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Paper Abstract and Keywords
Presentation 2010-03-05 16:10
A study on the probabilistic algorithm to solve the elliptic curve discrete logarithm problem
Osamu Ashihara, Yoshitaka Sagara, Nobuhide Sakuragi, Tomoko K. Matsushima (Polytechnic Univ.)
Abstract (in Japanese) (See Japanese page) 
(in English) In this paper a probabilistic algorithm to solve the ECDLP is presented. This scheme uses the symmetry of the elliptic curve, and can be applied to a wide class of elliptic curves. In the proposed scheme a large number of random integers $H_i \in \{0,1,\cdots, g-1\} \; (i=1,2,\cdots)$ are generated to calculate the points $Q+[H_i]P \; (i=1,2,\cdots)$, where $Q = [K_{secret}]P$ and $g$ is the order of the point $P$. Then, the scheme tries to find a pair of points $Q+[H_i]P$ and $Q+[H_j]P$ whose $X$-coordinates are the same value but $H_i \ne H_j$, i.e., one in the pair is the inverse of the other. If such a pair is found, the secret $K_{secret}$ can be calculated by $2K_{secret}+H_i+H_j \equiv 0 \pmod{g}$.
We investigate the probability to find such a pair among $m$ random points on an elliptic curve. The probability of finding one or more pairs would be 1/2 when $m$ is of the order of $\sqrt{g}$. Furthermore, we discuss the techniques to reduce the size of disk storage and to parallelize the operation with multiple computers. It is also shown that the proposed algorithm can be extended to solve the usual discrete logarithm problem (DLP).
Keyword (in Japanese) (See Japanese page) 
(in English) ECDLP / elliptic curve / discrete logarithm problem / public key cryptography / / / /  
Reference Info. IEICE Tech. Rep., vol. 109, no. 445, ISEC2009-137, pp. 373-380, March 2010.
Paper # ISEC2009-137 
Date of Issue 2010-02-25 (IT, ISEC, WBS) 
ISSN Print edition: ISSN 0913-5685  Online edition: ISSN 2432-6380

Conference Information
Committee IT ISEC WBS  
Conference Date 2010-03-04 - 2010-03-05 
Place (in Japanese) (See Japanese page) 
Place (in English) Nagano-Engineering Campus, Shinshu University 
Topics (in Japanese) (See Japanese page) 
Topics (in English)  
Paper Information
Registration To ISEC 
Conference Code 2010-03-IT-ISEC-WBS 
Language English (Japanese title is available) 
Title (in Japanese) (See Japanese page) 
Sub Title (in Japanese) (See Japanese page) 
Title (in English) A study on the probabilistic algorithm to solve the elliptic curve discrete logarithm problem 
Sub Title (in English)  
Keyword(1) ECDLP  
Keyword(2) elliptic curve  
Keyword(3) discrete logarithm problem  
Keyword(4) public key cryptography  
1st Author's Name Osamu Ashihara  
1st Author's Affiliation Polytechnic University (Polytechnic Univ.)
2nd Author's Name Yoshitaka Sagara  
2nd Author's Affiliation Polytechnic University (Polytechnic Univ.)
3rd Author's Name Nobuhide Sakuragi  
3rd Author's Affiliation Polytechnic University (Polytechnic Univ.)
4th Author's Name Tomoko K. Matsushima  
4th Author's Affiliation Polytechnic University (Polytechnic Univ.)
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Date Time 2010-03-05 16:10:00 
Presentation Time 25 
Registration for ISEC 
Paper # IEICE-IT2009-129,IEICE-ISEC2009-137,IEICE-WBS2009-108 
Volume (vol) IEICE-109 
Number (no) no.444(IT), no.445(ISEC), no.446(WBS) 
Page pp.373-380 
#Pages IEICE-8 
Date of Issue IEICE-IT-2010-02-25,IEICE-ISEC-2010-02-25,IEICE-WBS-2010-02-25 

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