Mixed Bases for Efficient Inversion in F((22)2)2 and Conversion Matrices of SubBytes of AES

Yasuyuki NOGAMI  Kenta NEKADO  Tetsumi TOYOTA  Naoto HONGO  Yoshitaka MORIKAWA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E94-A   No.6   pp.1318-1327
Publication Date: 2011/06/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E94.A.1318
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
AES,  inversion,  towering,  conversion matrix,  bases,  

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A lot of improvements and optimizations for the hardware implementation of SubBytes of Rijndael, in detail inversion in F28 have been reported. Instead of the Rijndael original F28, it is known that its isomorphic tower field F((22)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 F28 to some less efficient inversions in F((22)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((22)2)2 but also conversion matrices between the isomorphic fields F28 and F((22)2)2.