### 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 language | English |
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Title of host publication | Information Security Applications - 12th International Workshop, WISA 2011, Revised Selected Papers |

Pages | 98-108 |

Number of pages | 11 |

DOIs | |

Publication status | Published - Mar 15 2012 |

Event | 12th International Workshop on Information Security Applications, WISA 2011 - Jeju Island, Korea, Republic of Duration: Aug 22 2011 → Aug 24 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7115 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 12th International Workshop on Information Security Applications, WISA 2011 |
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Country | Korea, Republic of |

City | Jeju Island |

Period | 8/22/11 → 8/24/11 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

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 -