### Abstract

A lot of improvements and optimizations for the hardware implementation of SubBytes of Rijndael, in detail inversion in F_{2}
^{8} have been reported. Instead of the Rijndael original F_{2}
^{8} , it is known that its isomorphic tower field F((_{2}
^{2}) ^{2})^{2} has a more efficient inversion. Then, some conversion matrices are also needed for connecting these isomorphic binary fields. According to the previous works, it is said that the number of 1's in the conversion matrices is preferred to be small; however, they have not focused on the Hamming weights of the row vectors of the matrices. It plays an important role for the calculation architecture, in detail critical path delays. This paper shows the existence of efficient conversion matrices whose row vectors all have the Hamming weights less than or equal to 4. They are introduced as quite rare cases. Then, it is pointed out that such efficient conversion matrices can connect the Rijndael original F_{2}
^{8} to some less efficient inversions in F((_{2}
^{2})^{2})^{2} but not to the most efficient ones. In order to overcome these inconveniences, this paper next proposes a technique called mixed bases. For the towerings, most of previous works have used several kinds of bases such as polynomial and normal bases in mixture. Different from them, this paper proposes another mixture of bases that contributes to the reduction of the critical path delay of SubBytes. Then, it is shown that the proposed mixture contributes to the efficiencies of not only inversion in F((_{2}
^{2})^{2})^{2} but also conversion matrices between the isomorphic fields F_{2}
^{8} and F((_{2}
^{2})^{2})^{2} .

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

Pages (from-to) | 1318-1327 |

Number of pages | 10 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E94-A |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2011 |

### Fingerprint

### Keywords

- AES
- Bases
- Conversion matrix
- Inversion
- Towering

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Computer Graphics and Computer-Aided Design
- Applied Mathematics
- Signal Processing

### Cite this

_{2}

^{2})

^{2})

^{2}and conversion matrices of subbytes of AES.

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*,

*E94-A*(6), 1318-1327. https://doi.org/10.1587/transfun.E94.A.1318

**Mixed bases for efficient inversion in F(( _{2}
^{2}) ^{2})^{2} and conversion matrices of subbytes of AES.** / Nogami, Yasuyuki; Nekado, Kenta; Toyota, Tetsumi; Hongo, Naoto; Morikawa, Yoshitaka.

Research output: Contribution to journal › Article

_{2}

^{2})

^{2})

^{2}and conversion matrices of subbytes of AES',

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E94-A, no. 6, pp. 1318-1327. https://doi.org/10.1587/transfun.E94.A.1318

_{2}

^{2})

^{2})

^{2}and conversion matrices of subbytes of AES. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences. 2011 Jun;E94-A(6):1318-1327. https://doi.org/10.1587/transfun.E94.A.1318

}

TY - JOUR

T1 - Mixed bases for efficient inversion in F((2 2) 2)2 and conversion matrices of subbytes of AES

AU - Nogami, Yasuyuki

AU - Nekado, Kenta

AU - Toyota, Tetsumi

AU - Hongo, Naoto

AU - Morikawa, Yoshitaka

PY - 2011/6

Y1 - 2011/6

N2 - A lot of improvements and optimizations for the hardware implementation of SubBytes of Rijndael, in detail inversion in F2 8 have been reported. Instead of the Rijndael original F2 8 , it is known that its isomorphic tower field F((2 2) 2)2 has a more efficient inversion. Then, some conversion matrices are also needed for connecting these isomorphic binary fields. According to the previous works, it is said that the number of 1's in the conversion matrices is preferred to be small; however, they have not focused on the Hamming weights of the row vectors of the matrices. It plays an important role for the calculation architecture, in detail critical path delays. This paper shows the existence of efficient conversion matrices whose row vectors all have the Hamming weights less than or equal to 4. They are introduced as quite rare cases. Then, it is pointed out that such efficient conversion matrices can connect the Rijndael original F2 8 to some less efficient inversions in F((2 2)2)2 but not to the most efficient ones. In order to overcome these inconveniences, this paper next proposes a technique called mixed bases. For the towerings, most of previous works have used several kinds of bases such as polynomial and normal bases in mixture. Different from them, this paper proposes another mixture of bases that contributes to the reduction of the critical path delay of SubBytes. Then, it is shown that the proposed mixture contributes to the efficiencies of not only inversion in F((2 2)2)2 but also conversion matrices between the isomorphic fields F2 8 and F((2 2)2)2 .

AB - A lot of improvements and optimizations for the hardware implementation of SubBytes of Rijndael, in detail inversion in F2 8 have been reported. Instead of the Rijndael original F2 8 , it is known that its isomorphic tower field F((2 2) 2)2 has a more efficient inversion. Then, some conversion matrices are also needed for connecting these isomorphic binary fields. According to the previous works, it is said that the number of 1's in the conversion matrices is preferred to be small; however, they have not focused on the Hamming weights of the row vectors of the matrices. It plays an important role for the calculation architecture, in detail critical path delays. This paper shows the existence of efficient conversion matrices whose row vectors all have the Hamming weights less than or equal to 4. They are introduced as quite rare cases. Then, it is pointed out that such efficient conversion matrices can connect the Rijndael original F2 8 to some less efficient inversions in F((2 2)2)2 but not to the most efficient ones. In order to overcome these inconveniences, this paper next proposes a technique called mixed bases. For the towerings, most of previous works have used several kinds of bases such as polynomial and normal bases in mixture. Different from them, this paper proposes another mixture of bases that contributes to the reduction of the critical path delay of SubBytes. Then, it is shown that the proposed mixture contributes to the efficiencies of not only inversion in F((2 2)2)2 but also conversion matrices between the isomorphic fields F2 8 and F((2 2)2)2 .

KW - AES

KW - Bases

KW - Conversion matrix

KW - Inversion

KW - Towering

UR - http://www.scopus.com/inward/record.url?scp=79957989139&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957989139&partnerID=8YFLogxK

U2 - 10.1587/transfun.E94.A.1318

DO - 10.1587/transfun.E94.A.1318

M3 - Article

AN - SCOPUS:79957989139

VL - E94-A

SP - 1318

EP - 1327

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 6

ER -