Illegal Prime Number Unzips to DeCSS

Bob9113 writes: "A person named Phil Carmody has found a very interesting prime number. When converted to hexadecimal, the result is a gzip that contains a DeCSS implementation. I've posted a short bit of Java here that takes the prime as a command line parameter and dumps the result to standard out if you want to test it." Very clever, I just wish the background on that page wasn't headache inducing.

Hmm..

Will this number now be a prime suspect?

---
ticks = jiffies;
while (ticks == jiffies);
ticks = jiffies;

Re:Windows 2000 encoded to a single number!

You're right that anything could be encoded into a prime number (with a suitable prime->original conversion).

But using some prime->original decoder is no different from say, gunzip. It's decoding of information in on form back to it's "original" form.

So no, making copies of Win2K is not legal wether it's gzipped or encoded into a prime.

And distributing DeCSS as a prime number (or gzipped) doesn't change the legality either.

Subtracting 1 from the number then distributing that number just adds another layer on your decoder, it doesn't change what you're doing. gzipping something twice doesn't remove the licensing restrictions either.

However, this is interesting because it puts a new angle on the flawed notion of "Intellectual Property" (or, Intellectual Poverty as I like to call it, because that must be what we suffer from if we restrict other's access to ideas that are indeed just mathematics in some form or another).

Tomorrow's Headlines Today

RIAA Petitions Congress To Ban Number Theory
Mathematicians Declared "Enemy of Intellectual Property (and the American Way)"
Rambus Patents Prime Numbers

Any guesses about which one you'll see first? :)

This may be true...

But if so, nobody has proved it for Pi or e, at least. I don't know if it's been proven for "starting sequences" of prime numbers.

Beware of two things you're doing here: you're imagining that primes, Pi, and e are all sequences of "random" digits. They certainly look that way, but it isn't true, and some of that non-randomness may, for example, prevent a particular number from ever appearing in the digit sequence. Secondly, you're trying to make a mathematical argument from "common sense" rather than from axioms and logic. That doesn't work as often as you'd wish it would; common sense sucks.

Re:DeCSS old, but an illegal number is certainly i

There are already a few "forbidden numbers"

Note the Orthodoxes are against many countings and IDs because in some of them may appear the idoneous "666".

13 floor is non-existent in some places with predominant anglo-saxon population

2 is also for some cultures a "forbidden" or "bad" number. Btw never give two flowers to a girl from slavic culture.

Some think that the square root of two and pi were "demonic" numbers for Pitagorics, a mysthical sect of Antiquity and predecessors of many Christian ideas and with some love to play maths. The fact of the existence of real numbers was felt as a "fault" in the building of the Universe... Btw Pitagorics were responsible for the advent of prime numbers.

Easy--infinite number of primes

This is very easy.

If you want to find something in a prime number, you figure out what you're looking for--in this case, the gziped code. You then search for prime numbers that start with those digits. Since there are an infinite number of prime numbers, you will always be able to find one (given enough time).

You could also find DeCSS gzipped in a section of Pi or e, based on similar ideas.

Another illegal prime, efdtt.c

Inspired by Phil's effort, a prime number [cmu.edu] encoding of the source of efdtt.c [cmu.edu] has been contributed by Charles M. Hannum.

Re:Easy--infinite number of primes

You then search for prime numbers that start with those digits. Since there are an infinite number of prime numbers, you will always be able to find one (given enough time).

Wrong.

Tell me the number out of all odd numbers ending with 2. Or take all numbers which don't contain the digit 9 and ...

Just because something is infinite doesn't mean it contains everything.
One had to prove that for every number N there existed n, x such that

P = N*B^n + x

where x B^n and B is the base (10 for decimal, 16 for hexadecimal etc...).
I for one am not sure whether this is true or not. I guess it's true and could be proved analogous to the basic proof that there are infinite primes.

Hmmm...

A person named Phil Carmody has found a very interesting prime number. When converted to hexadecimal, the result is a gzip that contains a DeCSS implementation.

The odds of this happening in this order are slim to none. If you believe in this chain of the evenets I have some stock to sell you. What really happened was most certainly the reverse. He took gzip that contained DeCSS, converted it to hex and analyzed the number. The good geek karma dictated that this number should be a prime and the rest is now the history =)

Re:numbers and itellectual property

It's not that simple.

They couldn't trademark '80486' because it's a part number, and any other manufacturer could also call their chip an '80486'. Just like different word-processor makers can come out with version 7.0 at the same time.

And as any type of data can be converted to 'just a number'.... this won't hold up. It's still decss, just encoded and padded out to a prime.

Re:numbers

This can be persuaive because it shows a way to use a computer program (gzip) to circumvent CSS when that program was clearly never intended as a circumvention method in the first place. This is an attack on DMCA in the broad, rather than on CSS and MPAA in particular.

numbers and itellectual property

as i recall, numbers alone can never be considered intellectual property. that's what bit intel in the ass with the 486. all the companies that made knockoffs were calling them 486's, diluting the namespace. so intel came out with "pentium" to solve that problem.

the question now is whether the courts would consider this just a number, or an encoding of the decss data into a number.

7 lines != non attempt .

look at the 2 line program that implements RSA, which many people consider a very challenging thing to crack!

print pack"C*",split/\D+/,`echo "16iII*o\U@{$/=$z;[(pop,pop,unpack"H*",<>

)]}\EsMsKsN0[lN*1lK[d2%Sa2/d0<X+d*lMLa^*lN%0]ds Xx++lMlN/dsM0

are you saying that RSA is weaker then CSS because it only takes 2 lines?

Windows 2000 encoded to a single number!

The fact of the matter is that every piece of digital information is nothing but a sting of digits.

This one is interesting in that the number happens to be prime.
(Is this a mathematical trick? If not how on earth did the author make this discovery?)

My question for a lawyer is this; does Microsoft have legal copyright on some numbers?

If so, do they also own every number that can be derived mathematically from them?

If not, can we legally store any copyrighted files with say 1 subtracted from the number?

(Think of it as insecure encryption with a trivial key and algorithm.)

And finally if this act would be illegal, then surely as a copyright holder I own rights to all digital data as you can mathematically transform between any two numbers without much difficulty.

A-HA

So either God uses Linux, or maybe the MPAA is satanic after all, and he built DeCSS into the universe to make it crumble..

BTW, doesn't the MPAA's address have the number 666 in it? Or am I thinking of another corp.?

--Never trust a tech who tattoes his IP to his arm, especially if its DHCP.

You can reduce this further.

Prime numbers are countable. You in theory can be able to reduce this from 1400+ digits by saying it's the 12345...42153th prime (perhaps about 100 digits).

However determining this number would be (ludicrously) computionally expensive. Another quest for distributed.net?

Why work on the CSS code, why not the keys themselves? That would be more interesting.

Re:Isn't that whole DeCSS thing getting kind of ol

If DECSS is legal in Canada/Mexico, why not bring the lawsuit up as an illegal trade practice under NAFTA?

Re:Windows 2000 encoded to a single number!

does Microsoft have legal copyright on some numbers?

No, they don't. They have copyright on some particular piece of source or binary code. This copyright applies regardless of the form in which the code is stored, and applies to obvious derivatives as well.

Storing a piece of data as an integer is simply a different way of encoding and storing data, like ASCII or EBCDIC. Saying that performing a simple mathematical operation on an integer negates copyright is as preposterous as saying that, say, my novel is copyright when stored in ASCII but public domain when stored in EBCDIC. As to the number-transform question: if you encoded my novel and transformed it into "2", I would certainly not hold copyright on the number 2. That's because you couldn't logically argue that my novel is stored within the number "2". You could write a program to reverse whatever procedure you used to reduce my novel to 2, and when fed 2 it would spit out my novel. In that case, the combination of the data and your program would be illegal, because it's just another way of storing data.

In other words, as a copyright holder, you hold copyright on all digital data which can reasonably be seen as an encoding of your copyright data. It's quite straightforward, really, and semantic number games never end up meaning anything in the real world.

Reminds me of the Crystal Rod Encyclopedia

There was a short science fiction story that went something like this.

• Alien arrives on Earth.
• 
  Alien asks to view all Earth encyclopedias.
  Alien encodes all the content as a single very massive integer.
  Alien treats number as a fraction between 0 and 1.
  Alien takes out a crystal rod, measures, and makes a single mark on it.
  Alien goes home with the rod to decode later.

Of course, a few terabytes of digits would exceed the resolution of any atomic matter, but the idea was there.

Re:Eben Moglen would be happy...

When I first read this I laughed at the concept of a stream of numbers being copyrightable.

Umm... ALL software in binary form is just a stream of numbers number, and programs distributed as such have been successfully copyrighted for decades. .mp3's are just a series of numbers which was not even created by the copyright holder, but that's not saving Napster.

I always wondered if that would be a suitably confusing defense of copyright violation. "No your honor, I didn't 'pirate' this [item]. I merely copied a large series of seemingly random numbers, [refers to printout in binary form] see? It's just a bunch of 1's and 0's, not [item]." The counter arguments would just degenerate into semantics and the whole thing would just get ugly. :-)

That gets even more weird when you consider that the zipped form of the binary data in no way represents what was originally placed under copyright.

So, let's say you burn that to a CD. It's just a bunch of invisible pits on a disc. That's copyrighted? Even if you use the proper "device" (i.e. computer) to convert it into "human readable" form, it's still not the copyrighted material. You have to apply a second process to convert the data out of comressed form.

So, from one perspective (it's just a bunch of pits on a disc!) copyright seems silly. From another (I can see on a computer screen, using data extracted from those pits, an image which says 'Copyright (C) 2000 Microsoft (R)') it seems more reasonable, since you're actually producing order out of what would otherwise be random or imperceivable.

I wonder what happens if you use the Linux kernel as the XOR "key" to encrypt Windows? &ltow, brain hurt&gt

The Formula Used

The formula he used to "find" this prime number can be found here:
http://www.utm.edu/research/primes/glossary/Illega l.html [utm.edu]
Basically it says this:

First Carmody took the original anonymous version of the DeCSS C-code and gzip'ed it (a standard UNIX program for making files smaller). Suppose we call the resulting number k. By Dirichlet's theorem on primes in arithmetic progression, we know that for each fixed integer b relatively prime to k, there are infinitely many primes ak+b.

For technical reasons, if we choose a to be a power of 256 larger than b, the resulting number can still be unzipped to get the original file. This means there are infinitely many prime numbers which yield the same code. These include: k*256^2+2083 and k*256^211+99. At the time these were found they both were large enough to fit on the list of largest known primes (because of the method of proof).

Dlugar

Re:This may be true...

Transcendental numbers like Pi and e, while not being random, are indeed patternless. Given a sufficiently long string of digits (quite possibly longer than one could encode on a hard drive the size of the earth), one could eventually happen upon any given 1401 digit number they were looking for. I am guessing that the same holds true for all irrational numbers. I do suspect that the conversion to hex would be a little different, (Have you ever tried converting non-whole numbers to another base? It's a bit of a pain in the rear.) but you'd find it there too. If you search pi over a long enough range, you can find your phone number, your IP, your birth date, or the build ID for your browser. You can even find some rather long strings of zeros. There are websites for this. In fact, if you want, there are places where you can download pi ten megabytes at a time, and search yourself. Try it out. Do a statistical analysis. I suspect you'll see results very similar to those for random numbers.

Re:Windows 2000 encoded to a single number!

> This one is interesting in that the number happens to be prime.

It isn't even remotely interesting. There are a *lot* of prime numbers. About one on 3000, for a 1400 digits number.

Considering how easy it is to build valid variation of a gzippped file, it is a one banana thing.

Even if gzip would not accept any variation (ie: if a source file could give only one gzip file, and if any alteration would produce an error at output), then modify the C source file would be just too easy (put a '/*n*/' at the begining, and compress for n incrementing from zero. Would take about 1500 try), unless gzi pformat could never give you a prime number (ie: always finished with a binary '0')

Now, what is the size of w2k source code ? 1 Gygabyte ? There are prime numbers about every few billions in that range. By messing with 4 bytes (!) of the source, you can have resonable expectation that you can make the result a prime number (if windows source code ends with a binary '1', of course).

Cheers,

--fred

Re:Easy--infinite number of primes

Then you should be able to find the prime number you just found in the prime number you just found. Or can you find Pi and Pi... hmmm?

Not just DeCSS!

Courtesy of Segfault [segfault.org].

Open Source Transcendental Constant

In a revelation that could rock the foundations of science, a researcher in Pennsylvania has discovered that the digits of the transcendental constant PI encode a version of the Linux kernel. "I can't believe it," the researcher, Neil Hoffman, exclaimed. "And yet, here I am staring at what appears to be the source code for Linux kernel 5.0.0. Needless to say, my whole world-view has changed..."

Hoffman made the discovery accidentally. "I was trying to write a more efficient algorithm in C to calculate individual digits of PI. However, my relative lack of programming experience, combined with C's highly obfuscated syntax, led me to the discovery. Instead of calculating each digit and returning it as an int, my program was (for some reason I still haven't been able to figure out) converting it to its ASCII equivalent and returning it as a char."

"Then it hit me. What if some kind of secret messages, encoded in ASCII, was stored in the digits of PI? I set to work on the problem, and after several months of toil, have discovered the awesome truth. My algorithm, which applies several dozen conversions and manipulations of each digit of PI, spits out plain vanilla ASCII characters that happen to form the source code for the Linux kernel."

"I tried to compile the source code, but gcc choked on it. Apparently a later version of gcc is needed to compile the Linux 5.0.0 source code. It's too bad the code for gcc isn't encoded in another transcendental constant. Or is it? I wonder what would happen if I fed e through my algorithm..."

Many scientists are skeptical about Hoffman's discovery. One mathematician who has memorized the digits of PI to 10,000 places said, "This is the kind of nonsense one would expect to find in a tabloid such as the National Mathematics Enquirer. Or a nerd humor site. Hoffman's discovery' is obviously a hoax designed to secure government research grants."

Another scientist Segfault contacted said, "Hoffman's claim is filled with holes large enough to push Windows 95 through. Apply a little critical thinking and look at all the inconsistencies and problems with Hoffman's discovery'. ASCII is an arbitrary code. Why not EBCDIC? Also, the base 10 number system, which his PI-to-ASCII scheme is based on, is arbitrary. Why not binary numbers? Oh, and then there's the biggie: PI is infinitely long. The Linux source code is not (Windows NT, on the other hand...). Explain that, PI Boy!"

Hoffman will formally present his findings to the scientific community on March 14th at the Annual PI Day Conference and Exposition in Chicago. One conference attendee said, "Usually the PI Day expo is pretty boring, with some asinine workshops about 'The History of PI' and Teaching Techniques to Make Learning About PI More Fun for Remedial High School Students'. However, with the unfolding brouhaha surrounding the Linux-PI connection, this could be a very interesting convention. Then again, there's going to be several hundred mathematicians from around the world in attendance. It might not be that exciting after all."

In a related matter, Segfault has received an unconfirmed report that a region of the standard Mandelbrot fractal contains what appear to be the words "LINUS TORVALDS WAS HERE". In addition, the words "TRANSMETA: THIS SECRET MESSAGE IS NOT HERE YET" supposedly appear within the depths of the Julia Set.

Linus Torvalds and Benoit Mandelbrot were unavailable for comment at press time.

--

Re:Sans Tables?

There is an intricate mathematical reason why I did it without the tables. In short - the number is too damn big to prove _formally_ (I am a mathematician) using Elliptic Curve Primality Proving (ECPP), due to the O(n^6) runtime.

In my favour is the precedent set by the Think Geek T-shirt which has no tables either. Unless you're talking about the one with only tables, and that has no code. If ThinkGeek have an illegal T-shirt, then my prime number is just as illegal.

FatPhil
--

Re:or what if...

Your recipient has the message and all you transferred was two completely unrelated numbers.

You seem to have an odd definition of "unrelated". "Extract a sequence of pi starting at position X and continuing to position Y." is a fairly simple function, that can be defined as a decryption scheme. The numbers you find into that scheme are your encoded message and the result is your message. Just because your formula uses pi doesn't make your input unrelated to your output.

On the other hand, XOR does allow for some confusion. Imagine I take a purely random file (based off of measuring radioactive decay or some such) and then XOR it with DeCSS. Now I've got my random file and my encrypted DeCSS -- the catch is that there's no way to tell which is which. If I've got both files, I can XOR them and get DeCSS, but otherwise both files look like random noise and both files are treated "equally" by the decryption process.

To make things even more interesting, imagine two people, named Bob and Ted, who have online collections of files with random numbers in them. Now let's say Ted's a bit of a free speech advocate. So he takes a copy of DeCSS, XORs it with one of Bob's random number files, and posts it to his site as a collection of random numbers. How do you prove that it's Ted who's hosting the copy of DeCSS and not Bob? What if you force Bob to remove his set of random numbers, when someone else had used that set as an XOR decryption key for something else? What if that person had both the encrypted and unencrypted versions available (say, as a demonstration of using XOR to encrypt a file)? Using the encrypted and unencrypted versions for the third party, you could recreate Bob's (removed) key. Then you could use that key to decrypt Ted's encrypted DeCSS.

Sans Tables?

The article says that the source is "Sans Tables".. in other words, it's useless. So what's the point? Isn't it the encryption keys that are actually the "trade secrets" in question?

Woohoo

Math haters rejoice! Theory of prime number is now illegal under Digital Millenium Copyright Act.

Re:Proof of the existance of God

Ah, but you forgot something:

4856. . .74

66699. . .7166639. . .9966669. . .8766689. . .4766629. . .9443

Clearly, this is not a holy number. I predict that tomorrow's headline shall be Catholic Church Denounces DeCSS.

(Lameness filter, filter thyself! It's not an awful long string of letters, it's a number. It's not in all caps, it's a number. A number is a character with an ASCII value in the range of 48 to 57. Capital letters are from 65 to 90. Got it?)

Someone has to say it

The MPAA has issued a statement on the article posted by /. "All your base 16 are belong to us" --Joey

Compression does not work

A good approximation to pi(x), the number of all primes below x, which was first given by Gauss is obtained by taking as starting point the empirical fact that the frequency of prime numbers near a very large number x is almost exactly 1/log x. From this, the number of prime numbers up to x is approximately given by the logarithmic sum Ls(x) = 1/log 2 + 1/log 3 + ... + 1/log x which can be bounded from below by x/log x. So, if x has 1400 digits, the number of primes below x will have 1397 digits, give or take one. So you could save three bytes. Surely a contender for the prize for the most gratuitous use of all cpu time until the end of time.