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Journal of Mathematical Cryptology

Managing Editor: Magliveras, Spyros S. / Steinwandt, Rainer / Trung, Tran

Editorial Board: Blackburn, Simon R. / Blundo, Carlo / Burmester, Mike / Cramer, Ronald / Gilman, Robert / Gonzalez Vasco, Maria Isabel / Grosek, Otokar / Helleseth, Tor / Kim, Kwangjo / Koblitz, Neal / Kurosawa, Kaoru / Lauter, Kristin / Lange, Tanja / Menezes, Alfred / Nguyen, Phong Q. / Pieprzyk, Josef / Rötteler, Martin / Safavi-Naini, Rei / Shparlinski, Igor E. / Stinson, Doug / Takagi, Tsuyoshi / Williams, Hugh C. / Yung, Moti


CiteScore 2017: 1.43

SCImago Journal Rank (SJR) 2017: 0.293
Source Normalized Impact per Paper (SNIP) 2017: 1.117

Mathematical Citation Quotient (MCQ) 2017: 0.51

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ISSN
1862-2984
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Volume 4, Issue 1

Issues

The monodromy pairing and discrete logarithm on the Jacobian of finite graphs

Farbod Shokrieh
Published Online: 2010-07-08 | DOI: https://doi.org/10.1515/jmc.2010.002

Abstract

Every graph has a canonical finite abelian group attached to it. This group has appeared in the literature under a variety of names including the sandpile group, critical group, Jacobian group, and Picard group. The construction of this group closely mirrors the construction of the Jacobian variety of an algebraic curve. Motivated by this analogy, it was recently suggested by Norman Biggs that the critical group of a finite graph is a good candidate for doing discrete logarithm based cryptography. In this paper, we study a bilinear pairing on this group and show how to compute it. Then we use this pairing to find the discrete logarithm efficiently, thus showing that the associated cryptographic schemes are not secure. Our approach resembles the MOV attack on elliptic curves.

Keywords.: Discrete logarithm; graphs; Jacobian; monodromy pairing; generalized inverses; critical group; sandpiles

About the article

Received: 2009-07-30

Revised: 2010-03-02

Published Online: 2010-07-08

Published in Print: 2010-07-01


Citation Information: Journal of Mathematical Cryptology, Volume 4, Issue 1, Pages 43–56, ISSN (Online) 1862-2984, ISSN (Print) 1862-2976, DOI: https://doi.org/10.1515/jmc.2010.002.

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[2]
Fatemeh Mohammadi and Farbod Shokrieh
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[3]
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