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Power permutations on prime residue classes

  • Fischer Harald
  • Stingl Christian
Chapter
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT)

Abstract

Nöbauer proofed in (Nöbauer, 1954) that the power function xx k mod n is a permutation on Z n , for a positive integer n iff n is squarefree and (k, λ(n)) = 1, where λ(n) denotes the Carmichael function and (a, b) the greatest common divisor of a and b. The RSA-cryptosystem uses this property for n = pq where p, q are distinct primes. Hence the modal cannot be chosen arbitrarily. If we consider permutations on prime residue classes, there is no restriction for the module anymore. In order to find criteria for power permutations on Z n * we first deal with the fixed point problem. As a consequence we get the condition for k:
$$ (k,[\phi (p_1^{{\alpha _1}},...,p_r^{{\alpha _r}})]) = 1\quad for\quad n = \mathop \prod \limits_{i = 1}^r p_i^{{\alpha _i}} $$
where ø denotes the Euler totient function and [a, b]the least common multiple of a and b.

Keywords

Power permutations RSA-cryptosystems fixed points 

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Copyright information

© IFIP International Federation for Information Processing 1995

Authors and Affiliations

  • Fischer Harald
    • 1
  • Stingl Christian
    • 1
  1. 1.Institute of Mathematics, University of KlagenfurtUniversity of KlagenfurtKlagenfurtAustria

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