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Illegal prime (wikipedia.org)
152 points by niyazpk on Jan 11, 2010 | hide | past | favorite | 76 comments

Y'all totally missed what's going on here.

Primepages (http://primes.utm.edu/) lists the largest known primes of various forms. One of the categories is primes that have no special form, which are the hardest to prove prime.

Chris Caldwell, the primepages maintainer, called up Phil Carmody and said, "Phil, if you can turn DeCSS into a 'general prime' that's big enough for the top 20 ever discovered, I'll be happy to host it on primepages." (Just as he would host any prime, DeCSS or not, that was big enough for the top 20.) So Phil did, cuz he's a smart guy and had a lot of computrons, and the prime went in the database.

The point here is that if you're merely turning data into a number, you have no legitimate reason to distribute it other than whatever reason you had to distribute the data itself. However, primepages existed long before DeCSS became a problem, and was arguably simply doing what it always did, which is to list the top tens.

I don't know if that would hold up in court -- it was never tested -- but the argument is a lot more subtle than what people are criticizing here.

An aside: Phil and I worked together on some other primality records. I wrote most of that Wikipedia article back in 2002 or so, and they went and featured it.. this was all before the rules for citations/references on Wikipedia became tighter than a mouse's arsehole. So if anyone is in the mood to add some citations, that would be much appreciated.

This is actually a really interesting idea I can't believe I haven't heard of yet. No matter what the angry neckbeards might say, just because content is a number doesn't mean copyright is invalid.

Obviously, we can generalize this approach to more than primes. Still, I imagine any idea would be extremely hard to scale up to mp3/video/game sizes, so it looks like those content industries are safe for now.

I don't see how "a reason to distribute it" is necessary. What if I just choose a completely random integer with 2000 digits and host it? I don't have any reason to do so, and it's certainly not illegal. Then someone comes along and generates a program that's equivalent to that number, or some digital media rights developer happens to use that number as a key for encrypting some copyrighted media. Does my original act of hosting suddenly become illegal, because I had no good reason to host the number in the first place?

The probability of you doing that without piracy is so vanishingly small as to be irrelevant.

I think the argument that "how can a number be owned or its posession be illegal" is mostly bogus. One could translate any string of text into a series of bits and back to a single number and say that a book or a picture is just a number. Then one could say that noone invented that number, and that it's just a number, so you couldn't have copyright nor protection because everything could be randomly generated, etc, etc...

I think the real issue is that in the case of prime numbers made "illegal/protected", the absurd is not that they are protected, but that they are not encoding any meaningful information that a human being produced. They are a particular number (probably randomly generated) and the fact that they are used in a particular context (to encrypt data) makes them protected.

The argument is bogus, because it's not the number itself that is illegal: it is the number plus the necessary meta data required to turn the number into something that has been outlawed.

It is like saying that it is silly that we prosecute people for possessing television sets, when the case under consideration is someone possessing a television set previously in the possession of someone else. If you do that, you are hiding contextual information that changes the meaning of what is being said.

As in the case of the television set, someone would only be convicted of 'possessing' or 'distributing' the number, if it was beyond reasonable doubt that he intended to break the law using the number. The number itself doesn't break the law: the actions/intent of the user is what breaks the law.

it is the number plus the necessary meta data required to turn the number into something that has been outlawed.

Where in the DMCA does it say that? And even positing that it does (though I'm pretty sure it doesn't), would that mean my possession of this prime, a linux distro, and this article in my browser history constitutes a crime?

The number itself doesn't break the law: the actions/intent of the user is what breaks the law.

Again, you're not talking about the actual DMCA. The actual DMCA implies that the intent of the user may be irrelevant if the primary purpose of the thing itself is to circumvent DRM tech. That's the whole point: the law is so poorly written and broadly applicable that nobody really knows what you can and can't do under it.

Where in the DMCA does it say that

Where in the law does it say that you aren't allowed to stab someone with a screw driver? Nowhere. That doesn't mean it isn't strictly implied.

Secondly, the DMCA isn't of particular interest here: the number might as well constitute a secret pertaining to national security, distribution of which may get you the death penalty for espionage.

The DMCA is of particular interest because it's a potential counterexample of your assertion that other tools or intent are necessary to make distribution of a number illegal. The whole point is that it's a bad law, primarily because it's unclear what it prevents you from doing.

The law is pretty clear, though, on stabbing people and committing espionage.

"but that they are not encoding any meaningful information that a human being produced"

I was sort of with you up to that point; I don't agree, but I was with you. However, now I would say you need to read the article more carefully, as the entire point is that they do indeed encode something human-produced and usable. If you use Unix, you've almost certainly got the decoder for that number sitting right on your disk right now, it doesn't get much more concrete than that. Without that, you're missing the whole point. They aren't randomly generated at all.

Guess I might as well spell out the other objection I have, which is that you're subtly begging the question. You prove that the idea of owning a number is silly because the idea of owning a number is silly in your last sentence of your first paragraph. Let me give you a much better argument: flip your argument on its head, and say that of course it's perfectly sensible to own numbers.

All but the smallest works translate in any reasonable encoding scheme into enormous numbers, numbers that would simply never be chosen randomly. There 0 chance that sitting here and stringing together "random numbers" will ever produce anything like a movie or picture or even coherent sentence, that doesn't already exist in the encoding scheme. (Not literally 0, depending on the circumstances, but close enough.) Therefore, the actual problem is that it does indeed make sense to own numbers; in an infinite space when we are all choosing umpteen millions or billions or trillions or well beyond that of digits, one can very plausibly stake an ownership claim on the grounds that it is clear that if somebody else ends up with the exact same number, they got it from you.

Therefore, since it is sensible to own numbers, the objection that it is not collapses and then the rest of the argument against ownership does too.

As I said, I don't actually agree with that line of thought. There are a lot of intricacies in the issues of encoding, and some other objections I'm leaving out, but it's still a stronger line of argument. My feeling are more related to the classic bit color essay: http://ansuz.sooke.bc.ca/lawpoli/colour/2004061001.php If that doesn't seem related, think more; it is, it very much is.

> you're subtly begging the question.

You are the first person to correctly use the phrase "beg the question" in something I've read in the last three weeks.


yes it does represent something human-produced but the number is not said to be illegal because the human-produced thing is protected. The number is deemed illegal because there is a convoluted way in which it could be used illegally. My point is that the protected entity is a very abstract thing much disconnected from the number itself. And therefore it does not make sense to protect it.

Regarding your second point, it is obvious to me that it is related to the bit color issue. I was trying to make a similar point to yours, but maybe I failed. Let me try again: I don't think that making this rationale of "oh, it becomes a number therefore it can't be owned" is a proper argument. Exactly because anything can become a number and we do acknowledge properties to things (like a text of a book) that would be ridiculous for a pure number to have.

My point was that even though the "just a number" argument is bad, there are other points to be made on the stupidity of this particular protection.

The provision in the DMCA that this addresses says you can't make or distribute anything that is designed or primarily intended for breaking DRM tech.

A DVD decryption algorithm obviously is designed and primarily intended to circumvent DRM, and a prime number obviously is not. The fact that the former can be expressed as the latter effectively means the provision is either absurdly lenient or absurdly draconian. It's a law that leaves you in the dark as to what you're actually allowed to do without the threat of prosecution or legal harassment.

It is illegal to distribute a "device"(very broad definition) that can circumvent DRM.

The protest is against how broad and vague that prohibition is and that it prohibits items even if their normal/much more useful function has nothing to do with DRM. Such as prime numbers or Sharpie markers. A "Hey legislators, in your scramble to sell out to content industry you've just made these numbers illegal." kind of thing.

"One could translate any string of text into a series of bits and back to a single number and say that a book or a picture is just a number. Then one could say that noone invented that number, and that it's just a number, so you couldn't have copyright nor protection because everything could be randomly generated"

You could be on to something big there, maybe a new kind of science, even.

Except in NKOS-land, Wolfram has copyright on all the numbers that you might possibly ever generate...

And of course any numbers that can be derived from numbers that you might generate.

I think Wolfram should get it over with and simply copyright { Z }, that would at least make it clear where we stand on this.

On behalf of Stephen Wolfram, "Ouch".

A Danish Social security number consists of two numbers - the first one being four arbitrary digits, and the second one being the six digits of your birthday.

My two numbers aren't only prime, which in and of itself would be pretty unlikely. No they're Twin primes. (http://en.wikipedia.org/wiki/Twin_prime)

This is the only place I know where information like that would be considered pretty cool :-)

Now somebody guess my birthday.

Assuming that (1) you were born between 1950 and 1985 and (2) the danish date format is (dd-mm-yy), here are the possible DOBs:

    10-01-51    30-01-51    17-03-51    14-05-51    26-05-51    27-05-51
    13-06-51    16-06-51    16-07-51    18-07-51    25-07-51    25-09-51
    15-10-51    21-10-51    24-10-51    21-11-51    13-12-51    17-12-51
    19-12-51    28-12-51    10-01-53    17-03-53    27-05-53    16-07-53
    25-07-53    21-11-53    17-12-53    24-02-57    29-06-57    25-10-57
    13-02-59    24-02-59    21-03-59    29-06-59    14-07-59    17-07-59
    22-08-59    15-09-59    25-10-59    26-10-59    10-11-59    24-12-59
    13-02-61    10-03-61    21-03-61    31-03-61    27-04-61    23-05-61
    14-07-61    17-07-61    27-07-61    22-08-61    23-08-61    26-08-61
    15-09-61    12-10-61    18-10-61    21-10-61    26-10-61    10-11-61
    17-11-61    24-12-61    25-12-61    10-03-63    31-03-63    27-04-63
    23-05-63    27-07-63    23-08-63    26-08-63    12-10-63    18-10-63
    21-10-63    17-11-63    25-12-63    13-03-67    19-03-67    11-05-67
    19-06-67    15-07-67    14-08-67    20-08-67    16-09-67    25-09-67
    24-10-67    17-11-67    26-11-67    29-11-67    27-02-69    13-03-69
    17-03-69    19-03-69    22-04-69    11-05-69    19-06-69    29-06-69
    15-07-69    28-07-69    14-08-69    20-08-69    16-09-69    25-09-69
    22-10-69    24-10-69    12-11-69    15-11-69    17-11-69    26-11-69
    29-11-69    11-12-69    23-12-69    27-02-71    17-03-71    22-04-71
    29-04-71    29-06-71    23-07-71    28-07-71    22-10-71    12-11-71
    15-11-71    22-11-71    11-12-71    23-12-71    29-04-73    23-07-73
    22-11-73    15-03-77    11-04-77    19-05-77    16-08-77    22-08-77
    17-10-77    25-11-77    27-12-77    18-01-79    15-03-79    11-04-79
    19-05-79    11-08-79    16-08-79    22-08-79    17-10-79    14-11-79
    25-11-79    10-12-79    27-12-79    18-01-81    20-03-81    16-04-81
    30-05-81    14-06-81    11-08-81    15-08-81    24-08-81    14-11-81
    30-11-81    10-12-81    20-03-83    16-04-83    30-05-83    14-06-83
    15-08-83    24-08-83    30-11-83

Thanks for the list but he said 6 digits... So the format is probably dd-mm-yy.

Oops! Corrected.

Impossible? Danes write dates as dd-mm-yy according to http://en.wikipedia.org/wiki/Calendar_date, so your birthday can have at most one leading zero and thus is >= 10100. 9973 is the largest 4-digit prime number, and 9973 + 2 is 1) not prime and 2) not a 5-digit number.

We're talking about two sets of twin primes, each part of the social security number being in each set. I'm sorry I didn't make it clear in the OP.

Now someone can guess your entire social security number :P

Calculated risk

Assuming you're talking about splitting your SSN down the middle (two 5 digit numbers) and it looks something like this


The candidates are (unless I horked something up, these are pairs of 5 digit twin primes that start with a sensible 6 digit date and checksum as Danish SSNs)

0212702129 0608706089 2105721059 2210722109

That's assuming you were not born in '90 or '91.

Which two numbers? It's quite difficult for a six digit and four digit number to be twin primes. Some zany Danish date format?

This is pretty silly, isn't it? Is it that mind-boggling that data can be illegal to distribute, for example a movie or an unreleased game?

The prime number is a red herring. Given the prime number theorem, the probability that an n-digit number is prime scales as 1/log n, so it's easy to pad any data with random garbage that isn't executed until you get a prime. For a gigabyte of data that might be a few billion attempts. (Proving it's a prime will be harder.)

>> Is it that mind-boggling that data can be illegal to distribute...?

The DMCA has had an impact on the worldwide cryptography research community, since an argument can be made that any cryptanalytic research violates, or might violate, the DMCA.

- http://en.wikipedia.org/wiki/Digital_Millennium_Copyright_Ac...

That is mind-boggling.

I bring this up whenever I want to point out how law lags behind technology and ends up doing silly things like making a number illegal, or making cracking cyphers like rot13 a felony. That is to say, breaking even simple encryption a felony, not rot13 specifically.

People always get amused that a prime number could get them in trouble.

A more relevant example, as mentioned in a linked article, is that the binary data of images such as child pornography and state secrets are also numbers and as such may be considered illegal.

How can an image be a number? This is an honest question.

Isn't an image a list of numbers? These numbers could the RGB of each pixel, or the N higher Fourier coefficients, etc. From these numbers, one can compute a single number, sure. Nonetheless, the idea that an image is a number and, hence, can be made illegal, looks utterly ludicrous to me due to the fact that the image -> number mapping is certainly not injective.

Nope, that's just one interpretation and it's not more valid then any other. Your argument is equivalent to saying something like this "129 isn't a single number, it's actually 110^2+210^1+9*10^0 (aka 100 + 20 + 9)".

Anything that can be represented on a computer can be seen as a number, rather large numbers sometimes but still definitely numbers.

You are totally missing the point. The 3-tuple (1,2,9) may represent number 129 if you use the decimal expansion. What if you use a base-16 expansion, or a base-8 expansion? You can even choose the base so that the tuple -> number mapping is not injective (and, thus, not invertible).

So, you give me a number, and I can give you infinitely many n-tuples that can be transformed into that number. Unless you know the mapping, and unless the mapping is injective, a single number is useless. It could represent infinitely many images.

So, I ask again, how can an image be a number?

The simplest, and most common representation would be for a file to simply be a string of binary bits representing a large number.


   big_number = 0;

   while(!feof(file)) {
      big_number = (big_number<<8) | file.readByte();

I am pointing at the forest, and you're looking at the trees. We're not even talking about the same thing. I am taking a very high-level, abstract view of the problem. You're thinking of implementation.

Right. You're just saying there are an infinite number of ways you could convert a string of binary digits into a 'number'.

Well sure :/ Of course there are.

At the simplest form you could swap things around, represent a fractional part of a number etc, at the more complex side you just have encryption.

But those are different things to your question "So, I ask again, how can an image be a number?"

For an image to be a number, I demand a one-to-one correspondence between images and numbers. Lacking that, I merely say that an image can be represented by a number, but this is trivial. Hence, I repeat: we're not even discussing the same thing.

>> "So, I ask again, how can an image be a number?

>> "Lacking that, I merely say that an image can be represented by a number, but this is trivial"

I think you're arguing with yourself at this point.

Note that: "is" and "is represented by" are not the same on my book (which is obvious from the previous comments). If you want to counter-argument, try harder.

I wonder what you mean by "is". Nothing really "is" anything but itself. The only other "is", the one that real people actually use, is "is represented by" or "is a representation of", or "is an instance of", etc.

For example, this --> A <-- looks like the letter A. But it isn't the letter A. It "is" only a representation, a bit-pattern with decimal notation 65, a glyph drawn from a font, a set of pixel elements on a graphics surface, a pattern of light on a display, a bunch of photons traveling through space, a group of excited photoreceptor cells, a set of signals traveling through neurons, etc. And it is none of these things, because none of them capture the essence of the pattern, repeated over human experiences.

So it seems to me that if you're trying to win an argument by drawing a distinction between "is" and "is represented by", you are not going to convince anyone who's put much thought into it. "Ceci n'est pas une pipe"; but a representation of a pipe is a (more or less faithful) representation of a pipe, whether you are blind or not, and so a number is a digital image, and vice versa.

I think one must draw a distinction between the physical world and the metaphysical one. For a platonist, numbers do exist outside of any physical reality and, therefore, a number and its representation are not the same thing.

I am not a mathematician, but to me this discussion reminds me of a vector having different coordinates, depending on what coordinate system we choose. Those are all different representations of the same thing. Likewise, a number can be represented in many different ways (binary, oct, hex, etc), but they all represent the same number. Is the vector its representation? If so, then the vector is many things, which is a funny definition of "to be". If the vector is itself, then it cannot be equal to any of its representations, it's something beyond its representation.

I don't think it's at all a given that Hylaean Theoric Worlds exist independently the Arbran Causal Domain.

I think that the main reason you are being downvoted is not so much that you are technically wrong, but rather because you are being pedantic ad nauseum.

This is a programmers' forum, not a philosophers' forum. We should not be discussing the meaning of "is". That's too low-level and fundamental for our purposes. This does not imply it's not an interesting question, it's just not appropriate.

There is a one-to-one correspondence between numbers and their doubles, {0..n} and {0..2n}. Does that mean that numbers are their doubles?

It is more proper to talk about the number + a decoding scheme. For given triple (image, image data, encoding scheme) any one can be held constant and the other two will have an infinite number of possibilities. In particular, any image can be represented as any set of data, plus an encoding scheme for that data. Trivial (but relevant) proof: The encoding scheme can simply hard code the image in question and trigger returning that image when it sees the held-constant image data.

You could choose to represent an image as image data + decoding scheme, using an agreed representation of the decoding scheme such as a computer program. Mathematically we get nowhere because we are simply moving the encoding scheme around, but in practice, since we possess concrete encoding schemes it means we can practically discuss the situation better.

This is a long-winded away of agreeing with you, BTW.

Thanks for wording it more precisely than I did. Most people are assuming that the mapping tuple -> number is known, while I am not. I could come up with my own invertible mapping. The cool thing would be this:

- I carry a large number y with me

- the police demands that I disclose y, and I do; they apply inverse mapping f^{-1} and obtain x = f^{-1}(y) which is a porn image

- then I show the police that applying inverse mapping g^{-1}, one obtains a totally legit image z = g^{-1}(y)

Of course, finding such a mapping g could be extremely difficult, but if the same number can be obtained from two different images, how would the police be able to accuse me of anything?

It is fairly easy to find such mappings. Call the carried data the key. XOR the key data with any other image to get the encrypted file. Simply XOR the encrypted file with the key file to get the desired image.

Finding small mappings is another matter.

But the same argument applies to the original "illegal prime" under discussion. Your original comment indicated that you considered the case of an image to be different and you asked how a prime could represent an image.

In fact, I'd go as far as saying that you could apply your argument to any data stored on a computer. But someone I don't think a judge would be impressed by "there re infinity many things the data in this file could represent and only under one mapping is it an indecent image".

Think about Gödel numbering or Cantor diagonalization.

Let's assume discrete, rectangular representations of images (width, height in pixels and colorspace of #000000-#FFFFFF). Here's an injective mapping from natural numbers (the indices of the vector of images) to actual images:

    # In a C-like language where ** is 
    # the exponentiation operator, and 
    # ints are unbounded.
    int colors = 0xFFFFFF + 1;
    int i = 0;
    vector images = new vector();
    while(i++) {
      for(int width = 1; width < i; width++) {
        int height = i - width;
        int area = width * height;
        int pixels[area];
        int combinations = colors ** area;
        for(int c = 0; c < combinations; c++) {
          for(int p = 0; p < area; p++) {
            pixels[p] = c % (colors ** (p + 1));
          images.append(new image(width, height, pixels));

You could view it as a list of numbers, yes.

Alternatively, we all know that the file representing the image is nothing more than a collection of bytes. Bytes are nothing more than 8-element bit vectors. What if instead of treating the file as a collection of $n$ 8-element bit vectors we instead treated it as a single $n*8$-element bit vector. What we now have is a single (potentially VERY large) binary number.

To make any sense out of this number we'd have to pull out individual collections of bits, but this is no more complicated than doing a bitwise AND and bit shift.

Put another way: you can think of an IP-address as 4 sets of numbers between 0 and 255. For example, You can view that same IP-address as the hexidecimal number 0xC0A80101. The hex representation is convenient, as it makes elements in the dotted quad apparent. 0xc0 = 192, 0xa8 = 168, 0x01 = 1, 0x01 = 1.

Hope that helps.

That's not what I meant at all. Of course you can transform a list of bytes into a number. But can you transform a number into a list of bytes? That is the problem. If you know the "encoding" algorithm, then you can. If you do not, then you simply can't.

The point I was trying to make is that a number, itself, is nothing unless there's knowledge of the encoding algorithm. A CD-reader can make sense of the bits stored on the disk because it knows the encoding algorithm. If the CD-burner and CD-reader did not "talk the same language", it would be impossible to make sense of all those bits stored on the disk. Without knowing what interleaving and channel code was used, the CD-reader can't do its job.

An image is ultimately represented on disk as a binary string. Hence it is a number.

OK, Mr. "Ramanujan", let's play a game: I give you a binary string, and you tell me what image corresponds to it. Are you game?

I won't tell you how the bitstring was computed from the image. I don't tell you how many pixels the image has. I don't tell you anything. I only give you the bits. Wanna play for money?

Given that definition, you can't encode anything as a number. I think someone pointed out above it's the number + the decoding method that is needed for it to be useful and that applies to images or even the gzip and ELF examples in the article. If I give you the number, but don't tell you it's a gzip of a C program, you probably won't be able to decode it easily.

Not quite. One can encode anything into a number. Decoding is what it's hard, especially so if one does not know what the encoding scheme was. I suppose this is one of the reasons why NSA needs astronomical amounts of computational power.

I think his point is... the image is stored on disk in a binary form. Which can also be represented as a number (Decimal).

Geees, is this HN??? Am I the one who's too stupid to explain things properly, or is everyone under the weather because it's Monday morning?

An image is a 2-dimensional array. Say it's a M x N array. Suppose we use RGB so that each pixel is represented by 3 bytes. Then you have an M x N x 3 array of bytes. There are infinitely many mappings from the space of all M x N x 3 arrays of bytes into the reals. Of course you can transform the array into a number. But you can do it in many, many ways. If you know what mapping was used, and if the mapping is invertible, then you can obtain the M x N x 3 array of bytes from the number. If you don't know what mapping was used, or if the mapping is non-invertible, then you can't obtain the M x N x 3 array from the number.

So, I claim that an image is NOT a number. I claim that an image is a pair (y,f), where y is a number, and f is an invertible mapping from the space of M x N x 3 arrays into the reals. You guys are looking at y, and forgetting f. That's why an image is not a number, the image is x = f^{-1} (y). In short, one must know f, and f^{-1} must exist.

You're merely arguing there that a number is not necessarily an image, because it cannot be converted into one without knowing how. And you'd be right. But the image is still a number, because it's stored as a series of bits, and any series of bits is interpretable as a number.

I asked: "How can an image be a number?", when I should have asked instead: "How can a number be an image?"

To use the verb to be in this scenario, I demand a one-to-one correspondence between numbers and images. If a number y can represent two distinct images x1 and x2, then we can't use "is", we can only use "can be".

Such a strict definition prevents you for using "is" for anything at all! Is that your intention?

To quote a former President, it depends on what your definition of "is" is.

I think everyone here understands that we require information beyond the number itself to get the image. I agree that a more accurate (pedantic) verb is that a number can store or represent an image. That implies that the number alone is not sufficient to get the image.

The thing is that while f can be in theory anything, it will be one of few things in practice. We can say what f is very simply. While knowing f is important, it's also easy to the point of trivial, so some people here are comfortable being less accurate and just saying "a number can be an image."

This reminds me of philosophy of identity discussion me and a friend used to go through. He thought he could fit a Miata in his old-school Suburban - but maybe he'd have to take off the side mirrors. But if he's allowing the side mirrors to be taken off, then how much is too much before it's no longer a Miata?

Indeed. It seems that for most people here is means can be, while I demand a bit more. Yes, in practice, f will be one of a few mappings, but that does not change the fact that knowledge of f is required. In practice, one merely tries all possible mappings f until one works.

Such lack of precision is only tolerated because computation is fast and cheap. I would love to see people trying all f mappings with the help of an abacus! Let's be thankful that we live in this golden age.

We can go further, actually. We could argue even that (y, f) is not, actually, an image and that's it's just a number and a function. Rather, the image is what's on the screen and only exists when it's displayed.

I'm not trying to continue an argument, but rather just point out that the philosophy of identity can be an elusive thing.

To make things even more fun:

Take an image x, decimate it, and obtain a lower-resolution version of it, which we call z. Are x and z the same image?

Mathematically, no. But when displayed on the screen, then most people would say they are the same. Of course, it depends on the definition of is, but such discussion would be utterly pointless. If we use the same definition, then there's no ambiguity.

Indeed often info on which f you need is stored within y. which makes it even easier

But, f is fairly standardized (or rather, there's a short list of standard f's), and bitmaps aren't even hard to understand. Since the isomorphism is both natural and conventional, I would say that in practice it is not accurate to hold f to the same standards as y.

A list of numbers is a number, in the same way that a string of digits is a number.

Technically, a string of digits is only a representation of a number. And such representation may not even be unique.

This has nothing to do with law lagging behind technology.

Even before computers, the ability to express language in a number was well known. If someone turned a defamatory sentence into a number and spread that number together with the key to turn it into the sentence, then they would be just as guilty of defamation as when they had spread the original sentence.

I surely hope you do not advocate getting rid of all laws that deal with information that could be expressed in a number, because you feel it should not count as the same incriminating information, just because it was masked with some mathematics?

For all those commenting please read it carefully: the primes are not illegal, just number representations of apparently illegal programs.

I still don't get how a prime number can reveal hidden information. Can someone please explain me the process?

From the article: Its binary representation corresponds to a compressed version of the C source code of a computer program.

it doesn't reveal information except that it is a constant that is used as part of a process of encryption where without the constant cracking that encryption would be effectively impossible and with it cracking the encryption is trivial.

effectively the number is the encryption, as having the number makes the encryption useless. the rest of the algorithm is pointless without the number and vice versa.

That's a different number, not the one the article is about.

The one in question is a compressed version of a c program that breaks encryption.

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