Classical and Quantum Algorithms for Exponential Congruences

  • Wim van Dam
  • Igor E. Shparlinski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5106)


We discuss classical and quantum algorithms for solvability testing and finding integer solutions x,y of equations of the form af x  + bg y  = c over finite fields Open image in new window . A quantum algorithm with time complexity q 3/8 (logq) O(1) is presented. While still superpolynomial in logq, this quantum algorithm is significantly faster than the best known classical algorithm, which has time complexity q 9/8 (logq) O(1). Thus it gives an example of a natural problem where quantum algorithms provide about a cubic speed-up over classical ones.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Wim van Dam
    • 1
  • Igor E. Shparlinski
    • 2
  1. 1.Department of Computer Science, Department of PhysicsUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of ComputingMacquarie UniversityAustralia

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