranoid. You all think I'm paranoid, don't you!


------------------------------

From: Dave Howe <[EMAIL PROTECTED]>
Crossposted-To: 
alt.privacy,alt.security,alt.security.pgp,alt.security.scramdisk,alt.privacy.anon-server
Subject: Re: Welcoming another Anti-Evidence Eliminator stooge to USENET  (P.  Dulles 
/ AKA Loki)
Date: Mon, 04 Jun 2001 21:03:36 +0100

In our last episode (<alt.security.pgp>[Mon, 04 Jun 2001 01:59:03
GMT]), "Tom St Denis" <[EMAIL PROTECTED]> said :
>Take all primes and form a composite N.  Add one to N.  Now N is not
>divisible by any of the "known" primes.  Thus N+1 is a new prime not in the
>list.  Proof by contradiction.  We proved that "there are finite number of
>primes" is false.
<pedant>
or is divisible by a prime not in the original list
</pedant>

--== DaveHowe ( is at) Bigfoot dot com ==--

------------------------------

From: "Tom St Denis" <[EMAIL PROTECTED]>
Crossposted-To: 
alt.privacy,alt.security,alt.security.pgp,alt.security.scramdisk,alt.privacy.anon-server
Subject: Re: Welcoming another Anti-Evidence Eliminator stooge to USENET  (P.  Dulles 
/ AKA Loki)
Date: Mon, 04 Jun 2001 20:08:16 GMT


"Dave Howe" <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]...
> In our last episode (<alt.security.pgp>[Mon, 04 Jun 2001 01:59:03
> GMT]), "Tom St Denis" <[EMAIL PROTECTED]> said :
> >Take all primes and form a composite N.  Add one to N.  Now N is not
> >divisible by any of the "known" primes.  Thus N+1 is a new prime not in
the
> >list.  Proof by contradiction.  We proved that "there are finite number
of
> >primes" is false.
> <pedant>
> or is divisible by a prime not in the original list
> </pedant>

That's not possible.  Since we already have all consecutive primes...

3*5*7+1 = 106

106 is not divisible by any known prime (assume the only known primes are 3,
5, 7).  Compared to say

3*11*17= ...

Which is what you said but not what I used in my proof.

Tom



------------------------------

From: [EMAIL PROTECTED] (SCOTT19U.ZIP_GUY)
Subject: Re: Def'n of bijection
Date: 4 Jun 2001 20:02:53 GMT

[EMAIL PROTECTED] (John Savard) wrote in 
<[EMAIL PROTECTED]>:

>On Mon, 4 Jun 2001 16:58:23 GMT, "Douglas A. Gwyn" <[EMAIL PROTECTED]>
>wrote, in part:
>
>>Note, I was agreeing with D.Scott and disagreeing with Tom St. Denis.
>>Otherwise, what you say is correct and relevant.
>
>I suspected that might be the case, but since your post didn't
>indicate directly who you were agreeing or disagreeing with, I thought
>it made sense to note where the points were applicable.
>
>Given Mr. Scott's usual reputation, when one is saying that a fallacy
>has been committed in a thread that concerns his ideas, one kind of
>has to spell it out when the fallacy is not his. I didn't intend to
>claim you had things wrong, but I was concerned that your post as it
>stood, although unintentionally, might confuse some casual readers.
>
>John Savard


   Are you still being an ass about the fact when I rub you nose
in errors you kept them at your site for years. I doubt you will
find many fallacies except spelling and grammer. So what your problem
john don't undersrand bijective compress yet. Are you every going
to wirte code or is your site just to wave hands.

David A. Scott
-- 
SCOTT19U.ZIP NOW AVAILABLE WORLD WIDE "OLD VERSIOM"
        http://www.jim.com/jamesd/Kong/scott19u.zip
My website http://members.nbci.com/ecil/index.htm
My crypto code http://radiusnet.net/crypto/archive/scott/
MY Compression Page http://members.nbci.com/ecil/compress.htm
**NOTE FOR EMAIL drop the roman "five" ***
Disclaimer:I am in no way responsible for any of the statements
 made in the above text. For all I know I might be drugged or
 something..
 No I'm not paranoid. You all think I'm paranoid, don't you!


------------------------------

From: [EMAIL PROTECTED] (SCOTT19U.ZIP_GUY)
Subject: Re: Best, Strongest Algorithm (gone from any reasonable topic)
Date: 4 Jun 2001 20:08:17 GMT

[EMAIL PROTECTED] (Tom St Denis) wrote in
<ARPS6.25421$[EMAIL PROTECTED]>: 

>
>Because if the cipher is any good a change in any bit should have
>drastic confusing consequences.  Seriously Tim may I suggest you do some
>research before posting?  I know I've zinged myself a few times but at
>least I try! 
>
>Tom
>
>

  Tommy the hell you try. He was trying to see why your so screwed up.
You have no idea what security means. You make false claims about
betting money. You don't understand Unicity distance compression or
encryption. And you wonder why you couldn't keep your last job. Well
I thinks it your in ability to reason. If your previous boss could
not even get you to reason on simple things you do wrong. If you
refuse to think. Why should he have kept you on.


David A. Scott
-- 
SCOTT19U.ZIP NOW AVAILABLE WORLD WIDE "OLD VERSIOM"
        http://www.jim.com/jamesd/Kong/scott19u.zip
My website http://members.nbci.com/ecil/index.htm
My crypto code http://radiusnet.net/crypto/archive/scott/
MY Compression Page http://members.nbci.com/ecil/compress.htm
**NOTE FOR EMAIL drop the roman "five" ***
Disclaimer:I am in no way responsible for any of the statements
 made in the above text. For all I know I might be drugged or
 something..
 No I'm not paranoid. You all think I'm paranoid, don't you!


------------------------------

From: [EMAIL PROTECTED]
Subject: Re: Dynamic Transposition Revisited Again (long)
Date: 4 Jun 2001 20:11:35 GMT

In article <[EMAIL PROTECTED]>, [EMAIL PROTECTED] 
(Mok-Kong Shen) wrote:

> > Yes - so that the biased bit patterns within those inverted
> > bytes approximately negate the patterns in the unmodified
> > ones. The intention is to force the attacker to try all
> > 2**n permutations (n=block size in bits). Having said that,
> > I think that a substitution cipher should be applied as well.
> 
> Consider your inverted bytes (their number is half of 
> the total in the message). These bytes are transformed
> by one and the same substitution isn't it? So you are
> applying a monoalphabetical substitution to one half
> of the number of bytes of the message, right? If
> that is the case, wouldn't it be better to have a
> polyalphabetical substitution table and have a PRNG
> to generate a key sequence to determine which byte
> of the message is to be substituted according to which 
> column of the substitution table? 

The last sentence was a throw away comment as to further 
strengthening of the cipher. 

The transposition of ASCII bytes and bit balancing were the main 
theme. After transposition, substitution is obviously not on the 
same ASCII bytes, but bytes that should look fairly random, 
without any bias in their bit values.

In a known plaintext attack, a 1-byte or 1-bit change can be used 
to analyse the transposition. Substitution causes a degree of 
diffusion and makes analysis more difficult.

Keith.

------------------------------

From: "Joseph Ashwood" <[EMAIL PROTECTED]>
Subject: Re: Help with Comparison Of Complexity of Discrete Logs, Knapsack, and Large 
Primes
Date: Mon, 4 Jun 2001 12:39:52 -0700


"sisi jojo" <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]...
> You can't win $200,000 with QS and NFS, can you?  :)

Actually a varient of NFS is the most likely candidate for winning the
$200,000 right now, so I'd say you're wrong.

> Actually the math in cryptography is not that hard, but the mathematical
> language is very cryptic.

Then please feel free to explain the Number Field Sieve, it is not overly
complex (certainly not on the order of some of the proofs of Fermat's work),
and it is clearly cryptographically important, so please feel free to
elighten us.

> If someone just re-writes all the papers in human
> language, even teenagers can become good cryptographers. Otherwise, you
need
> years of schooling just to understand the math notations.

There are solid reasons for much of the notation in use. To understand what
is needed for various types of cryptography, to make things secure requires
a large body of knowledge. Do you really think that the RSA algorithm was
easy to come up with? How about the Wide Trail Strategy? I know maybe
Differential Cryptanalysis is the easy one? If they are easy why did it take
several thousand years of cryptographic innovation to find them? Yes we can
make things easy, RSA looks easy enough, once the Wide Trail Strategy has
been developed it makes development fairly straight forward. OTOH
Differential Cryptanalysis takes a college level education to really
understand, and the proofs behind it are even deeper.

It may take only a high school education to understand your view of
cryptography, but your view is not fully enlightened yet. Take a simpler
problem 1+1=2, everyone learns that in 1st grade (some earlier, some a
little later), but it takes a doctorate in mathematics, and a few hundred
pages of very intricate math to prove it without assuming things. With
addition we have an innate understanding of it because we can easily
visualize it. We have no such built in ability with cryptography,
cryptography is much more complex, very few people can actually visualize
even something as seemingly simple as RSA, fewer still can visualize the
Rijndael. Cryptgraphy is not a transparent art, yes anyone can dabble in it,
and I encourage anyone to dabble in it, but they also need to understand
that even the best of us don't know all the rules, some just have a feel for
the right thing to do, others are doomed to be just dabblers for their
entire life, but you don't know until you try.
                            Joe



------------------------------

Subject: Re: Def'n of bijection
From: [EMAIL PROTECTED]
Date: 04 Jun 2001 16:24:21 -0400

Anton Stiglic <[EMAIL PROTECTED]> writes:
>
> Yes.  One property that does hold is that the cardinality of the 
> sets are equal for finite sets.

For *all* sets.

> (like you can't have a bijection between an infinite enumerable set
> and an infinite non-enumerable set...)

Correct. But note: ``infinite'' is not a cardinality. Denumarable sets
have cardinality \aleph_0, and an infinite set has countable
cardinality if and only if there exists a bijection with the set of
integers. The real numbers have cardinality at least \aleph_1 (where
equality holds if the continuum hypothesis is true).

Len.


-- 
The BIND company isn't just reinventing the wheel. It's reinventing the
pushcart in a world full of cars.
                                        -- Dan Bernstein

------------------------------

From: [EMAIL PROTECTED] (SCOTT19U.ZIP_GUY)
Subject: Re: National Security Nightmare?
Date: 4 Jun 2001 20:11:30 GMT

[EMAIL PROTECTED] (Douglas A. Gwyn) wrote in <[EMAIL PROTECTED]>:

>> Ahh, yes, FOIA: ...
>
>It's still your best available option.  Unfortunately, a lot of
>FIOA office staff work has had to go into researching trash like
>UFO references.  That is now somewhat under control, so perhaps
>response time will improve.  If undue delays keep resulting in
>court actions and complaints to Congressmen, one would expect
>that more resources will be directed toward this function.

   And from what little they get back it shows that FIOA's
are a joke. I know people who swear somthing happend at Roswell
and that the government is hiding it.

David A. Scott
-- 
SCOTT19U.ZIP NOW AVAILABLE WORLD WIDE "OLD VERSIOM"
        http://www.jim.com/jamesd/Kong/scott19u.zip
My website http://members.nbci.com/ecil/index.htm
My crypto code http://radiusnet.net/crypto/archive/scott/
MY Compression Page http://members.nbci.com/ecil/compress.htm
**NOTE FOR EMAIL drop the roman "five" ***
Disclaimer:I am in no way responsible for any of the statements
 made in the above text. For all I know I might be drugged or
 something..
 No I'm not paranoid. You all think I'm paranoid, don't you!


------------------------------

From: [EMAIL PROTECTED] (SCOTT19U.ZIP_GUY)
Crossposted-To: comp.compression
Subject: Re: WEB PAGES
Date: 4 Jun 2001 20:16:01 GMT

[EMAIL PROTECTED] (SCOTT19U.ZIP_GUY) wrote in 
<[EMAIL PROTECTED]>:

>[EMAIL PROTECTED] (Mikael Lundqvist) wrote in
><[EMAIL PROTECTED]>: 
>
>>
>>
>>"SCOTT19U.ZIP_GUY" wrote:
>>
>>>   Dear Anyone
>>> I have been having a terible time even accessing my site
>>> at http://members.nbci.com/ecil/index.htm
>>> it seems problems are only getting worse. Does
>>> anyone have recomendations as to where a alternative
>>> free webpage hosting occurs.
>>> Thank You.
>>>
>>
>>Hi David.
>>You should talk with them first. Your homepage seems to have been
>>removed.
>>But if you're tired of all the trouble, is here a list:
>>http://directory.google.com/Top/Computers/Internet/Web_Design_and_Develop
>>ment/Hosting/Free/Personal/ 
>>
>>I don't know what could be the best choice. It's up to you.
>>
>>Regards,
>>--
>
>  I have written them several times about the only response
>is they are sorry for the email trouble. Its like they
>don't eve bother to anwser why it down or not. I don't they
>they care since its free. I don't care about the email account.
>I don't even ask about the email.
>
>
>David A. Scott

   Well I posted this with better english than what appeared
appartendly when I am on another line a go back to fix the
line it looks ok but that appears not to be what I sent.
Any way they don't anwer there mail.

David A. Scott
-- 
SCOTT19U.ZIP NOW AVAILABLE WORLD WIDE "OLD VERSIOM"
        http://www.jim.com/jamesd/Kong/scott19u.zip
My website http://members.nbci.com/ecil/index.htm
My crypto code http://radiusnet.net/crypto/archive/scott/
MY Compression Page http://members.nbci.com/ecil/compress.htm
**NOTE FOR EMAIL drop the roman "five" ***
Disclaimer:I am in no way responsible for any of the statements
 made in the above text. For all I know I might be drugged or
 something..
 No I'm not paranoid. You all think I'm paranoid, don't you!


------------------------------

From: Kyle Paskewitz <[EMAIL PROTECTED]>
Crossposted-To: 
alt.privacy,alt.security,alt.security.pgp,alt.security.scramdisk,alt.privacy.anon-server
Subject: Re: Welcoming another Anti-Evidence Eliminator stooge to USENET 
Date: Mon, 4 Jun 2001 16:48:22 -0400

Tom - 

You've forgotten that 2 is also prime.  If you take the product of any
number of consecutive primes beginning with 2 (the first prime) and add 1,
you will get another prime.  E.G.

2*3 + 1 = 7
2*3*5 + 1 = 31
2*3*5*7 + 1 = 211 , etc...

-- Kyle Paskewitz --




------------------------------


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