Illegal prime
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An illegal prime is a prime number which contains information forbidden by law to possess or distribute.
The first illegal prime to be announced, when interpreted a particular way, describes a computer program which bypasses copyright protection schemes on some DVDs. Because that program has been found illegal by courts in the United States of America, this has produced debate about whether the number itself could be considered illegal.
This question has never been tested in court, and it is possible that the number itself and its possession would be found to be legal, but not a particular interpretation of it.
The first illegal prime number was generated on 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 scheme. Such programs are illegal to possess or distribute under the Digital Millennium Copyright Act.
The existence of infinitely many such primes is guaranteed by Dirichlet's theorem.
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Background
DeCSS.PNG
Protest against the indictment of DeCSS author Jon 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. The primality of a number is a fundamental property, one outside the scope of the law.
The large prime database of the prime pages 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
By exploitation of the fact that the gzip compression program ignores bytes after the end of a null terminated compressed file, a set of candidate primes was generated, each of which would result in the DeCSS C code when unzipped. Of these 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. At the time of discovery, this 1401 digit number was the tenth largest prime found using ECPP.
Following this, Carmody also created another prime, this one directly executable machine language implementing the same functionality. According to a strict reading of the DMCA, this number would appear to be illegal in the USA.
The first illegal prime number
The Register gives the number as: 48565 07896 57397 82930 98418 94694 28613 77074 42087 35135 79240 19652 07366 86985 13401 04723 74469 68797 43992 61175 10973 77770 10274 47528 04905 88313 84037 54970 99879 09653 95522 70117 12157 02597 46669 93240 22683 45966 19606 03485 17424 97735 84685 18855 67457 02571 25474 99964 82194 18465 57100 84119 08625 97169 47970 79915 20048 66709 97592 35960 61320 72597 37979 93618 86063 16914 47358 83002 45336 97278 18139 14797 95551 33999 49394 88289 98469 17836 10018 25978 90103 16019 61835 03434 48956 87053 84520 85380 45842 41565 48248 89333 80474 75871 12833 95989 68522 32544 60840 89711 19771 27694 12079 58624 40547 16132 10050 06459 82017 69617 71809 47811 36220 02723 44827 22493 23259 54723 46880 02927 77649 79061 48129 84042 83457 20146 34896 85471 69082 35473 78356 61972 18622 49694 31622 71666 39390 55430 24156 47329 24855 24899 12257 39466 54862 71404 82117 13812 43882 17717 60298 41255 24464 74450 55834 62814 48833 56319 02725 31959 04392 83873 76407 39168 91257 92405 50156 20889 78716 33759 99107 88708 49081 59097 54801 92857 68451 98859 63053 23823 49055 80920 32999 60323 44711 40776 01984 71635 31161 71307 85760 84862 23637 02835 70104 96125 95681 84678 59653 33100 77017 99161 46744 72549 27283 34869 16000 64758 59174 62781 21269 00735 18309 24153 01063 02893 29566 58436 62000 80047 67789 67984 38209 07976 19859 49364 63093 80586 33672 14696 95975 02796 87712 05724 99666 69805 61453 38207 41203 15933 77030 99491 52746 91835 65937 62102 22006 81267 98273 44576 09380 20304 47912 27749 80917 95593 83871 21000 58876 66892 58448 70047 07725 52497 06044 46521 27130 40432 11826 10103 59118 64766 62963 85849 50874 48497 37347 68614 20880 529443.
See also
External links
- The prime pages (http://primes.utm.edu/)
- Prime glossary - Illegal prime (http://primes.utm.edu/glossary/page.php?sort=Illegal)
- Prime Curios - Illegal prime (http://primes.utm.edu/curios/page.php?number_id=953)
- The first illegal prime (http://web.archive.org/web/20011212144451/fatphil.org/maths/illegal1.html)
- DVD descrambler encoded in ‘illegal’ prime number (http://www.theregister.co.uk/2001/03/19/dvd_descrambler_encoded_in_illegal/) (Thomas C. Greene, The Register, Mon 19 March 2001)de:Illegale Primzahl