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Factoring and Testing Primes in Small Space

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SOFSEM 2009: Theory and Practice of Computer Science (SOFSEM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5404))

Abstract

We discuss how much space is sufficient to decide whether a unary number n is a prime. We show that O(log log n) space is sufficient for a deterministic Turing machine, if it is equipped with an additional pebble movable along the input tape, and also for an alternating machine, if the space restriction applies only to its accepting computation subtrees. That is, un-Primes is in pebble-DSPACE(log log n) and also in accept-ASPACE(log log n), where un-primes={1n:n is a prime}. Moreover, if the given n is composite, such machines are able to find a divisor of n. Since O(log log n) space is too small to write down a divisor which might require Ω(log n) bits, the witness divisor is indicated by the input head position at the moment when the machine halts.

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Author information

Authors and Affiliations

  1. Department of Computer Science, P. J. Šafárik University, Jesenná 5, 04001, Košice, Slovakia

    Viliam Geffert

  2. Department of Computer Science, Comenius University, 84248, Bratislava, Slovakia

    Dana Pardubská

Authors
  1. Viliam Geffert
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  2. Dana Pardubská
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Editor information

Editors and Affiliations

  1. Aarhus Graduate School of Science, University of Aarhus, Ny Munkegade Bldg 1521-110, DK-8000, Aarhus C, Denmark

    Mogens Nielsen

  2. Faculty of Informatics, Masaryk University, Botanická 68a, 60200, Brno, Czech Republic

    Antonín Kučera

  3. Department of Computer Science, University of Aarhus, IT-parken, Aabogade 34, DK-8200, Aarhus N., Denmark

    Peter Bro Miltersen

  4. Ecole Polytechnique, Rue de Saclay, 91128, Palaiseau Cedex, France

    Catuscia Palamidessi  & Frank Valencia  & 

  5. Faculty of Mathematics and Physics, Charles University, Malostranské námestí 25, 118 00, Prague 1, Malá Strana, Czech Republic

    Petr Tůma

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© 2009 Springer-Verlag Berlin Heidelberg

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Geffert, V., Pardubská, D. (2009). Factoring and Testing Primes in Small Space. In: Nielsen, M., Kučera, A., Miltersen, P.B., Palamidessi, C., Tůma, P., Valencia, F. (eds) SOFSEM 2009: Theory and Practice of Computer Science. SOFSEM 2009. Lecture Notes in Computer Science, vol 5404. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-95891-8_28

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  • DOI: https://doi.org/10.1007/978-3-540-95891-8_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-95890-1

  • Online ISBN: 978-3-540-95891-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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