Illegal number

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An illegal number is a number that represents information which is illegal to possess, utter, propagate, or otherwise transmit in some legal jurisdiction. Any piece of digital information is representable as a number; consequently, if communicating a specific set of information is illegal in some way, then the number may be illegal as well.[1][2][3]

Free Speech flag, from the HD DVD AACS case



A number may represent some type of classified information or trade secret, legal to possess only by certain authorized persons. An AACS encryption key (09 F9 11 02 9D 74 E3 5B D8 41 56 C5 63 56 88 C0) that came to prominence in May 2007 is an example of a number claimed to be a secret, and whose publication or inappropriate possession is claimed to be illegal in the United States. It allegedly assists in the decryption of any HD DVD or Blu-ray Disc released before this date. The issuers of a series of cease-and-desist letters claim that the key itself is therefore a copyright circumvention device,[4] and that publishing the key violates Title 1 of the US Digital Millennium Copyright Act.

In part of the DeCSS court order[5] and in the AACS legal notices, the claimed protection for these numbers is based on their mere possession and the value or potential use of the numbers. This makes their status and legal issues surrounding their distribution quite distinct from that of copyright infringement.[5]

The PlayStation 3 edition of the free speech flag

Any image file or an executable program[6] can be regarded as simply a very large binary number. In certain jurisdictions, there are images that are illegal to possess,[7] due to obscenity or secrecy/classified status, so the corresponding numbers could be illegal.[1][8]

In 2011 Sony sued George Hotz and members of fail0verflow for jailbreaking the PlayStation 3.[9] Part of the lawsuit complaint was that they had published PS3 keys. Sony also threatened to sue anyone who distributed the keys.[10] Sony later accidentally retweeted an older dongle key through its fictional Kevin Butler character.[11]

Flags and steganography

The word "Wikipedia" translated into colors via hex codes

As a protest of the DeCSS case, many people created "steganographic" versions of the illegal information (i.e. hiding them in some form in flags etc.). Dave Touretzky of Carnegie Mellon University created a "Gallery of DeCSS descramblers". In the AACS encryption key controversy, a "free speech flag" was created. Some illegal numbers are so short that a simple flag (shown in the image) could be created by using triples of components as describing red-green-blue colors. The argument is that if short numbers can be made illegal, then any representation of those numbers also becomes illegal, like simple patterns of colors, etc.

In the Sony Computer Entertainment v. Hotz case, many bloggers (including one at Yale Law School) made a "new free speech flag" in homage to the AACS free speech flag. Most of these were based on the "dongle key" rather than the keys Hotz actually released.[12] Several users of other websites posted similar flags.[13]

Illegal primes


An illegal prime is an illegal number which is also prime. One of the earliest illegal prime numbers was generated in March 2001 by Phil Carmody. Its binary representation corresponds to a compressed version of the 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.[14]

Protests against the indictment of DeCSS author Jon Lech Johansen and legislation prohibiting publication of DeCSS code took many forms.[15] 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 PrimePages 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.

Other examples


There are other contexts in which smaller numbers have run afoul of laws or regulations, or drawn the attention of authorities.

See also



  1. ^ a b Carmody, Phil. "An Executable Prime Number?". Archived from the original on February 16, 2014. Retrieved December 30, 2018. Maybe I was reading something between the lines that wasn't there, but if arbitrary programs could be expressed as primes, the immediate conclusion is that all programs, including ones some people wished didn't exist, can too. I.e. the so called 'circumvention devices' of which my previous prime exploit was an example.
  2. ^ Greene, Thomas C. (March 19, 2001). "DVD descrambler encoded in 'illegal' prime number". The Register. Archived from the original on November 27, 2010. Retrieved December 30, 2018. The question, of course, is whether an interesting number is illegal merely because it can be used to encode a contraband program.
  3. ^ "The Prime Glossary: illegal prime". Archived from the original on February 28, 2023. Retrieved December 30, 2018. The bottom line: If distributing code is illegal, and these numbers contain (or are) the code, doesn't that make these number illegal?
  4. ^ "AACS licensor complains of posted key". Lumen. April 17, 2007. Archived from the original on October 8, 2023. Retrieved December 30, 2018. Illegal Offering of Processing Key to Circumvent AACS Copyright Protection [...] are thereby providing and offering to the public a technology, product, service, device, component, or part thereof that is primarily designed, produced, or marketed for the purpose of circumventing the technological protection measures afforded by AACS (hereafter, the "circumvention offering"). Doing so constitutes a violation of the anti-circumvention provisions of the Digital Millennium Copyright Act (the "DMCA")
  5. ^ a b "Memorandum Order, in MPAA v. Reimerdes, Corley and Kazan". February 2, 2000. Archived from the original on June 1, 2023. Retrieved December 30, 2018.
  6. ^ "Prime Curios: 48565...29443 (1401-digits)". Archived from the original on January 4, 2023. Retrieved December 30, 2018. What folks often forget is a program (any file actually) is a string of bits (binary digits)—so every program is a number.
  7. ^ "Criminal Justice Act 1988". Archived from the original on December 30, 2018. Retrieved December 30, 2018.
  8. ^ Wells, David (2011). "Illegal prime". Prime Numbers: The Most Mysterious Figures in Math. Wiley. pp. 126–127. ISBN 9781118045718.
  9. ^ Patel, Nilay (January 12, 2011). "Sony follows up, officially sues Geohot and fail0verflow over PS3 jailbreak". Engadget. Archived from the original on October 19, 2017. Retrieved December 30, 2018.
  10. ^ Kravets, David (February 8, 2011). "Sony lawyers now targeting anyone who posts PlayStation 3 hack". Ars Technica. Archived from the original on June 1, 2023. Retrieved December 30, 2018.
  11. ^ Miller, Ross (February 9, 2011). "PS3 'jailbreak code' retweeted by Sony's Kevin Butler, no punchline needed". Engadget. Archived from the original on August 16, 2017. Retrieved December 30, 2018.
  12. ^ S., Ben (March 1, 2011). "46-dc-ea-d3-17-fe-45-d8-09-23-eb-97-e4-95-64-10-d4-cd-b2-c2". Yale Law Tech. Archived from the original on March 10, 2011. Retrieved December 30, 2018.
  13. ^ See File:Free-speech-flag-ps3.svg description.
  14. ^ "Prime glossary - Illegal prime". 1999-10-06. Archived from the original on 2021-05-09. Retrieved 2013-03-26.
  15. ^ Hamilton, David P. "Banned Code Lives in Poetry and Song" Archived 2021-10-29 at the Wayback Machine
  16. ^ MacKinnon, Mark (June 4, 2012). "Banned in China on Tiananmen anniversary: 6, 4, 89 and 'today'". The Globe and Mail. Archived from the original on April 18, 2023. Retrieved December 30, 2018.
  17. ^ Meyer, Jeremy P. (September 5, 2012). "Greeley school ban on gang numbers includes Peyton Manning's 18". The Denver Post. Archived from the original on April 18, 2023. Retrieved December 30, 2018.
  18. ^ "Police charge leader of Slovak far-right party with extremism". Reuters. July 28, 2017. Archived from the original on August 9, 2017. Retrieved December 30, 2018.