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An '''illegal prime''' is a [[prime number]] that represents information whose possession or distribution is forbidden in some [[legal jurisdiction]]. One of the first illegal primes was found in 2001. When interpreted in a particular way, it describes a [[computer program]] that bypasses the [[digital rights management]] scheme used on [[DVD]]s. Distribution of such a program in the [[United States]] is illegal under the [[Digital Millennium Copyright Act]]. An illegal prime is a kind of [[illegal number]].

==Background==
[[Image:DeCSS.PNG|right|250px|thumbnail|The [[DeCSS]] code can be used by a computer to circumvent a DVD's [[copy protection]].]]

One of the earliest illegal prime numbers was generated in March 2001 by [[Phil Carmody]]. Its [[binary numeral system|binary]] representation corresponds to a [[data compression|compressed]] version of the [[C (programming language)|C]] [[source code]] of a [[computer program]] implementing the [[DeCSS]] decryption algorithm, which can be used by a computer to circumvent a DVD's [[copy protection]].

Protests against the indictment of DeCSS author [[Jon Lech Johansen]] and legislation prohibiting publication of DeCSS code took many forms. One of them was the representation of the illegal code in a form that had an ''intrinsically archivable'' quality. Since the bits making up a computer program also represent a number, the plan was for the number to have some special property that would make it archivable and publishable (one method was to print it on a T-shirt). The [[primality]] of a number is a fundamental property of [[number theory]] and is therefore not dependent on legal definitions of any particular jurisdiction.

The large prime database of The [[Prime Pages]] website records the top 20 primes of various special forms; one of them is proof of primality using the [[elliptic curve primality proving]] (ECPP) [[algorithm]]. Thus, if the number were large enough and proved prime using ECPP, it would be published.

==Discovery==
Specifically, Carmody applied [[Dirichlet's theorem on arithmetic progressions|Dirichlet's theorem]] to several prime candidates of the form
''k''·256''n'' + ''b'', where ''k'' was the [[decimal]] representation of the original compressed file. Multiplying by a power of 256 adds as many trailing [[null character]]s to the [[gzip]] file as indicated in the [[exponentiation|exponent]] which would still result in the DeCSS C code when unzipped.

Of those prime candidates, several were identified as [[probable prime]] using the [[open source]] program OpenPFGW, and one of them was proved prime using the ECPP algorithm implemented by the Titanix software. Even at the time of discovery in 2001, this 1401-digit number, of the form ''k''·2562 + 2083, was too small to be mentioned, so Carmody created a 1905-digit prime, of the form ''k''·256211 + 99, that was the tenth largest prime found using ECPP, a remarkable achievement by itself and worthy of being published on the lists of the highest prime numbers.<ref name=gloss/> In a way, by having this number independently published for a completely unrelated reason to the DeCSS code, he had been able to evade legal responsibility for the original software.

Following this, Carmody also discovered another prime, this one being directly [[executable]] machine language for [[Linux]] [[Intel 80386|i386]], implementing the same functionality.

==See also==
*[[AACS encryption key controversy]]
*[[HDCP#Master key release|HDCP master key release]]
*[[The Library of Babel]]
*[[Normal number]]
*[[PlayStation_3_homebrew#Private key compromised|PlayStation 3 private key compromised]]
*[[Prior art]]
*[[Streisand effect]]
*[[Texas Instruments signing key controversy]]

==Source==

[http://wikipedia.org/ http://wikipedia.org/]
[[Category:Cryptography]]

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