Solving a DLP with auxiliary input with the ρ-algorithm

Yumi Sakemi, Tetsuya Izu, Masahiko Takenaka, Masaya Yasuda

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

Abstract

The discrete logarithm problem with auxiliary input (DLPwAI) is a problem to find a positive integer α from elements G, αG, α d G in an additive cyclic group generated by G of prime order r and a positive integer d dividing r -1. In 2011, Sakemi et al. implemented Cheon's algorithm for solving DLPwAI, and solved a DLPwAI in a group with 128-bit order r in about 131 hours with a single core on an elliptic curve defined over a prime finite field which is used in the TinyTate library for embedded cryptographic devices. However, since their implementation was based on Shanks' Baby-step Giant-step (BSGS) algorithm as a sub-algorithm, it required a large amount of memory (246 GByte) so that it was concluded that applying other DLPwAIs with larger parameter is infeasible. In this paper, we implemented Cheon's algorithm based on Pollard's ρ-algorithm in order to reduce the required memory. As a result, we have succeeded solving the same DLPwAI in about 136 hours by a single core with less memory (0.5 MByte).

Original languageEnglish
Title of host publicationInformation Security Applications - 12th International Workshop, WISA 2011, Revised Selected Papers
Pages98-108
Number of pages11
DOIs
Publication statusPublished - Mar 15 2012
Event12th International Workshop on Information Security Applications, WISA 2011 - Jeju Island, Korea, Republic of
Duration: Aug 22 2011Aug 24 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7115 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other12th International Workshop on Information Security Applications, WISA 2011
CountryKorea, Republic of
CityJeju Island
Period8/22/118/24/11

Fingerprint

Discrete Logarithm Problem
Data storage equipment
Integer
Cyclic group
Elliptic Curves
Galois field

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Sakemi, Y., Izu, T., Takenaka, M., & Yasuda, M. (2012). Solving a DLP with auxiliary input with the ρ-algorithm. In Information Security Applications - 12th International Workshop, WISA 2011, Revised Selected Papers (pp. 98-108). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7115 LNCS). https://doi.org/10.1007/978-3-642-27890-7_8

Solving a DLP with auxiliary input with the ρ-algorithm. / Sakemi, Yumi; Izu, Tetsuya; Takenaka, Masahiko; Yasuda, Masaya.

Information Security Applications - 12th International Workshop, WISA 2011, Revised Selected Papers. 2012. p. 98-108 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7115 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Sakemi, Y, Izu, T, Takenaka, M & Yasuda, M 2012, Solving a DLP with auxiliary input with the ρ-algorithm. in Information Security Applications - 12th International Workshop, WISA 2011, Revised Selected Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7115 LNCS, pp. 98-108, 12th International Workshop on Information Security Applications, WISA 2011, Jeju Island, Korea, Republic of, 8/22/11. https://doi.org/10.1007/978-3-642-27890-7_8
Sakemi Y, Izu T, Takenaka M, Yasuda M. Solving a DLP with auxiliary input with the ρ-algorithm. In Information Security Applications - 12th International Workshop, WISA 2011, Revised Selected Papers. 2012. p. 98-108. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-27890-7_8
Sakemi, Yumi ; Izu, Tetsuya ; Takenaka, Masahiko ; Yasuda, Masaya. / Solving a DLP with auxiliary input with the ρ-algorithm. Information Security Applications - 12th International Workshop, WISA 2011, Revised Selected Papers. 2012. pp. 98-108 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{3d8c0b5b7ebd4ca7be8448e83d8264bc,
title = "Solving a DLP with auxiliary input with the ρ-algorithm",
abstract = "The discrete logarithm problem with auxiliary input (DLPwAI) is a problem to find a positive integer α from elements G, αG, α d G in an additive cyclic group generated by G of prime order r and a positive integer d dividing r -1. In 2011, Sakemi et al. implemented Cheon's algorithm for solving DLPwAI, and solved a DLPwAI in a group with 128-bit order r in about 131 hours with a single core on an elliptic curve defined over a prime finite field which is used in the TinyTate library for embedded cryptographic devices. However, since their implementation was based on Shanks' Baby-step Giant-step (BSGS) algorithm as a sub-algorithm, it required a large amount of memory (246 GByte) so that it was concluded that applying other DLPwAIs with larger parameter is infeasible. In this paper, we implemented Cheon's algorithm based on Pollard's ρ-algorithm in order to reduce the required memory. As a result, we have succeeded solving the same DLPwAI in about 136 hours by a single core with less memory (0.5 MByte).",
author = "Yumi Sakemi and Tetsuya Izu and Masahiko Takenaka and Masaya Yasuda",
year = "2012",
month = "3",
day = "15",
doi = "10.1007/978-3-642-27890-7_8",
language = "English",
isbn = "9783642278891",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
pages = "98--108",
booktitle = "Information Security Applications - 12th International Workshop, WISA 2011, Revised Selected Papers",

}

TY - GEN

T1 - Solving a DLP with auxiliary input with the ρ-algorithm

AU - Sakemi, Yumi

AU - Izu, Tetsuya

AU - Takenaka, Masahiko

AU - Yasuda, Masaya

PY - 2012/3/15

Y1 - 2012/3/15

N2 - The discrete logarithm problem with auxiliary input (DLPwAI) is a problem to find a positive integer α from elements G, αG, α d G in an additive cyclic group generated by G of prime order r and a positive integer d dividing r -1. In 2011, Sakemi et al. implemented Cheon's algorithm for solving DLPwAI, and solved a DLPwAI in a group with 128-bit order r in about 131 hours with a single core on an elliptic curve defined over a prime finite field which is used in the TinyTate library for embedded cryptographic devices. However, since their implementation was based on Shanks' Baby-step Giant-step (BSGS) algorithm as a sub-algorithm, it required a large amount of memory (246 GByte) so that it was concluded that applying other DLPwAIs with larger parameter is infeasible. In this paper, we implemented Cheon's algorithm based on Pollard's ρ-algorithm in order to reduce the required memory. As a result, we have succeeded solving the same DLPwAI in about 136 hours by a single core with less memory (0.5 MByte).

AB - The discrete logarithm problem with auxiliary input (DLPwAI) is a problem to find a positive integer α from elements G, αG, α d G in an additive cyclic group generated by G of prime order r and a positive integer d dividing r -1. In 2011, Sakemi et al. implemented Cheon's algorithm for solving DLPwAI, and solved a DLPwAI in a group with 128-bit order r in about 131 hours with a single core on an elliptic curve defined over a prime finite field which is used in the TinyTate library for embedded cryptographic devices. However, since their implementation was based on Shanks' Baby-step Giant-step (BSGS) algorithm as a sub-algorithm, it required a large amount of memory (246 GByte) so that it was concluded that applying other DLPwAIs with larger parameter is infeasible. In this paper, we implemented Cheon's algorithm based on Pollard's ρ-algorithm in order to reduce the required memory. As a result, we have succeeded solving the same DLPwAI in about 136 hours by a single core with less memory (0.5 MByte).

UR - http://www.scopus.com/inward/record.url?scp=84858036337&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858036337&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-27890-7_8

DO - 10.1007/978-3-642-27890-7_8

M3 - Conference contribution

AN - SCOPUS:84858036337

SN - 9783642278891

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 98

EP - 108

BT - Information Security Applications - 12th International Workshop, WISA 2011, Revised Selected Papers

ER -