Strong pseudoprime

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Template:Base-relative pseudoprimality property

Definition

Suppose is an odd composite natural number and is an integer relatively prime to . We say that is a strong pseudoprime (also called Miller-Rabin pseudoprime, Rabin-Miller pseudoprime, Miller-Rabin strong pseudoprime, Rabin-Miller strong pseudoprime) to base if the following holds.

Write </math>n-1 = 2^k s</math> where is odd and is a nonnegative integer. Then, either one of these conditions should hold:

  • .
  • . Further, consider the smallest for which . Then, . In other words, the last value before becoming should be .

The name strong pseudoprime is because the above condition is satisfied for all primes, and is a particularly strong condition for which finding composite numbers is hard.

Relation with other properties

Weaker properties