# This Week's Finds in Mathematical Physics (Week 226)

1. Nov 4, 2006

### John Baez

Also available at http://math.ucr.edu/home/baez/week226.html

February 23, 2003
This Week's Finds in Mathematical Physics (Week 226)
John Baez

This month I'm hanging out at CIRM, the "Centre International de Recontres
Mathematiques" near Marseille. It's like a little hotel with a classroom,
library and computers, set at the edge of a forest that borders the
region of limestone hills and cliffs called the Calanques on the coast
of the Mediterranean. All they do here is hold conferences: guests come
in, listen to math talks, and eat nice French meals prepared by the
overworked staff.

This February they're having a conference on logic and computation,
with a strong slant towards the use of diagrams:

1) Geometry of Computation 2006 (Geocal06), http://iml.univ-mrs.fr/geocal06/

The first week they had lots of talks on "higher-dimensional rewriting",
where you use diagrams to draw ways of rewriting ways of rewriting ways
of... rewriting strings of symbols. This is inherently connected to
n-categories, since n-categories show up whenever you consider "processes
that take a process and turn it into another process" - and iterate
this notion until your eyes start bugging out.

The first week was pretty cool, and I hope to talk about it someday.
The second week I've been recovering from the first week. But today
cryptography, random sequences, and logic.

Let's zoom in on NGC 1097. Here's an amateur astronomer's photograph
of this galaxy. It's a barred spiral galaxy that's colliding with a smaller
elliptical galaxy called NGC 1097A:

2) Daniel Verschatse, Antilhue - Chile, NGC 1097 in Fornax,
http://www.astrosurf.com/antilhue/ngc_1097_in_fornax.htm

NGC 1097 is called a "Seyfert galaxy" because its center emits lots
of radio waves, but not enough to be considered a full-fledged "quasar".
As you can see, the center also emits visible light:

3) Spiral galaxy NGC 1097, European Southern Observatory,
http://www.eso.org/outreach/press-rel/pr-2004/pr-28-04.html#phot-35d-04

Recently the VLT has gotten a really close look at the center of
NGC 1097. VLT stands for "Very Large Telescope". It's actually
four 8-meter telescopes and three smaller auxiliary ones, all of
which can function as a single unit, making it the biggest
telescope in the world. It's run by Europeans as part of
the "European Southern Observatory" - but it's based on a
mountain called Cerro Paranal in the driest part of the Atacama
Desert in northern Chile, which makes for wonderful viewing
conditions. They had to carry the enormous mirrors across
rugged desert roads to build this observatory:

4) Mirror Transport, European Southern Observatory,
http://www.eso.org/outreach/press-rel/pr-1997/phot-35g-97.jpg

but the view of the night sky up there makes it all worthwhile:

5) The Southern sky Above Paranal, European Southern Observatory,
http://www.eso.org/outreach/press-rel/pr-2005/images/phot-40b-05-normal.jpg

and now they have everything they need to take advantage of that location:

6) The VLT Array on Paranal Mountain, European Southern Observatory,
http://www.eso.org/outreach/press-rel/pr-2000/phot-14a-00-normal.jpg

So, what did they see when they looked at NGC 1097?

In the middle there seems to be a black hole, emitting radiation
as filaments of gas and dust spiral in. This has caused a ring
of new stars to form, wich are ionizing the hydrogen in their vicinity:

7) Feeding the Monster, European Southern Observatory
http://www.eso.org/outreach/press-rel/pr-2005/phot-33-05.html

This ring is about 5,500 light-years across! That sounds big, but the
galaxy is 45 million light-years away, so this is a stunningly detailed
photo. So, we can examine in great detail the process by which black
holes eat galaxies.

Anyway....

Talking to the logicians and computer scientists here, I'm hearing
lots of gossip that I don't usually get. For example, I just learned
that in 2004 there was a successful collision attack on MD5.

Huh? Well, it sounds very technical, but it boils down to this: it
means somebody took a function f that is known not to be one-to-one,
and found x and y such that f(x) = f(y).

You wouldn't think this would make news! But such functions, called
"cryptographic hash functions", are used throughout the computer security
business. The idea is that you can take any file and apply the hash
function to compute a string of, say, 128 bits. It's supposed to be hard
to find two files that give the same bit string. This lets you use the
bit string as a kind of summary or "digest" of the file. It's also
supposed to be hard to guess the contents of a file from its digest.
This lets you show someone the digest of a file without giving away
any secrets.

MD5 is a popular hash function invented by Ron Rivest in 1991. People
use it for checking the integrity of files: first you compute the digest
of a file, and then, when you send the file to someone, you also send
the digest. If they're worried that the file has been corrupted or
tampered with, they compute its digest and compare it to what you sent them.

People also use MD5 and other hash functions for things like digital
fingerprinting and copy protection. To illustrate this I'll give a
silly example that's easy to understand.

Suppose Alice proves Goldbach's conjecture in February and wants to
take her time writing a nice paper about it... but she wants to be
able to show she was the first to solve this problem, in case anyone
else solves it while she's writing her paper.

To do this, she types up a quick note explaining her solution, feeds
this file into a cryptographic hash function, and posts the resulting
128-bit string to the newsgroup sci.math in an article entitled
"I PROVED GOLDBACH'S CONJECTURE!" A dated copy appears on Google for
everyone to see.

Now, if Bob solves the problem in July while Alice is still writing up
her solution, Alice can reveal her note. If anyone doubts she wrote
her note back in February, they can apply the cryptographic hash function
and check that yes, the result matches the bit string she posted on Google!

For this to work, it had better be hard for a nasty version of Alice
to take Bob's solution and cook up some note summarizing it whose hash
function equals some bit string she already posted.

So, it's very good if the hash function is resistant to a "preimage
attack". A preimage attack is where for a given x you have a trick
for finding y such that f(y) = f(x).

Nobody has carried out a successful preimage attack on MD5. But,
people have carried out a successful "collision attack". This is
where you can cook up pairs x, y such that f(x) = f(y). This isn't
as useful, since you don't have control over *either* x or y. But,
there do exist fiendish schemes for conning people using collision attacks:

8) Magnus Daum and Stefan Lucks, Attacking hash functions by poisoned messages:
"The Story of Alice and Her Boss", http://www.cits.rub.de/MD5Collisions/

On this webpage you can see a letter of recommendation for Alice and
a letter granting her a security clearance which both have the same MD5
digest! It also explains how Alice could use these to do evil deeds.

For more on cryptographic hash functions and their woes, try these:

9) Cryptographic hash function, Wikipedia,
http://en.wikipedia.org/wiki/Cryptographic_hash

Steve Friedl, An illustrated guide to cryptographic hashes,
http://www.unixwiz.net/techtips/iguide-crypto-hashes.html

Now, if you're a mathematician, the whole idea of a cryptographic
hash function may seem counterintuitive. Just for a change of pace,
take another important cryptographic hash function, called SHA-1.
This is a function that takes any string of up to 2^64 bits and
gives a digest that's 160 bits long. So, it's just some function

f: S -> T

from a set S of size 2^{2^64 + 1} to a set T of size 2^160.

The first set is vastly bigger. So, the function f must be far from
one-to-one! So why in the world is anyone surprised, much less dismayed,
when people find a way to generate two elements in the first set that map
to the same element in the second?

One reason is that while the first set is much bigger than the second,
the second is mighty big too!

Suppose you're trying to do a preimage attack. Someone hands you an
element of T and asks you to find an element of S that maps to it.
The brute-force approach, where you keep choosing elements of S,
applying the function, and seeing if you get the desired element,
will on average take about 2^160 tries. That's infeasible.

Note that the huge size of S is irrelevant here; what matters most
is the size of T.

Or, suppose you're trying a brute-force collision attack. You start
looking through elements of S, trying to find two that map to the same
thing. On average it will take 2^80 tries - the square root of the
size of T. That's a lot less, but still infeasible.

(Why so much less? This is called the "birthday paradox": it's a lot
easier to find two people at a big party who share the same birthday
than to find someone with the same birthday as the host.)

Of course, a smarter approach is to use your knowledge about the function
f to help you find pairs of elements that map to the same thing. This
is what Xiaoyun Wang, Yiqun Lisa Yin, and Hongbo Yu actually did in
February 2005. Based on trials with watered-down versions of SHA-1,
they argued that they could do a collision attack that would take only
2^69 tries instead of the expected 2^80.

They didn't actually carry out this attack - with the computer power
they had, it would have taken them 5 million years. But their
theoretical argument was already enough to make people nervous!

And indeed, in August 2005, an improved version of their strategy
reduced the necessary number of tries to 2^63 - in other words,
reducing the time to just 78 thousand years, after only a few months
of work.

So, people are getting wary of SHA-1. This is serious, because it's
a widely used government standard. You can see the algorithm for this
function here:

10) SHA hash functions, Wikipedia,
http://en.wikipedia.org/wiki/SHA_hash_functions

People still hope that good hash functions exist. They hope that if
a function f is cleverly chosen, f(x) will depend in a "seemingly
random" way on x, so that given f(x), it's hard to compute some y
with f(y) = f(x). People call this a "one-way function".

Now we're finally getting to the really interesting math. It takes work
to make the concept of "one-way function" precise, but it can be done.
For example, Chapter 2 of this book starts out by defining "strong" and
"weak" one-way functions:

11) Oded Goldreich, Foundations of Cryptography, Cambridge U.
Press, Cambridge, 2004. Older edition available at
http://www.wisdom.weizmann.ac.il/~oded/frag.html

Since you can look them up and they're a bit gnarly, I won't give
these definitions here. But *roughly*, a "strong one-way function"
is a function f that satisfies two conditions:

A) You can compute f in "polynomial time": in other words,
there's an algorithm that computes it in a number of steps
bounded by some polynomial in the length of the input.

B) Given some input x, any polynomial-time probabilistic algorithm
has a very low chance of finding y with f(y) = f(x). Here a
"probabilistic algorithm" is just an algorithm equipped with

Condition B is related to the idea of a "preimage attack".
We allow the attacker to use a probabilistic algorithm because
it seems this can help them, and a strong one-way function
should be able to resist even really nasty attacks.

Unfortunately, nobody has proved that a one-way function exists!

In fact, the existence of a one-way function would imply that
"P does not equal NP". But, proving or disproving this claim is one
of the most profound unsolved math problems around. If you settle it,
you'll get a million dollars from the Clay Mathematics Institute:

11) Clay Mathematics Institute, P vs NP problem,
http://www.claymath.org/millennium/P_vs_NP/

But if you prove that P *does* equal NP, you might make more money by
breaking cryptographic hash codes and setting yourself up as the Napoleon
of crime.

The existence of one-way-functions is also closely related to the
existence of "pseudorandom" sequences. These are sequences of
numbers that "seem" random, but can be computed using deterministic
algorithms - so they're not *really* random.

To see the relation, recall that a good cryptographic hash code
shouldn't let you guess a message from its digest. So, for example,
if my message is the EMPTY STRING - no message at all! - its digest
should be a pseudorandom sequence.

For example, if we apply SHA-1 to the empty string we get the
following 160 bits, written as 40 hexadecimal digits:

SHA1("") = "da39a3ee5e6b4b0d3255bfef95601890afd80709"

Seems mighty random to me! But of course it comes out the same every time.

Indeed, if you've ever seen a "random number generator" while messing
with computers, it probably was a pseudorandom number generator. There
are a few people who generate random numbers from physical phenomena
like atmospheric noise. In fact, there's a website called random.org

12) Random.org: random integer generator, http://random.org/

There's also a website that offers *nonrandom* numbers:

13) NoEntropy.net: your online source for truly deterministic numbers,
http://www.noentropy.net/

But joking aside, it's a really tantalizing and famous problem to figure
out what "random" means, and what it means for something to "seem"
random even if it's the result of a deterministic process. These are
huge and wonderful philosophico-physico-mathematical questions with
serious practical implications. Much ink has been spilt regarding them,
and I don't have the energy to discuss them carefully, so I'll just
say some stuff to pique your curiosity, and then give you some references.

We can define a "random sequence" to be one that no algorithm can guess
with a success rate better than chance would dictate. By virtue of this
definition, no algorithm can generate truly random sequences.
It's easy to prove that most sequences are random - but it's also
easy to prove that you can never exhibit any one *particular* sequence
and prove it's random! Chaitin has given a marvelous definition of a
particular random sequence of bits called Omega using the fact that no
algorithm can decide which Turing machines halt... but this random sequence is
uncomputable, so you can't really "exhibit" it:

14) Gregory Chaitin, Paradoxes of randomness, Complexity 7 (2002), 14-21.
Also available as http://www.umcs.maine.edu/~chaitin/summer.html

So, true randomness is somewhat elusive. It seems hard to come by except
in quantum mechanics. For example, the time at which a radioactive
atom decays is believed to be *really* random. I'll be pissed off if
it turns out that God (or his henchman Satan) is fooling us by simulating
quantum mechanics with some cheap pseudorandom number generator!

Similarly, we can define a "pseudorandom sequence" to be one that no
*efficient* algorithm can guess with a success rate higher than chance
would dictate.

Efficiency is a somewhat vague concept. It's popular to define it by
saying an algorithm is "efficient" if it runs in "polynomial time": the
time it takes to run is bounded by some polynomial function of the
size of the input data. If the polynomial is

p(n) = 1000000000000000 n^1000000000000000 + 1000000000000000

most people wouldn't consider the algorithm efficient, but this definition
is good for proving theorems, and usually in practice the polynomial
turns out to be more reasonable.

A "pseudorandom number generator" needs to be defined carefully if
we want to find efficient algorithms that do this job. After all,
no efficient algorithm can produce a sequence that no efficient
algorithm can guess: *it* can always guess what it's going to do!

So, the basic idea is that a pseudorandom number generator should be
an efficient algorithm that maps short truly random strings to long
pseudorandom strings: we give it a short random "seed" and it cranks
out a lot of digits that no efficient algorithm can guess with a
success rate higher than chance would dictate.

If you want a more precise definition and a bunch of theorems, try these:

15) Oded Goldreich, papers and lecture notes on pseudorandomness
available at http://www.wisdom.weizmann.ac.il/~oded/pp_pseudo.html

M. Luby, Pseudorandomness and Cryptographic Applications.
Princeton, NJ: Princeton University Press, 1996.

Unfortunately, nobody has proved that pseudorandom number generators
exist! So, this whole subject is a bit like axiomatic quantum field
theory, or the legendary Ph.D thesis where the student couldn't produce
any examples of the mathematical gadgets he was studying. It's a
But in the case of pseudorandom number generators, the subject is
too important not to take the chance.

One of the most interesting things about pseudorandom number generators is
that they let us mimic probabilistic algorithms with deterministic ones.
them for you.

As I already mentioned, a "probabilistic algorithm" is just a deterministic
generator. Just imagine a computer with the ability to reach out and
flip a coin when it wants to. A problem is said to be in "BPP" -
"bounded-error probabilistic polynomial time" - if you can find
polynomial-time probabilistic algorithms that solve this problem
with arbitrarily high chance of success.

It's a fascinating question whether randomness actually helps you compute
stuff. I guess most computer scientists think it does. But, it's tricky.
For example, consider the problem of deciding whether an integer is prime.
Nobody knew how to do this in polynomial time... but then in 1977, Solovay
and Strassen showed this problem was in BPP. This is one of the results
that got people really excited about probabilistic algorithms.

However, in 2002, Maninda Agrawal, Neeraj Kayal and Nitin Saxena showed
that deciding whether numbers are prime is in P! In other words, it can
be solved in polynomial time by a plain old deterministic algorithm:

16) Anton Stiglic, Primes is in P little FAQ,
http://crypto.cs.mcgill.ca/~stiglic/PRIMES_P_FAQ.html

Is BPP = P? Nobody knows! But if good enough pseudorandom number
generators could be shown to exist, we would have BPP = P, since then we
could use these pseudorandom numbers as a substitute for truly random ones.
This is not obvious, but it was proved by Nisan and Wigderson in 1994.

Here's a great review article that discusses their result:

17) Luca Trevisian, Pseudorandomness and combinatorial constructions,
available as cs.CC/0601100.

I recommend reading it along with this:

18) Scott Aaronson, Is P versus NP formally independent?, available
at http://www.scottaaronson.com/papers/pnp.pdf

This is a delightfully funny and mindblowing crash course on logic.
It starts with a review of first-order logic and Goedel's theorems,
featuring a dialog between a mathematician and the axiom system
ZF+not(Con(ZF)). Here ZF is the popular Zermelo-Fraenkel axioms
for set theory and not(Con(ZF)) is a statement asserting that ZF
is not consistent. Thanks to Goedel's second incompleteness theorem,
ZF+not(Con(ZF)) is consistent if ZF is! The mathematician is naturally
puzzled by this state of affairs, but in this dialog, the axiom system
explains how it works.

Then Aaronson zips on through Loeb's theorem, which is even weirder than
Goedel's first incompleteness theorem. Goedel's first incompleteness
theorem says that the statement

This statement is unprovable in ZF.

is not provable in ZF, as long as ZF is consistent. Loeb's theorem says
that the statement

This statement is provable in ZF.

*is* provable in ZF.

Then Aaronson gets to the heart of the subject: a history of the P vs. NP
question. This leads up to the amazing 1993 paper of Razborov and Rudich,
which I'll now summarize.

The basic point of this paper is that if P is not equal to NP, as most
mathematicians expect, then this fact is hard to prove! Or, as Aaronson
more dramatically puts it, this conjecture "all but asserts the titanic
difficulty of finding its own proof".

Zounds! But let's be a bit more precise. A Boolean circuit is a gizmo
built of "and", "or" and "not" gates, without any loops. We can think
of this as computing a Boolean function, meaning a function of the form:

f: {0,1}^n -> {0,1}

Razborov and Rudich start by studying a common technique for proving lower
bounds on the size of a Boolean circuit that computes a given function.
The technique goes like this:

A) Invent some way of measuring the complexity of a Boolean function.

B) Show that any Boolean circuit of a certain size can compute only
functions of complexity less than some amount.

C) Show that the function f has high complexity.

They call this style of proof a "natural" proof.

The P versus NP question can be formulated as a question about the size
of Boolean circuits - but Razborov and Rudich show that, under certain
assumptions, there is no "natural" proof that P is not equal to NP.
What are these assumptions? They concern the existence of good
pseudorandom number generators. However, the existence of these
pseudorandom number generators would follow from P = NP! So, if
P = NP is true, it has no natural proof.

Aaronson says this is the deepest insight into the P versus NP question
so far. I would like to understand it better - I explained it very
sketchily because I don't really understand it yet. Aaronson recommends
us to these papers for more details:

19) A. A. Razborov, Lower bounds for propositional proofs and independence
results in bounded arithmetic, in Proceedings of ICALP 1996, 1996, pp. 48-62.
Also available at http://genesis.mi.ras.ru/~razborov/icalp.ps

20) R. Raz, P =/= NP, propositional proof complexity, and resolution
lower bounds for the weak pigeonhole principle, in Proceedings of ICM 2002,
Vol. III, 2002, pp. 685-693. Also available at
http://www.wisdom/weizmann.ac.il/~ranraz/publication/Pchina.ps

21) S. Buss, Bounded arithmetic and propositional proof complexity,
in Logic of Computation, ed. H. Schwictenberg, Springer-Verlag, 1997,
pp. 67-122. Also available at
http://math.ucsd.edu/~sbuss/ResearchWeb/theoria/index.html

(Why don't these guys use the arXiv??)

Also, here are some lecture notes on Boolean circuits that might help:

22) Uri Zwik, Boolean circuit complexity,
http://www.math.tau.ac.il/~zwick/scribe-boolean.html

Aaronson wraps up by musing on the possibility that the P versus NP
question independent from strong axiom systems like Zermelo-Fraenkel
set theory. It's possible... and it's possible that this is true and
unprovable!

So, there is a fascinating relationship between one-way functions,
pseudorandom numbers, and incompleteness - but it's a relationship
shrouded in mystery... and perhaps inevitably so. Perhaps it will
always remain unknown whether this mystery is inevitable... and perhaps
this is inevitable too! And so on.

Here's a question for you experts out there: have people studied Goedel-like
self-referential sentences of this form?

The shortest proof of this statement has n lines.

I hear that people *have* studied ones like

The shortest proof that 0 = 1 has n lines.

and they've proved that the shortest proof of *this* has to keep getting
longer with larger n, as long as one is working in a sufficiently powerful
and consistent axiom system. This is fairly obvious if you know how
Goedel's second incompleteness theorem is proved... but it's possible that
some interesting, nonobvious lower bounds have been proved. If so, I'd
like to know!

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

"Anyone who considers arithmetical methods of producing random digits
is, of course, in a state of sin." - John von Neumann

"He is the Napoleon of crime, Watson. He is the organizer of half
that is evil and of nearly all that is undetected in this great city.
He is a genius, a philosopher, an abstract thinker...." - Sherlock Holmes

-----------------------------------------------------------------------
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http://math.ucr.edu/home/baez/twfcontents.html

A simple jumping-off point to the old issues is available at

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If you just want the latest issue, go to

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2. Nov 4, 2006

### tessel@um.bot

On Sat, 11 Feb 2006, John Baez mentioned that md5sum was "broken" about a
year ago. I just wanted to add:

1. If I am not mistaken, sha-1 and md5sum are different algorithms (IIRC,
both are known to be insecure). Anyone interested can probably figure
this out from the official specs:

SHA-1 hash algorithm:
http://www.ietf.org/rfc/rfc3174.txt

md5sum message-digest algorithm:

http://www.faqs.org/rfcs/rfc1321.html

2. The latest versions of the open source utility gpg supports a more
secure algorithm, SHA-512, which AFAIK has not been broken; see

Tony Stieber, "GnuPG Hacks", Linux Journal, March 2006.

3. Even insecure checksum utilities are probably better than none at all.
Indeed, checking the given example

gpg --print-md md5 letter_of_rec.ps order.ps
A2 5F 7F 0B 29 EE 0B 39 68 C8 60 73 85 33 A4 B9
A2 5F 7F 0B 29 EE 0B 39 68 C8 60 73 85 33 A4 B9

Oh NOOOOOO!!! But wait, there's more:

gpg --print-md sha1 letter_of_rec.ps order.ps
0783 5FDD 04C9 AFD2 8304 6BD3 0A36 2A65 16B7 E216
3548 DB4D 0AF8 FD2F 1DBE 0228 8575 E8F9 F539 BFA6

gpg --print-md RIPEMD160 letter_of_rec.ps order.ps
9069 8ACC 6D67 6608 657B 9C26 F047 59A1 DC0E 6CA1
C1BB DE12 B312 EAAD DD3D D3B8 4CA1 CB1B BA47 DD13

Ah HAAAAAA!!! Gotcha, Alice!

> For more on cryptographic hash functions and their woes, try these:
>
> 9) Cryptographic hash function, Wikipedia,
> http://en.wikipedia.org/wiki/Cryptographic_hash

Of course, bear in mind that anyone can edit, blah, blah (including
unregistered users, contrary to widely publicized news reports based upon
a fundamental misunderstanding).

> So, people are getting wary of SHA-1.

Something to think about when you are reading articles at that website
which anyone can <s>poison</s> edit.

> These are huge and wonderful philosophico-physico-mathematical questions
> with serious practical implications.

You mean the Weyl curvature hypothesis? :-/

But while we should never neglect incompleteness entirely, I was
fascinated to discover from my readings a few years back that even first
order logic has its fascinations!

Joel Spencer, The Strange Logic of Random Graphs, Springer 2001

Here's a thought: "Everyone knows" that if on day D, mathematician M is
studying an example of size S in class C, he is more likely to be studying
a "secretly special" representative R than a generic representative G of
size S. Why? Because the secretly special reps show up in disguise in
other areas, and M was probably hacking through the jungle from one of
those places when he got lost and ate a poisoned cache.

Kinda like the "random appearing" md5sum of the empty file.

Is scientific creativity nothing but a hash sum hack?

Bite me, Brian!

"T. Essel"

3. Nov 4, 2006

### John Baez

Here are some corrections and clarifications, mostly thanks to a
friend who usually prefers to remain anonymous:

In article <dsirq1$pjg$1@glue.ucr.edu>,
John Baez <baez@math.removethis.ucr.andthis.edu> wrote:

>MD5 is a popular hash function invented by Ron Rivest in 1991.

This is what it says in Wikipedia:

http://en.wikipedia.org/wiki/MD5

with a big picture of Rivest right on top of the article,
but my friend says

"I think it's usually credited to a small set of coinventors,
and I think Ralph Merkle is a coinventor either of MD5 or one
of its immediate ancestors."

>People use it for checking the integrity of files: first you compute the
>digest of a file, and then, when you send the file to someone, you also send
>the digest. If they're worried that the file has been corrupted or
>tampered with, they compute its digest and compare it to what you sent them.

Of course, if deliberate tampering is what you fear, you have
to send the digest by a different channel than the original file,
or use some other trick.

>But if you prove that P *does* equal NP, you might make more money by
>breaking cryptographic hash codes and setting yourself up as the Napoleon
>of crime.

Or, you could make lots of money by solving problems that nobody
else can solve. This could be a more sustainable lifestyle... but
I wanted to work in a reference to that Sherlock Holmes quote, to
play off against the von Neumann quote.

>We can define a "random sequence" to be one that no algorithm can guess
>with a success rate better than chance would dictate.

Here "generate" would be clearer than "guess", since I was
trying to allude to the usual notion of randomness from
algorithmic information theory (in an informal sort of way).

> Chaitin has given a marvelous definition of a
>particular random sequence of bits called Omega using the fact that no
>algorithm can decide which Turing machines halt... but this random
> sequence is uncomputable, so you can't really "exhibit" it:

On the other hand, Wolfgang Brand points out this paper:

http://www.cs.auckland.ac.nz/~cristian/Calude361_370.pdf

where the first 64 bits of Omega have been computed. (There's
no contradiction, as the paper explains.)

>Then Aaronson gets to the heart of the subject: a history of the P vs. NP
>question. This leads up to the amazing 1993 paper of Razborov and Rudich,
>which I'll now summarize.

Here's the paper:

Alexander A. Razborov and Steven Rudich, Natural proofs, in
Journal of Computer and System Sciences, Vol. 55, No 1, 1997, pages 24-35.
Available at http://www-2.cs.cmu.edu/~rudich/papers/natural.ps and
http://genesis.mi.ras.ru/~razborov/int.ps

Aaronson says it was written in 1993 even though the date of publication
and the date on the paper itself (1996 or 1999, depending on which copy
you look at) are later. I believe him; you have to be careful when using
the /today command in LaTeX, since if you LaTeX the same paper 6 years
later, you'll get a new date.

>The P versus NP question can be formulated as a question about the size
>of Boolean circuits - but Razborov and Rudich show that, under certain
>assumptions, there is no "natural" proof that P is not equal to NP.
>What are these assumptions? They concern the existence of good
>pseudorandom number generators. However, the existence of these
>pseudorandom number generators would follow from P = NP! So, if
>P = NP is true, it has no natural proof.

Aargh!

In both cases here when I wrote P = NP, I meant P *not* equal to NP.

4. Nov 4, 2006

### John Baez

In article <Pine.LNX.4.61.0602102025360.19201@zeno1.math.washington.edu>,
<tessel@um.bot> wrote:

>On Sat, 11 Feb 2006, John Baez mentioned that md5sum was "broken" about a
>year ago. I just wanted to add:

>1. If I am not mistaken, sha-1 and md5sum are different algorithms (IIRC,
>both are known to be insecure).

Yeah, I said SHA-1 and MD5 are different, and I said they were both vulnerable
to collision attacks. MD5 is very vulnerable in practice, while the
vulnerability of SHA-1 is still theoretical: you'd have to have big
computers or lots of time or another clever idea to exploit it.
(Guess who's likely to have all three!)

I gave this reference for MD5:

Magnus Daum and Stefan Lucks, Attacking hash functions by poisoned
messages: "The Story of Alice and Her Boss",
http://www.cits.rub.de/MD5Collisions/

and this one for SHA-1:

SHA hash functions, Wikipedia,
http://en.wikipedia.org/wiki/SHA_hash_functions

The latter has some good links, including this nice review:

Arjen K. Lenstra, Further progress in hashing cryptanalysis,
February 26, 2005, http://cm.bell-labs.com/who/akl/hash.pdf

>> For more on cryptographic hash functions and their woes, try these:
>>
>> 9) Cryptographic hash function, Wikipedia,
>> http://en.wikipedia.org/wiki/Cryptographic_hash

>Of course, bear in mind that anyone can edit, blah, blah (including
>unregistered users, contrary to widely publicized news reports based upon
>a fundamental misunderstanding).

Yes, I could have tried to give references to dated versions of
Wikipedia articles, so I could be sure they're good... but I'm an optimist.

>> These are huge and wonderful philosophico-physico-mathematical questions
>> with serious practical implications.

>You mean the Weyl curvature hypothesis? :-/

Heh, no - I mean stuff like whether there's such a thing as a provably
good cryptographic hash code function, or cipher.

> Joel Spencer, The Strange Logic of Random Graphs, Springer 2001

>Here's a thought: "Everyone knows" that if on day D, mathematician M is
>studying an example of size S in class C, he is more likely to be studying
>a "secretly special" representative R than a generic representative G of
>size S. Why? Because the secretly special reps show up in disguise in
>other areas, and M was probably hacking through the jungle from one of
>those places when he got lost and ate a poisoned cache.

Interesting.

>Bite me, Brian!

I have no idea what that's a reference to.

Here's some more stuff, from my email correspondence.
It's more related to physics.

I wrote:

> Allan Erskine wrote:

>> I enjoyed week 226! Algorithmic complexity was the area I studied
>> in... Your readers might find "The Tale of One-way Functions" by
>> Leonid Levin an enjoyable read:
>>
>> http://arxiv.org/abs/cs.CR/0012023

>Hey, that's great! I'm printing it out now... Levin and I have
>argued against Greg Kuperberg and others on sci.physics.research:
>we tend to think that quantum computers are infeasible *in principle*.

>> As for your "shortest proof of this statement has n lines" question,
>> you may have noticed that Chaitin asks a very similar question about
>> the shortest proofs that a LISP program is "elegant" (most short) and
>> proves a strong incompleteness result with an actual 410 + n
>> character LISP program! Crazy...
>>
>> http://www.cs.auckland.ac.nz/CDMTCS/chaitin/lisp.html

>Yes. You might like the following related article below.
>
>Best,
>jb

It's an old one...

From: b...@math.ucr.edu (john baez)
Subject: Re: compression, complexity, and the universe
Date: 1997/11/20
Message-ID: <652c5t$62g$1@agate.berkeley.edu>#1/1
X-Deja-AN: 291100089
References: <64nsqo$8rg$1@agate.berkeley.edu> <346fd86d.1059260@news.demon.co.uk> <64t2ar$qcs@charity.ucr.edu> Originator: bunn@pac2 Organization: University of California, Riverside Newsgroups: sci.physics.research,comp.compression.research In article <651lm1$q3...@agate.berkeley.edu>,
Aaron Bergman <aaron.berg...@yale.edu> wrote:

>The smallest number not expressable in under ten words

Hah! This, by the way, is the key to that puzzle I laid out:
prove that there's a constant K such that no bitstring can be
proved to have algorithmic entropy greater than K.

I won't give away the answer to the puzzle; anyone who gets
stuck can find the answer in Peter Gacs' nice survey, "Lecture
notes on descriptional complexity and randomness", available at

http://www.cs.bu.edu/faculty/gacs/

In my more rhapsodic moments, I like to think of K as the
"complexity barrier". The world *seems* to be full of incredibly
complicated structures --- but the constant K sets a limit on our
ability to *prove* this. Given any string of bits, we can't rule
out the possibility that there's some clever way of printing it
out using a computer program less than K bits long. The Encyclopedia
Brittanica, the human genome, the detailed atom-by-atom recipe for
constructing a blue whale, or for that matter the entire solar system
--- we are unable to prove that a computer program less than K bits
long couldn't print these out. So we can't firmly rule out the
reductionistic dream that the whole universe evolved mechanistically
starting from a small "seed", a bitstring less than K bits long.
(Maybe it did!)

So this raises the question, how big is K?

It depends on ones axioms for mathematics.

Recall that the algorithmic entropy of a bitstring is defined
as the length of the shortest program that prints it out. For
any finite consistent first-order axiom system A exending the
usual axioms of arithmetic, let K(A) be the constant such that
no bitstring can be proved, using A, to have algorithmic entropy
greater than K(A). We can't compute K(A) exactly, but there's a
simple upper bound for it. As Gacs explains, for some constant c
we have:

K(A) < L(A) + 2 log_2 L(A) + c

where L(A) denotes the length of the axiom system A, encoded as
bits as efficiently as possible. I believe the constant c is
computable, though of course it depends on details like what
universal Turing machine you're using as your computer.

What I want to know is, how big in practice is this upper
bound on K(A)? I think it's not very big! The main problem
is to work out a bound on c.

5. Nov 4, 2006

### Stephen Riley

In message <dsn1ji$sfj$1@glue.ucr.edu>, John Baez
<baez@math-cl-n03.math.ucr.edu> writes

>"I think it's usually credited to a small set of coinventors,
>and I think Ralph Merkle is a coinventor either of MD5 or one
>of its immediate ancestors."

The book "Applied Cryptography" by Bruce Schneier goes into detail on
these algorithms and has loads of references. Schneier credits Rivest as
the designer of MD4, saying Bert den Boer and Antoon Bosselaears
successfully crpytanalysed the last of the algorithms three rounds,
while Ralph Merkle successfully attacked the first two rounds. Others
are credited (with attacks and analysis on MD4) too. Schneier credits
Rivest as strengthening MD4 with the result being MD5. While Rivest
himself outlined the improvement of MD5 over MD4 in
"The MD5 Message Digest Algorithm", RFC 1321, Apr 1992

If you want more references, Schneier's book has another 1652 of them :)

A quick google also turns up citations like "At Crypto '91 Ronald L.
Rivest introduced the MD5 Message Digest Algorithm as a strengthened
version of MD4, differing from it on six points".

>Of course, if deliberate tampering is what you fear, you have
>to send the digest by a different channel than the original file,
>or use some other trick.

By coincidence my Masters thesis, soon due for submission, was on the
use of one of those other tricks, in business: digital signatures.
Nothing interesting mathematically, it was only as interesting a subject
as I could manage to get a degree in IT to be :)

>>But if you prove that P *does* equal NP, you might make more money by
>>breaking cryptographic hash codes and setting yourself up as the Napoleon
>>of crime.

For a bit of fun I once wrote a program that would generate random
programs, spitting out and testing something like 50,000 a minute. It
would quickly output programs for polynomial time problems, like finding
greatest common divisors or whatever, but the only ones it spat out for
NP problems like factoring were polynomial solutions, such as iterating
through odd numbers until the square root of n. As I say it was nothing
scientific and it would probably be unlikely to spit out anything useful
anyway, just too many combinations. Not sure if that was a new idea, but
probably not.

--
Stephen Riley

6. Nov 4, 2006

### jmfbah@aol.com

In article <dsn1ji$sfj$1@glue.ucr.edu>,
baez@math-cl-n03.math.ucr.edu (John Baez) wrote:

<snip>

<snip>

/BAH

7. Nov 4, 2006

### Squark

Hello John!

John Baez wrote:
> Hah! This, by the way, is the key to that puzzle I laid out:
> prove that there's a constant K such that no bitstring can be
> proved to have algorithmic entropy greater than K.
> ...
> Recall that the algorithmic entropy of a bitstring is defined
> as the length of the shortest program that prints it out.

because it's 4 AM). What does it mean for a program to "print
out" a bit string? I see two interpretations:
1) As a substring of its output
2) As it's precise output
If we use interpretation 1 the statement is obviously correct
because I can construct a finite program that prints out any
finite bit string: it just prints out all of the natural
numbers in binary form, one after another. This is hardly
interesting, though.
If we use interpretation 2 the statement seems to be false.
Suppose such a constant K exists. Then, let us write down all
of the programs of length at most K and run them in parallel.
Given there are N such programs, we can choose some M with
2^M > N, and wait until each of them prints the M-th bit of
the output or finishes execution. Then, there is surely a
string of length M which neither of them printed and for
which this can be proved (by the very experiment I am
describing).
Maybe you're using another interpretation of "print out":
3) As a substring positioned at the end of the output
but that would be somewhat strange.

Best regards,
Squark

8. Nov 4, 2006

### Ralph Hartley

Squark wrote:
> Hello John!
>
> John Baez wrote:
>
>>Hah! This, by the way, is the key to that puzzle I laid out:
>>prove that there's a constant K such that no bitstring can be
>>proved to have algorithmic entropy greater than K.
>>...
>>Recall that the algorithmic entropy of a bitstring is defined
>>as the length of the shortest program that prints it out.

>
> What does it mean for a program to "print
> out" a bit string? I see two interpretations:

...
> 2) As it's precise output

This is the intended interpretation. It is usually required that the
machine print the string, and nothing else, and then halt.

> If we use interpretation 2 the statement seems to be false.
> Suppose such a constant K exists. Then, let us write down all
> of the programs of length at most K and run them in parallel.
> Given there are N such programs, we can choose some M with
> 2^M > N, and wait until each of them prints the M-th bit of
> the output or finishes execution.

But those are not the only two possibilities. Some programs will
continue running forever never printing any bits at all.

For some of those it cannot be proved that they run forever.

For some of those, there is no bitstring they can be proved not to print.

> Then, there is surely a
> string of length M which neither of them printed and for
> which this can be proved (by the very experiment I am
> describing).

Of course there are strings none of the programs of length K print
(infinitely many).

But none can be proved to have that property. Your procedure fails,
because no matter how long you run your simulation you can never be sure
that there are not some programs that would eventually print your string.

Perhaps this is too abstract. We can construct a *particular* program
with that property.

Proof (of the original theorem):

Consider the sequence of programs P_m that do the following:

For each proof (in order of length) {
If the proof is of a statement of the form
"bitstring x has complexity greater than m"
then {
print x
halt
}
}

P_m has length C + log(m), for some C.

If there exists a bitstring with complexity provably greater than m,
then P_m will print one and halt, otherwise P_m runs forever without
printing anything.

Let K be a number such that K > C + log(K)

If P_K halts, then it would be a program of length less than K which
prints a string that is provably not printed by any program of length
less than K.

Therefore, unless false statements can be proved, P_K does not halt.

So K has the required property.

Ralph Hartley