### Abstract

As one of problems that guarantee the security of pairing-based cryptography, pairing inversion problem is studied. Some recent works have reduced fixed argument pairing inversion (FAPI) problem to exponentiation inversion (EI) problem. According to the results, FAPI problem is solved if EI problem of exponent (q^{k} - 1)/Φ_{k} (q) is solved, where q, k, and r are the characteristic, embedding degree, and order of pairing group, respectively. Φ_{k}(x) is the cyclotomic polynomial of order k. This paper shows an approach for reducing the exponent of EI problem to q - 1 especially on Ate pairing. For many embedding degrees, it is considerably reduced from the previous result (q^{k} - 1)/Φ_{k}(q). After that, the difficulty of the reduced EI problem is discussed based on the distribution of correct (q - 1)-th roots on a small example.

Original language | English |
---|---|

Pages (from-to) | 240-249 |

Number of pages | 10 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 8639 LNCS |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### Keywords

- Barreto-Naehrig curve
- pairing inversion problem
- trace

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*8639 LNCS*, 240-249. https://doi.org/10.1007/978-3-319-09843-2_18

**Exponentiation inversion problem reduced from fixed argument pairing inversion on twistable ate pairing and its difficulty.** / Akagi, Shoichi; Nogami, Yasuyuki.

Research output: Contribution to journal › Article

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 8639 LNCS, pp. 240-249. https://doi.org/10.1007/978-3-319-09843-2_18

}

TY - JOUR

T1 - Exponentiation inversion problem reduced from fixed argument pairing inversion on twistable ate pairing and its difficulty

AU - Akagi, Shoichi

AU - Nogami, Yasuyuki

PY - 2014

Y1 - 2014

N2 - As one of problems that guarantee the security of pairing-based cryptography, pairing inversion problem is studied. Some recent works have reduced fixed argument pairing inversion (FAPI) problem to exponentiation inversion (EI) problem. According to the results, FAPI problem is solved if EI problem of exponent (qk - 1)/Φk (q) is solved, where q, k, and r are the characteristic, embedding degree, and order of pairing group, respectively. Φk(x) is the cyclotomic polynomial of order k. This paper shows an approach for reducing the exponent of EI problem to q - 1 especially on Ate pairing. For many embedding degrees, it is considerably reduced from the previous result (qk - 1)/Φk(q). After that, the difficulty of the reduced EI problem is discussed based on the distribution of correct (q - 1)-th roots on a small example.

AB - As one of problems that guarantee the security of pairing-based cryptography, pairing inversion problem is studied. Some recent works have reduced fixed argument pairing inversion (FAPI) problem to exponentiation inversion (EI) problem. According to the results, FAPI problem is solved if EI problem of exponent (qk - 1)/Φk (q) is solved, where q, k, and r are the characteristic, embedding degree, and order of pairing group, respectively. Φk(x) is the cyclotomic polynomial of order k. This paper shows an approach for reducing the exponent of EI problem to q - 1 especially on Ate pairing. For many embedding degrees, it is considerably reduced from the previous result (qk - 1)/Φk(q). After that, the difficulty of the reduced EI problem is discussed based on the distribution of correct (q - 1)-th roots on a small example.

KW - Barreto-Naehrig curve

KW - pairing inversion problem

KW - trace

UR - http://www.scopus.com/inward/record.url?scp=84907373112&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907373112&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-09843-2_18

DO - 10.1007/978-3-319-09843-2_18

M3 - Article

AN - SCOPUS:84907373112

VL - 8639 LNCS

SP - 240

EP - 249

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -