### Abstract

In 1984, Ong, Schnorr and Shamir proposed an efficient signature scheme (OSS signature scheme) using a bivariate quadratic equation. Its security was believed to be based on the difficulty of integer factorization. However, an efficient attack without integer factorization was subsequently found. In 2008, Hashimoto and Sakurai proposed an extended scheme (HS scheme), based on OSS signature scheme that used multivariate and non-commutative ring. HS scheme uses a composite number as a modulus in the same manner as OSS signature scheme. In this paper, we redefine HS scheme in such a way that it deals with not only integers modulo a composite number, but also elements of a finite field. In the case of a finite field, it becomes a scheme in the multivariate public key cryptosystem. In fact, its public key is constructed by a version of Rainbow in which all the components in the parameter are equal. (We call such a Rainbow a uniformly-layered Rainbow.) In particular, our scheme is a candidate for post-quantum cryptography. If a non-commutative ring used in the proposed scheme is chosen by the group ring associated to dihedral group, the speed of the signature generation can be accelerated by about 50% in comparison with the corresponding Rainbow. We analyze the security of the extended HS scheme against some attacks and conclude that if its base field is GF(256), then the dimension of a non-commutative ring must be more than 10 in order to be secure.

Original language | English |
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Title of host publication | Post-Quantum Cryptography - 4th International Workshop, PQCrypto 2011, Proceedings |

Pages | 275-294 |

Number of pages | 20 |

DOIs | |

Publication status | Published - Dec 12 2011 |

Event | 4th International Workshop on Post-Quantum Cryptography, PQCrypto 2011 - Taipei, Taiwan, Province of China Duration: Nov 29 2011 → Dec 2 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 | 7071 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 4th International Workshop on Post-Quantum Cryptography, PQCrypto 2011 |
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Country | Taiwan, Province of China |

City | Taipei |

Period | 11/29/11 → 12/2/11 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Post-Quantum Cryptography - 4th International Workshop, PQCrypto 2011, Proceedings*(pp. 275-294). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7071 LNCS). https://doi.org/10.1007/978-3-642-25405-5_18

**A security analysis of uniformly-layered rainbow : Revisiting Sato-Araki's non-commutative approach to Ong-Schnorr-Shamir signature towards postquantum paradigm.** / Yasuda, Takanori; Sakurai, Kouichi.

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

*Post-Quantum Cryptography - 4th International Workshop, PQCrypto 2011, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7071 LNCS, pp. 275-294, 4th International Workshop on Post-Quantum Cryptography, PQCrypto 2011, Taipei, Taiwan, Province of China, 11/29/11. https://doi.org/10.1007/978-3-642-25405-5_18

}

TY - GEN

T1 - A security analysis of uniformly-layered rainbow

T2 - Revisiting Sato-Araki's non-commutative approach to Ong-Schnorr-Shamir signature towards postquantum paradigm

AU - Yasuda, Takanori

AU - Sakurai, Kouichi

PY - 2011/12/12

Y1 - 2011/12/12

N2 - In 1984, Ong, Schnorr and Shamir proposed an efficient signature scheme (OSS signature scheme) using a bivariate quadratic equation. Its security was believed to be based on the difficulty of integer factorization. However, an efficient attack without integer factorization was subsequently found. In 2008, Hashimoto and Sakurai proposed an extended scheme (HS scheme), based on OSS signature scheme that used multivariate and non-commutative ring. HS scheme uses a composite number as a modulus in the same manner as OSS signature scheme. In this paper, we redefine HS scheme in such a way that it deals with not only integers modulo a composite number, but also elements of a finite field. In the case of a finite field, it becomes a scheme in the multivariate public key cryptosystem. In fact, its public key is constructed by a version of Rainbow in which all the components in the parameter are equal. (We call such a Rainbow a uniformly-layered Rainbow.) In particular, our scheme is a candidate for post-quantum cryptography. If a non-commutative ring used in the proposed scheme is chosen by the group ring associated to dihedral group, the speed of the signature generation can be accelerated by about 50% in comparison with the corresponding Rainbow. We analyze the security of the extended HS scheme against some attacks and conclude that if its base field is GF(256), then the dimension of a non-commutative ring must be more than 10 in order to be secure.

AB - In 1984, Ong, Schnorr and Shamir proposed an efficient signature scheme (OSS signature scheme) using a bivariate quadratic equation. Its security was believed to be based on the difficulty of integer factorization. However, an efficient attack without integer factorization was subsequently found. In 2008, Hashimoto and Sakurai proposed an extended scheme (HS scheme), based on OSS signature scheme that used multivariate and non-commutative ring. HS scheme uses a composite number as a modulus in the same manner as OSS signature scheme. In this paper, we redefine HS scheme in such a way that it deals with not only integers modulo a composite number, but also elements of a finite field. In the case of a finite field, it becomes a scheme in the multivariate public key cryptosystem. In fact, its public key is constructed by a version of Rainbow in which all the components in the parameter are equal. (We call such a Rainbow a uniformly-layered Rainbow.) In particular, our scheme is a candidate for post-quantum cryptography. If a non-commutative ring used in the proposed scheme is chosen by the group ring associated to dihedral group, the speed of the signature generation can be accelerated by about 50% in comparison with the corresponding Rainbow. We analyze the security of the extended HS scheme against some attacks and conclude that if its base field is GF(256), then the dimension of a non-commutative ring must be more than 10 in order to be secure.

UR - http://www.scopus.com/inward/record.url?scp=82955190472&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82955190472&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-25405-5_18

DO - 10.1007/978-3-642-25405-5_18

M3 - Conference contribution

AN - SCOPUS:82955190472

SN - 9783642254048

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

SP - 275

EP - 294

BT - Post-Quantum Cryptography - 4th International Workshop, PQCrypto 2011, Proceedings

ER -