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Journal of Mathematical Cryptology

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

Editorial Board Member: Blackburn, Simon R. / Blundo, Carlo / Burmester, Mike / Cramer, Ronald / Dawson, Ed / 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

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SCImago Journal Rank (SJR) 2015: 0.313
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Impact per Publication (IPP) 2015: 0.396

Mathematical Citation Quotient (MCQ) 2015: 0.24

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Time-memory trade-offs for index calculus in genus 3

Kim Laine
  • Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA
  • :
/ Kristin Lauter
  • Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA
  • :
Published Online: 2015-05-21 | DOI: https://doi.org/10.1515/jmc-2014-0033


In this paper, we present a variant of Diem's O˜(q) index calculus algorithm to attack the discrete logarithm problem (DLP) in Jacobians of genus 3 non-hyperelliptic curves over a finite field 𝔽q. We implement this new variant in C++ and study the complexity in both theory and practice, making the logarithmic factors and constants hidden in the O˜-notation precise. Our variant improves the computational complexity at the cost of a moderate increase in memory consumption, but we also improve the computational complexity even when we limit the memory usage to that of Diem's original algorithm. Finally, we examine how parallelization can help to reduce both the memory cost per computer and the running time for our algorithms.

Keywords: Discrete logarithm problem; index calculus; double large prime; higher genus; genus 3; non-hyperelliptic curve; quartic curve; plane curve; time-memory trade-off

MSC: 11Y16; 11T71

Received: 2014-09-12

Accepted: 2015-04-07

Published Online: 2015-05-21

Published in Print: 2015-06-01

Citation Information: Journal of Mathematical Cryptology. Volume 9, Issue 2, Pages 95–114, ISSN (Online) 1862-2984, ISSN (Print) 1862-2976, DOI: https://doi.org/10.1515/jmc-2014-0033, May 2015

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