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The Power of Leibniz-Like Functions as Oracles

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Computer Science – Theory and Applications (CSR 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12159))

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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].

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Notes

  1. 1.

    For sufficiently large fields.

  2. 2.

    If \(N = \prod \limits _{i=1}^k p_i^{\alpha _i}\) then its “square-free” part (or “radical”) is defined as \(\prod \limits _{i=1}^k p_i\).

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Acknowledgements

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

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Authors and Affiliations

  1. CSE Division, University of Michigan, Ann Arbor, MI, USA

    Jaeyoon Kim, Ilya Volkovich & Nelson Xuzhi Zhang

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  1. Jaeyoon Kim
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  2. Ilya Volkovich
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  3. Nelson Xuzhi Zhang
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Correspondence to Ilya Volkovich .

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Editors and Affiliations

  1. University of Trier, Trier, Germany

    Henning Fernau

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Kim, J., Volkovich, I., Zhang, N.X. (2020). The Power of Leibniz-Like Functions as Oracles. In: Fernau, H. (eds) Computer Science – Theory and Applications. CSR 2020. Lecture Notes in Computer Science(), vol 12159. Springer, Cham. https://doi.org/10.1007/978-3-030-50026-9_19

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  • DOI: https://doi.org/10.1007/978-3-030-50026-9_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-50025-2

  • Online ISBN: 978-3-030-50026-9

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