Industrial-grade prime: Difference between revisions

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Industrial-grade primes are integers for which primality has not been certified (i.e. rigorously proven), but have undergone a test such as the Miller-Rabin primality test, which has a positive, but impossibly low, failure rate.

Industrial-grade primes are often used instead of certified primes in algorithms such as RSA encryption, which require the user to generate large prime numbers. Certifying the primality of large numbers (over 100 digits for instance) would be extremely difficult, even using modern methods. Trying to prove primality by simply testing every possible prime factor less than it at a trillion trial divisions per second would take over 100000000000000000000 (10²⁰) times the age of the universe. However, an industrial-grade prime test can accurately test this number for primality almost instantly within an impossibly low failure rate. In other words, the number is certified to be prime with very high, but not absolute, confidence.

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 '''Industrial-grade primes''' are [[integer]]s for which [[primality]] has not been certified (i.e. rigorously proven), but have undergone a test such as the [[Miller-Rabin primality test]], which has a positive, but impossibly low, failure rate.
-Industrial-grade primes are often used instead of certified primes in [[algorithms]] such as [[RSA encryption]], which require the user to generate large [[prime numbers]].  Say, for example, that I need to test a number that is 100 digits long for primality.  Certifying primality for this number would be extremely difficult, even using modern methods.  (Trying to prove primality by simply testing every possible prime factor less than it at a [[trillion]] trial divisions per second would take over 100000000000000000000 (10<sup>20</sup>) times the [[age of the universe]]).  However, an industrial-grade prime test can accurately test this number for primality almost instantly within an impossibly low [[failure rate]].
+Industrial-grade primes are often used instead of certified primes in [[algorithms]] such as [[RSA encryption]], which require the user to generate large [[prime numbers]].  Certifying the primality of large numbers (over 100 digits for instance) would be extremely difficult, even using modern methods.  Trying to prove primality by simply testing every possible prime factor less than it at a [[trillion]] trial divisions per second would take over 100000000000000000000 (10<sup>20</sup>) times the [[age of the universe]].  However, an industrial-grade prime test can accurately test this number for primality almost instantly within an impossibly low [[failure rate]]. In other words, the number is certified to be prime with very high, but not absolute, confidence.
 == See also ==