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The Power of Leibniz-Like Functions as Oracles

  • Jaeyoon Kim
  • Ilya VolkovichEmail author
  • Nelson Xuzhi Zhang
Conference paper
  • 81 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12159)

Abstract

A Leibniz-like function \(\chi \) is an arithmetic function (i.e., \(\chi : \mathbb {N}\rightarrow \mathbb {N}\)) satisfying the product rule (which is also known as “Leibniz’s rule”): \(\chi (MN) = \chi (M) \cdot N + M \cdot \chi (N)\). In this paper we study the computational power of efficient algorithms that are given oracle access to such functions. Among the results, we show that certain families of Leibniz-like functions can be use to factor integers, while many other families can used to compute the radicals of integers and other number-theoretic functions which are believed to be as hard as integer factorization [1, 2].

Keywords

Integer factorization Number-theoretic functions Square-free integers Möbius function Oracles 

Notes

Acknowledgements

The authors would also like to thank the anonymous referees for their detailed comments and suggestions.

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.CSE DivisionUniversity of MichiganAnn ArborUSA

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