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Paper Abstract and Keywords
Presentation 2017-03-07 13:30
[Invited Talk] Beating Brute Force for Systems of Polynomial Equations over Finite Fields
Daniel Lokshtanov (U. Bergen), Ramamohan Paturi (UC San Diego), Suguru Tamaki (Kyoto U.), Ryan Williams (MIT), Huacheng Yu (Stanford U.) COMP2016-53
Abstract (in Japanese) (See Japanese page) 
(in English) Solving systems of multivariate polynomial equations over finite fields is a fundamental problem in mathematics, science and engineering.
Let $q$ be the order of the underlying field and $n$ be the number of variables.
Then, the problem can be solved by the brute force search over $q^n$ possible solutions.
We present the first algorithm that solves the problem in time faster than $q^n$ in the worst-case.
Our algorithm can count the number of solutions as well.
The running time of the algorithm is roughly of the form $q^{n(1-1/O(d))}$, where $d$ is the maximum degree of polynomials.
We also consider a generalization of the problem, where each equation is defined by a certain type of arithmetic circuit, and give an algorithm for the generalized problem.
Keyword (in Japanese) (See Japanese page) 
(in English) exponential time algorithm / polynomial / low degree / arithmetic circuit / / / /  
Reference Info. IEICE Tech. Rep., vol. 116, no. 503, COMP2016-53, pp. 19-19, March 2017.
Paper # COMP2016-53 
Date of Issue 2017-02-28 (COMP) 
ISSN Print edition: ISSN 0913-5685    Online edition: ISSN 2432-6380
Copyright
and
reproduction		 All rights are reserved and no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Notwithstanding, instructors are permitted to photocopy isolated articles for noncommercial classroom use without fee. (License No.: 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034)
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Conference Information
Committee COMP  
Conference Date 2017-03-07 - 2017-03-07 
Place (in Japanese) (See Japanese page) 
Place (in English) Nanzan University 
Topics (in Japanese) (See Japanese page) 
Topics (in English)  
Paper Information
Registration To COMP 
Conference Code 2017-03-COMP 
Language Japanese 
Title (in Japanese) (See Japanese page) 
Sub Title (in Japanese) (See Japanese page) 
Title (in English) Beating Brute Force for Systems of Polynomial Equations over Finite Fields 
Sub Title (in English)  
Keyword(1) exponential time algorithm  
Keyword(2) polynomial  
Keyword(3) low degree  
Keyword(4) arithmetic circuit  
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1st Author's Name Daniel Lokshtanov  
1st Author's Affiliation University of Bergen (U. Bergen)
2nd Author's Name Ramamohan Paturi  
2nd Author's Affiliation University of California, San Diego (UC San Diego)
3rd Author's Name Suguru Tamaki  
3rd Author's Affiliation Kyoto University (Kyoto U.)
4th Author's Name Ryan Williams  
4th Author's Affiliation Massachusetts Institute of Technology (MIT)
5th Author's Name Huacheng Yu  
5th Author's Affiliation Stanford University (Stanford U.)
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Speaker Author-3 
Date Time 2017-03-07 13:30:00 
Presentation Time 60 minutes 
Registration for COMP 
Paper # COMP2016-53 
Volume (vol) vol.116 
Number (no) no.503 
Page p.19 
#Pages 1 
Date of Issue 2017-02-28 (COMP) 

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