Relating undisturbed bits to other properties of substitution boxes

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Tezin Türü: Yüksek Lisans

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2014

Öğrenci: RUSYDİ HASAN MAKARIM

Danışman: ALİ DOĞANAKSOY

Özet:

Recently it was observed that for a particular nonzero input difference to an S-Box, some bits in all the corresponding output differences may remain invariant. This specific invariant bit is called undisturbed bit. Undisturbed bit can also be seen as a truncated differential with probability 1 for an S-Box. The existence of undisturbed bits was found in the S-Box of PRESENT and its inverse. A 13-round improbable differential attack on PRESENT was provided by Tezcan (2013) and without using the undisturbed bits in the S-Box an attack of this type can only reach 7 rounds. Although the observation and the cryptanalytic application of undisturbed bits are given, its relation with other properties of an S-Box remain unknown. This thesis presents some results on mathematical properties of S-Boxes having undisturbed bits. We show that an S-Box has undisturbed bits if any of its coordinate function has a nonzero linear structure. The relation of undisturbed bits with other cryptanalytic tools such as difference distribution table (DDT) and linear approximation table (LAT) are also given. We show that autocorrelation table is proven to be a more useful tool, compared to DDT, to obtain all nonzero input differences that yield undisturbed bits. Autocorrelation table can then be viewed as a counterpart of DDT for truncated differential cryptanalysis. Given an n x m balanced S-Box, we state that the S-Box has undisturbed bit whenever the degree of any of its coordinate function is quadratic.