Cryptography-Digest Digest #716, Volume #13 Mon, 19 Feb 01 16:13:01 EST
Contents:
Re: Is there an algorithm to sequentially enumerate all transcendental numbers?
("Henrick Hellstr�m")
Re: Is there an algorithm to sequentially enumerate all transcendental (Philip
Anderson)
Re: Is there an algorithm to sequentially enumerate all transcendental numbers?
("Henrick Hellstr�m")
Re: Is there an algorithm to sequentially enumerate all transcendental (Dave Seaman)
Re: Authentication before Key Exchange (Mike Rosing)
Re: Metallurgy and Cryptography (Mike Rosing)
Re: RSA on FPGA (Mike Rosing)
Re: Rnadom Numbers ("Simon Johnson")
Re: Is there an algorithm to sequentially enumerate all transcendental numbers?
("Henrick Hellstr�m")
Re: relative key strength private vs public key ("Simon Johnson")
What's a KLB-7? (Richard Outerbridge)
Re: Is there an algorithm to sequentially enumerate all transcendental numbers?
("Doom the Mostly Harmless")
Re: Is there an algorithm to sequentially enumerate all transcendental numbers?
(Steve Leibel)
Given any arbitrary numbers a and b. (jtnews)
PGP Author Philip Zimmermann Quits NAI (gm)
Re: Big Numbers in C/C++ ("Dann Corbit")
Re: Given any arbitrary numbers a and b. Can I ALWAYS find a transcendental
number between a and b? ("Henrick Hellstr�m")
Re: Given any arbitrary numbers a and b.Can I ALWAYS find a (Jan Kristian Haugland)
Re: Given any arbitrary numbers a and b. Can I ALWAYS find a transcendental
number between a and b? ("Paul Lutus")
Re: Could someone compile it for me? ("Ryan M. McConahy")
Rijndael vb and vbscript code ("Phil Fresle")
----------------------------------------------------------------------------
From: "Henrick Hellstr�m" <[EMAIL PROTECTED]>
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
numbers?
Date: Mon, 19 Feb 2001 18:15:20 +0100
Well, trivially you could leave out omega, 2*omega etc, but use e.g.
omega+1, 2*omega+1 etc. But you can NOT map the entire set of transcendental
numbers injectively into the open set [0..omega), if that's what you mean.
It seems as if we agree on the definition of "enumerate". But usually the
term "induction" is used to denote what you describe with the expression
"present a recursive function". ;-)
--
Henrick Hellstr�m
StreamSec HB
"stanislav shalunov" <[EMAIL PROTECTED]> skrev i meddelandet
news:[EMAIL PROTECTED]...
> "Henrick Hellstr�m" <[EMAIL PROTECTED]> writes:
>
> > Theoretically, you could assign a unique ordinal value to each
> > transcendental number, but you would have to use ordinal values
> > without predecessors (omega, 2*omega, ... , etc) so you cannot
> > enumerate the set of transcendental numbers.
>
> You can map transcendental numbers into ordinals with predecessors
> injectively just fine.
>
> "Enumerate", however, has a specific meaning: to present a recursive
> function whose image is the set in question. For uncountable sets
> there's obviously nothing of the sort by definition (of "recursive
> function" and "uncountable").
>
> --
> Stanislav Shalunov http://www.internet2.edu/~shalunov/
>
> The United States now sleeps under a Soviet moon. -- Nikita Khrushchev
------------------------------
Date: Mon, 19 Feb 2001 17:39:37 +0000
From: Philip Anderson <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
Paul Lutus wrote:
>
<snip>
> I think the problem is in how you are describing the problem, not the
> problem itself. You want to list an arbitrarily large number of
> transcendental numbers, not "all possible."
No, I think he wants more than that. Avoiding the use of the word "all"
which you balk at, he wants an algorithm which (given enough time &
resources) will eventually output any particular transcendental number;
ie it is working through an infinite, complete list and while it will
never reach the end it will reach any number in the list. (If each step
could be done in half the time of its predecessor, I'd argue it could be
finished <g>)
Any countable set can be so enumerated, but an uncountable one cannot
and the transcendental numbers are uncountable.
But it is possible "to list an arbitrarily large number of
transcendental numbers" by choosing a countable subset of transcendental
numbers, eg e, e+1, e+2, e+3 ...
--
hwyl/cheers,
Philip Anderson
Alenia Marconi Systems
Cwmbr�n, Cymru/Wales
------------------------------
From: "Henrick Hellstr�m" <[EMAIL PROTECTED]>
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
numbers?
Date: Mon, 19 Feb 2001 18:54:19 +0100
"jtnews" <[EMAIL PROTECTED]> skrev i meddelandet
news:[EMAIL PROTECTED]...
> jtnews wrote:
> >
> > Is there an algorithm to sequentially enumerate
> > all possible transcendental numbers?
> >
> > I want to be able to generate very long
> > passphrases but at the same time be able
> > to express them succinctly in the form of mathematical
> > expressions.
>
>
> Let me clarify what I mean by sequentially enumerate.
> By sequentially enumerate I mean successively enumerate
> all possible transcendental numbers starting from zero
> to infinity.
>
> It doesn't matter that the set itself is infinite.
It seems as if you are looking for an inductive method to enumerate all
transcendental numbers. Apart from the fact that the set is infinite (which
actually doesn't matter from a strictly theoretical point of view) the
answer is "no", since the set of transcendental numbers is not an inductive
set.
--
Henrick Hellstr�m
StreamSec HB
------------------------------
From: [EMAIL PROTECTED] (Dave Seaman)
Crossposted-To: sci.math
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
Date: 19 Feb 2001 12:55:22 -0500
In article <[EMAIL PROTECTED]>,
jtnews <[EMAIL PROTECTED]> wrote:
>Let me clarify what I mean by sequentially enumerate.
>By sequentially enumerate I mean successively enumerate
>all possible transcendental numbers starting from zero
>to infinity.
>It doesn't matter that the set itself is infinite.
Agreed.
>My question is analagous to:
>Is there an algorithm to sequentially enumerate all
>prime numbers?
>Here the answer is yes. Even though the set of all
>prime numbers is infinite, there is an algorithm to
>enumerate the entire set starting from zero to infinity.
There is a fundamental difference. The prime numbers are countable. The
transcendentals are uncountable.
This means there is a function mapping the natural numbers N onto the set
of primes. There is no function mapping N onto the set of transcendental
numbers. This follows from the fact that the real numbers are
uncountable and the algebraic numbers are countable.
--
Dave Seaman [EMAIL PROTECTED]
Amnesty International calls for new trial for Mumia Abu-Jamal
<http://www.amnestyusa.org/abolish/reports/mumia/>
------------------------------
From: Mike Rosing <[EMAIL PROTECTED]>
Subject: Re: Authentication before Key Exchange
Date: Mon, 19 Feb 2001 12:34:14 -0600
George wrote:
>
> Several days ago I made a post inquiring about the security of Diffei
> Helman for key exchange. One of the users here brought to my attention
> it's very secure, but a good authentication algorithm must be chosen to
> ensure Alice is communicating with Bob. This past week I researched
> many different authentication protocols that use only Alice and Bob and
> NO trusted third party. In all of the protocols to pick from, each one
> seemed to have a weakness for an MITM attack. Could someone please
> dirrect my attention to a place where I can find more information about
> secure authentication protocols that use only Alice and Bob and no
> trusted third party? What is the best authentication algorithm/protocol
> to use to meet these requirements? Thank you for your time.
True authentication requires an "out of band" exchange. That means the
two sides have to communicate some information via an alternate route
to the one used for key and data exchange. You have to *start* by knowing
there's no MITM. Otherwise you can't know there isn't! there are lots of
ways to do that: face to face key exchange, newspaper published key finger
print, a telephone call, etc.
One of my favorite authentication/key exchange algorithms is MQV. This
generates a secret key and authenticates at the same time. But you still
have to know that the permenent keys are from who you think they are.
Patience, persistence, truth,
Dr. mike
------------------------------
From: Mike Rosing <[EMAIL PROTECTED]>
Subject: Re: Metallurgy and Cryptography
Date: Mon, 19 Feb 2001 12:37:39 -0600
David Eppstein wrote:
>
> In article <ryRj6.79650$[EMAIL PROTECTED]>, "Tad
> Johnson" <[EMAIL PROTECTED]> wrote:
>
> > What the heck is going on here?
>
> I think it's just an ironic coincidence.
could also be an ionic coincidence too :-)
Patience, persistence, truth,
Dr. mike
------------------------------
From: Mike Rosing <[EMAIL PROTECTED]>
Subject: Re: RSA on FPGA
Date: Mon, 19 Feb 2001 12:43:31 -0600
ajd wrote:
>
> Anyone ever done it? Which bits of the algorithm were implemented?
Check here:
http://www.ece.wpi.edu/Research/crypt/ches/ches2000/program2000.html
You might look at the previous year, and see if there's anything coming
in May.
Patience, persistence, truth,
Dr. mike
------------------------------
From: "Simon Johnson" <[EMAIL PROTECTED]>
Subject: Re: Rnadom Numbers
Date: Mon, 19 Feb 2001 18:42:55 -0000
Would I right in saying that if a crypto-secure pseudo-random generator was
unable to be cryptanalysed until x bytes of stream were recovered then all
of the bytes previous to the x'th byte must be statistically random?
Or are there cases when this isn't true?
Simon
=========
Viktor CK Pilpenok wrote in message <968kvd$s48$[EMAIL PROTECTED]>...
>Hi Everybody!
>
>Is there any algorithm that allows to estimate the randomness of a
>stream of numbers?
>
>Thanks in advance. Viktor P
>
>
>Sent via Deja.com
>http://www.deja.com/
------------------------------
From: "Henrick Hellstr�m" <[EMAIL PROTECTED]>
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
numbers?
Date: Mon, 19 Feb 2001 19:46:25 +0100
"Dave Seaman" <[EMAIL PROTECTED]> skrev i meddelandet
news:96rmma$[EMAIL PROTECTED]...
> This means there is a function mapping the natural numbers N onto the set
> of primes. There is no function mapping N onto the set of transcendental
> numbers. This follows from the fact that the real numbers are
> uncountable and the algebraic numbers are countable.
That would be a circular proof. One way to prove it is to use the theorem
that card(N) < card(P(N)), where P(N) is the set of all subsets of N, and
find a map f from P(N) onto the set of real numbers, e.g. f(M) = g(h(M)),
where g(x) = arctan(pi(x-0.5)), and h(M) is a number in the interval [0..1)
such that if and only if n is in M, then the binary decimal at position n of
h(M) is 1.
--
Henrick Hellstr�m
StreamSec HB
------------------------------
From: "Simon Johnson" <[EMAIL PROTECTED]>
Subject: Re: relative key strength private vs public key
Date: Mon, 19 Feb 2001 18:56:18 -0000
Tom St Denis wrote in message <95ufuo$shk$[EMAIL PROTECTED]>...
>In article <[EMAIL PROTECTED]>,
> [EMAIL PROTECTED] (DJohn37050) wrote:
> Just for the record, for 256 bit AES keys, I have heard ideas that
> RSA keys should be 20,000 bits or even more. I prefer to call the
>numbers I use NIST numbers, as that is where they came from. It is
>EXACTLY because I MAY be identified as anti-RSA (which I am >NOT, but that
is another matter) that I defer to NIST and their >numbers. Similarly,
others may be dentified with other interests and >results can be skewed but
still be reasonable, by always choosing >the appropriate plausible value
from the range of estimation that >gets the calculation to come out as one
wishes.
Hrm, factoring 20,000-bit RSA would probably require more energy than this
universe contains. 256-bit is probably okay with around galaxy worth of
energy. This based on a few unstated assumptions, of course.
Simon.
------------------------------
From: Richard Outerbridge <[EMAIL PROTECTED]>
Subject: What's a KLB-7?
Date: Mon, 19 Feb 2001 12:44:38 -0500
=====BEGIN PGP SIGNED MESSAGE=====
2001-02-19 17:29:00 GMT
On board the H.M.S. Belfast today (a heavy-light cruiser
museum piece moored in the London Pool) in the Electronic
Warfare Shack, behind glass, I observed an obvious piece
of crypto equipment which proclaimed itself an instance
of NSA "KLB-7/T SEC" S#12405, best-delivery-before date
sometime in 1990 to somewhere in Whitehall.
Can anyone say what this hardware was for? The Belfast
saw 'angry' service up to the end of the Korean War.
outer
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--
<[EMAIL PROTECTED]> :
Just an eccentric soul "with a curiosity for the bizarre".
Payloads to: A902/MCE307/3/17TPU-28413618 (or thereabouts)
------------------------------
From: "Doom the Mostly Harmless" <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
numbers?
Date: Mon, 19 Feb 2001 19:50:28 GMT
: Let me clarify what I mean by sequentially enumerate.
: By sequentially enumerate I mean successively enumerate
: all possible transcendental numbers starting from zero
: to infinity.
:
: It doesn't matter that the set itself is infinite.
It is impossible to create an ordinal set of trancendental numbers because
between any two given, there is an infinite number. Let's examine a trivial
example:
t1 = 3.14159....
t2 = 3.15149....
To generate a trancendental number t3 between t1 and t2 can be done as
follows:
Copy the numbers to the first difference:
t3 = 3.1....
Pick the smaller number (or any in the range exclusive)
t3 = 3.14....
Increment the next non-9 in the sequence t1:
t3 = 3.142....
Finish however you like:
t3 = 3.1421234567890.....
This can be repeated for any two trancendentals, so that we can find an
infinite number of them between any other two. That is, we can find an
infinite number between t1 and t2 by finding t3, then finding two
trancendentals t4 and t5, t1 < t4 < t3 < t5 < t2, ad infinitum. It's
tedious, but it works.
Hope this clarifies.
To air is human....
--Doom.
------------------------------
From: Steve Leibel <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
numbers?
Date: Mon, 19 Feb 2001 19:56:40 GMT
In article <8Oek6.8603$[EMAIL PROTECTED]>, "Doom the
Mostly Harmless" <[EMAIL PROTECTED]> wrote:
> : Let me clarify what I mean by sequentially enumerate.
> : By sequentially enumerate I mean successively enumerate
> : all possible transcendental numbers starting from zero
> : to infinity.
> :
> : It doesn't matter that the set itself is infinite.
>
> It is impossible to create an ordinal set of trancendental numbers
> because
> between any two given, there is an infinite number. Let's examine a
What about the rationals? Between any two there are infinitely many
other rationals, yet we can easily generate all the rationals using an
algorithm -- just follow the arrows in the standard demonstration that
the rationals are countable.
------------------------------
Date: Mon, 19 Feb 2001 15:01:18 -0500
From: jtnews <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Given any arbitrary numbers a and b.
Since my previous post seemed to be not to clear
to some people. Let me restate the problem another
way which I hope will be simpler to understand.
Given any arbitrary numbers a and b where a and
b are NOT the same number.
Can I ALWAYS find a transcendental number
between a and b?
------------------------------
From: gm <[EMAIL PROTECTED]>
Subject: PGP Author Philip Zimmermann Quits NAI
Date: Mon, 19 Feb 2001 14:12:25 -0600
Posted this morning in alt.security.pgp
=====BEGIN PGP SIGNED MESSAGE=====
Hash: SHA1
A note to PGP users:
As most PGP users know, Network Associates Inc (NAI) acquired my
company, PGP Inc, in December 1997. For three years after that, I
stayed on with NAI as Senior Fellow, to provide technical guidance
for PGP's continued development, and to ensure PGP's cryptographic
integrity. But I can't stay on forever. In the past three years,
NAI has developed a different vision for PGP's future, and it's time
for me to move on to other projects more fitting with my own
objectives to protect personal privacy.
Let me assure all PGP users that all versions of PGP produced by NAI,
and PGP Security, a division of NAI, up to and including the current
(January 2001) release, PGP 7.0.3, are free of back doors. In all
previous releases, up through PGP 6.5.8, this has been proven by the
release of complete source code for public peer review. New senior
management assumed control of PGP Security in the final months of
2000, and decided to reduce how much PGP source code they would
publish. If NAI ever publishes the complete PGP 7.0.3 source code, I
am confident that the public will be able to see that there are still
no back doors. Until that time, I can offer only my own assurances
that this version of PGP was developed on my watch, and has no back
doors. In fact, I believe it to be the most secure version of PGP
produced to date.
While it is true that NAI holds the PGP trademark and the source
code for the NAI implementation of PGP, I'd like to point out that
PGP is defined by an IETF open standard called OpenPGP, embodied in
IETF RFC 2440, which any company may implement freely into its
products. I will be working with other companies to support
implementations of the OpenPGP standard, to turn it into a real
industry standard supported by multiple vendors. I think the
emergence of more than one strong commercial implementation of the
OpenPGP standard is necessary for the long term health of the PGP
movement, and will, incidentally, ultimately benefit NAI.
To this end, I will be assisting the makers of HushMail, Hush
Communications (http://www.hush.com), to implement the OpenPGP
standard in their future products. They will be doing their own
announcement of this new relationship.
In addition, I will be assisting Veridis (http://www.veridis.com), a
recent spin-off of Highware (http://www.highware.com), to create
other OpenPGP compliant products, including software for certificate
authorities for the OpenPGP community.
I am also launching the OpenPGP Consortium (http://openpgp.org), to
facilitate interoperability of different vendors' implementations of
the OpenPGP standard, as well as to help guide future directions of
the OpenPGP standard.
This coming June marks the 10 year anniversary of the 1991 release of
PGP to the public. PGP was originally designed for human rights
applications, and to protect privacy and civil liberties in the
information age. By proliferating the OpenPGP standard, we can renew
that promise, and continue the commitment to personal privacy that
captured the imagination and participation of millions around the
world.
Philip Zimmermann
19 Feb 2001
[EMAIL PROTECTED]
http://web.mit.edu/prz
tel. +1 650 347-9743
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=====END PGP SIGNATURE=====
--
======================================================
Philip R Zimmermann http://web.mit.edu/prz
tel +1 650 347-9743 [EMAIL PROTECTED]
fax +1 650 348-4849
------------------------------
From: "Dann Corbit" <[EMAIL PROTECTED]>
Subject: Re: Big Numbers in C/C++
Date: Mon, 19 Feb 2001 12:16:23 -0800
"Douglas A. Gwyn" <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]...
> Paul Schlyter wrote:
> > I also believe MIRACL is somewhat more accurate - why? Because it
> > implements reals not as the usual floating-point numbers but as
> > rational numbers: A/B where A and B both are integers. ...
> > MIRACL includes a full set of transcendental functions
> > (logs/trigs/etc) for MIRACL hi-precision real numbers.
>
> Pray tell, how does MIRACL represent sin(1/2), log(5), etc.?
> Those are not rational numbers.
With an approximation. You only get exact results when the fractions are
exact (obviously). It does remarkably well at inverting a Hilbert matrix for
obvious reasons.
The one minor limitation I see with MIRACL is the lack of an exponent field.
So to represent really large numbers you need a really large object. It would
be nice if it could normalize blank trailing bits into an exponent field.
If you have never downloaded MIRACL to play with it, it is well worth a go.
The best thing about it is all of the pre-programmed solutions to interesting
problems. Also, because it uses C++ classes, to get some new idea working is
incredibly easy. I fool with it a bit so that it will print larger numbers
(some internal buffers are rather small). Here is a build of the factor
program and also pari that I made for Win32 systems. You can compare the
speed of solving large problems and MIRACL has a favorable result:
ftp://cap.connx.com/pub/factor.exe
ftp://cap.connx.com/pub/pari.exe
When you have problems that are represented with rational numbers, MIRACL is a
very natural solution (example: Recursive Monotone Stable numeric integration
technique by Favotti, Lotti, and Romani.)
--
C-FAQ: http://www.eskimo.com/~scs/C-faq/top.html
"The C-FAQ Book" ISBN 0-201-84519-9
C.A.P. FAQ: ftp://cap.connx.com/pub/Chess%20Analysis%20Project%20FAQ.htm
------------------------------
From: "Henrick Hellstr�m" <[EMAIL PROTECTED]>
Subject: Re: Given any arbitrary numbers a and b. Can I ALWAYS find a
transcendental number between a and b?
Date: Mon, 19 Feb 2001 21:19:17 +0100
Sorry, but now it is even less clear what you mean. You can't "find" any
transcendental number at all. But, yes, if that was what you meant, if a < b
then the set of transcendental numbers x such that a < x < b, has the same
cardinality as the entire set of real numbers.
--
Henrick Hellstr�m
StreamSec HB
"jtnews" <[EMAIL PROTECTED]> skrev i meddelandet
news:[EMAIL PROTECTED]...
> Since my previous post seemed to be not to clear
> to some people. Let me restate the problem another
> way which I hope will be simpler to understand.
>
> Given any arbitrary numbers a and b where a and
> b are NOT the same number.
>
> Can I ALWAYS find a transcendental number
> between a and b?
------------------------------
From: Jan Kristian Haugland <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b.Can I ALWAYS find a
Date: Mon, 19 Feb 2001 21:22:55 +0100
jtnews wrote:
> Since my previous post seemed to be not to clear
> to some people. Let me restate the problem another
> way which I hope will be simpler to understand.
>
> Given any arbitrary numbers a and b where a and
> b are NOT the same number.
>
> Can I ALWAYS find a transcendental number
> between a and b?
Yes. If c is a rational number with 0 < c < b-a,
then one of (a+b)/2, (a+b)/2 + c/pi is transcendental.
This, however, doesn't show that the set of
transcendentals is uncountable (see other comments
about the rationals).
--
Jan Kristian Haugland
http://home.hia.no/~jkhaug00
------------------------------
From: "Paul Lutus" <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b. Can I ALWAYS find a
transcendental number between a and b?
Date: Mon, 19 Feb 2001 12:29:01 -0800
"jtnews" <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]...
> Since my previous post seemed to be not to clear
> to some people. Let me restate the problem another
> way which I hope will be simpler to understand.
>
> Given any arbitrary numbers a and b where a and
> b are NOT the same number.
>
> Can I ALWAYS find a transcendental number
> between a and b?
If A and B are integers, yes.
1. Pi is transcendental.
2. Therefore, by definition, Pi+n is transcendental if n = integer.
3. Therefore this sequence can be constructed:
A, A+Pi-3, B.
As usual, I am more than happy to be corrected on this sequence of steps by
those more skilled than I am.
Is this the answer to your question?
--
Paul Lutus
www.arachnoid.com
------------------------------
From: "Ryan M. McConahy" <[EMAIL PROTECTED]>
Subject: Re: Could someone compile it for me?
Date: Mon, 19 Feb 2001 15:36:09 -0500
Pllllllleeeeeeeeeeeeeeeeeeeeeeaseee???
------------------------------
From: "Phil Fresle" <[EMAIL PROTECTED]>
Subject: Rijndael vb and vbscript code
Date: Mon, 19 Feb 2001 20:45:17 -0000
Reply-To: "Phil Fresle" <[EMAIL PROTECTED]>
I have posted vb and vbscript (asp) code implementing Rijndael on my web
site http://www.frez.co.uk
------------------------------
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