Factoring and testing primes in small space

Viliam Geffert; Dana Pardubská

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2013)

  • Volume: 47, Issue: 3, page 241-259
  • ISSN: 0988-3754

Abstract

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We discuss how much space is sufficient to decide whether a unary given 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. In other words, the language is a prime is in pebble–DSPACE(log log n) and also in accept–ASPACE(log log n). 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.

How to cite

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Geffert, Viliam, and Pardubská, Dana. "Factoring and testing primes in small space." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 47.3 (2013): 241-259. <http://eudml.org/doc/273058>.

@article{Geffert2013,
abstract = {We discuss how much space is sufficient to decide whether a unary given 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. In other words, the language is a prime is in pebble–DSPACE(log log n) and also in accept–ASPACE(log log n). 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.},
author = {Geffert, Viliam, Pardubská, Dana},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {prime numbers; factoring; sublogarithmic space; computational complexity},
language = {eng},
number = {3},
pages = {241-259},
publisher = {EDP-Sciences},
title = {Factoring and testing primes in small space},
url = {http://eudml.org/doc/273058},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Geffert, Viliam
AU - Pardubská, Dana
TI - Factoring and testing primes in small space
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 3
SP - 241
EP - 259
AB - We discuss how much space is sufficient to decide whether a unary given 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. In other words, the language is a prime is in pebble–DSPACE(log log n) and also in accept–ASPACE(log log n). 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.
LA - eng
KW - prime numbers; factoring; sublogarithmic space; computational complexity
UR - http://eudml.org/doc/273058
ER -

References

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