On Primes Recognizable in Deterministic Polynomial Time

  • Sergei Konyagin
  • Carl Pomerance
Part of the Algorithms and Combinatorics book series (AC, volume 13)


In this paper we present several algorithms that can find proofs of primality in deterministic polynomial time for some primes. In particular we show this for any prime p for which the complete prime factorization of p − 1 is given. We can also show this when a completely factored divisor of p − 1 is given that exceeds p l/4+ε. And we can show this if p − 1 has a factor F exceeding p ε with the property that every prime factor of F is at most (log p)2/ε. Finally, we present a deterministic polynomial time algorithm that will prove prime more than x 1-ε primes up to x, The key tool we use is the idea of a smooth number, that is, a number with only small prime factors. We show an inequality for their distribution that perhaps has independent interest.


Factorization Method Binary Search Deterministic Algorithm Primitive Root Nontrivial Factorization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Sergei Konyagin
    • 1
  • Carl Pomerance
    • 2
  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Department of MathematicsUniversity of GeorgiaAthensUSA

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