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Paper Abstract and Keywords
Presentation 2012-07-20 14:20
An Improved Algorithm for Approximate GCD Problems
Atsushi Takayasu, Noboru Kunihiro (UT) ISEC2012-35 SITE2012-31 ICSS2012-37 EMM2012-27
Abstract (in Japanese) (See Japanese page) 
(in English) In this paper, we analyze multivariate approximate common divisor problem(ACDP), given approximate multiples of the integer and we recover the integer.
So far, via lattice based Coppersmith's theorem, the polynomial time ACDP algorithms have been proposed.
In 2001, Howgrave-Graham proposed the polynomial time algorithms for ACDP as long as the tolerated errors are sufficiently small, when given two approximate multiples.
In 2011, Cohn and Heninger gave the multivariate generalization of Howgrave-Graham's algorithm, when given plural approximate multiples.
In this paper, we analyze only partially approximate common divisor problem(PACDP) when given one exact multiple of the integer.
We propose the improved polynomial time algorithm by Cohn and Heninger when given plural approximate multiples.
Cohn and Heninger revealed that PACDP can be solved when $ (\alpha _1+\cdots+\alpha _n)/n<\beta^{(n+1)/n}$.
We change the lattice construction by Cohn and Heninger's algorithm and improve the condition $ \sqrt[n]{\alpha _1 \cdots \alpha _n}<\beta^{(n+1)/n}$.
In the left hand side of the inequality, where is the arithmetic mean of $ \alpha _1,\ldots,\alpha _n$ in Cohn and Heninger's condition
become the geometric mean of $ \alpha _1,\ldots,\alpha _n$ in our condition.
Our algorithm can be applied to broader context.
Moreover, considering the multivariate generalization of Howgrave-Graham's algorithm, our condition is more natural than Cohn and Heninger's condition.
Keyword (in Japanese) (See Japanese page) 
(in English) approximate-GCD / lattice / Coppersmith's theorem / / / / /  
Reference Info. IEICE Tech. Rep., vol. 112, no. 126, ISEC2012-35, pp. 189-194, July 2012.
Paper # ISEC2012-35 
Date of Issue 2012-07-12 (ISEC, SITE, ICSS, EMM) 
ISSN Print edition: ISSN 0913-5685  Online edition: ISSN 2432-6380
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. (No. 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034)
Download PDF ISEC2012-35 SITE2012-31 ICSS2012-37 EMM2012-27

Conference Information
Conference Date 2012-07-19 - 2012-07-20 
Place (in Japanese) (See Japanese page) 
Place (in English)  
Topics (in Japanese) (See Japanese page) 
Topics (in English) Security 
Paper Information
Registration To ISEC 
Conference Code 2012-07-EMM-ISEC-SITE-ICSS-CSEC-SPT 
Language Japanese 
Title (in Japanese) (See Japanese page) 
Sub Title (in Japanese) (See Japanese page) 
Title (in English) An Improved Algorithm for Approximate GCD Problems 
Sub Title (in English)  
Keyword(1) approximate-GCD  
Keyword(2) lattice  
Keyword(3) Coppersmith's theorem  
1st Author's Name Atsushi Takayasu  
1st Author's Affiliation The University of Tokyo (UT)
2nd Author's Name Noboru Kunihiro  
2nd Author's Affiliation The University of Tokyo (UT)
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Date Time 2012-07-20 14:20:00 
Presentation Time 25 
Registration for ISEC 
Paper # IEICE-ISEC2012-35,IEICE-SITE2012-31,IEICE-ICSS2012-37,IEICE-EMM2012-27 
Volume (vol) IEICE-112 
Number (no) no.126(ISEC), no.127(SITE), no.128(ICSS), no.129(EMM) 
Page pp.189-194 
#Pages IEICE-6 
Date of Issue IEICE-ISEC-2012-07-12,IEICE-SITE-2012-07-12,IEICE-ICSS-2012-07-12,IEICE-EMM-2012-07-12 

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