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Mixed Bases for Efficient Inversion in F((2^{2})^{2})^{2} and Conversion Matrices of SubBytes of AES
Yasuyuki NOGAMI Kenta NEKADO Tetsumi TOYOTA Naoto HONGO Yoshitaka MORIKAWA
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E94A
No.6
pp.13181327 Publication Date: 2011/06/01
Online ISSN: 17451337
DOI: 10.1587/transfun.E94.A.1318
Print ISSN: 09168508 Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications) Category: Keyword: AES, inversion, towering, conversion matrix, bases,
Full Text: PDF>>
Summary:
A lot of improvements and optimizations for the hardware implementation of SubBytes of Rijndael, in detail inversion in F_{28} have been reported. Instead of the Rijndael original F_{28}, 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 F_{28} 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 F_{28} and F_{((22)2)2}.

