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Enforcing and Defying Associativity, Commutativity, Totality, and Strong Noninvertibility for One-Way Functions in Complexity Theory

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Theoretical Computer Science (ICTCS 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3701))

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Abstract

Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist one-way functions (i.e., p-time computable, honest, p-time noninvertible functions) that are total, commutative, and associative but not strongly noninvertible. In this paper we improve the sufficient condition to P ≠ NP.

More generally, in this paper we completely characterize which types of one-way functions stand or fall together with (plain) one-way functions—equivalently, stand or fall together with P ≠ NP. We look at the four attributes used in Rabi and Sherman’s seminal work on algebraic properties of one-way functions (see [RS97,RS93]) and subsequent papers—strongness (of noninvertibility), totality, commutativity, and associativity—and for each attribute, we allow it to be required to hold, required to fail, or “don’t care.” In this categorization there are 34 = 81 potential types of one-way functions. We prove that each of these 81 feature-laden types stand or fall together with the existence of (plain) one-way functions.

Work supported in part by the NSF under grant NSF-CCF-0426761 and by the DFG under grant RO 1202/9-1. Work done in part while the first two authors were visiting Julius-Maximilians-Universität Würzburg and while the second author was visiting the University of Rochester.

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Author information

Authors and Affiliations

  1. University of Rochester, USA

    Lane A. Hemaspaandra

  2. Heinrich-Heine-Universität Düsseldorf, Germany

    Jörg Rothe

  3. La Trobe University, Australia

    Amitabh Saxena

Authors
  1. Lane A. Hemaspaandra
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  2. Jörg Rothe
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  3. Amitabh Saxena
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica, Università di Torino,  

    Mario Coppo

  2. Dipartimento di Scienze Matematiche e Informatiche, Università di Siena, Pian dei Mantellini 44, 53100, Siena, Italy

    Elena Lodi

  3. Dipartimento di Matematica e Informatica, Università di Cagliari, P.O. Box

    G. Michele Pinna

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© 2005 Springer-Verlag Berlin Heidelberg

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Hemaspaandra, L.A., Rothe, J., Saxena, A. (2005). Enforcing and Defying Associativity, Commutativity, Totality, and Strong Noninvertibility for One-Way Functions in Complexity Theory. In: Coppo, M., Lodi, E., Pinna, G.M. (eds) Theoretical Computer Science. ICTCS 2005. Lecture Notes in Computer Science, vol 3701. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11560586_22

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  • DOI: https://doi.org/10.1007/11560586_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29106-0

  • Online ISBN: 978-3-540-32024-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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