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Factoring and testing primes in small space∗

Published online by Cambridge University Press:  30 July 2013

Viliam Geffert  and
Dana Pardubská
Viliam Geffert
Affiliation:
Department of Computer Science, P. J. Šafárik University, Jesenná 5, 04001 Košice, Slovakia.. viliam.geffert@upjs.sk
Dana Pardubská
Affiliation:
Department of Computer Science, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia.; pardubska@dcs.fmph.uniba.sk
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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.

Keywords

Prime numbersfactoringsublogarithmic spacecomputational complexity
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 47 , Issue 3 , July 2013 , pp. 241 - 259
Copyright
© EDP Sciences 2013

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