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Title:
Distinguishing Properties of Higher Order Derivatives of Boolean Functions
Authors: Ming Duan, Xuejia Lai, Mohan Yang, Xiaorui Sun, Bo Zhu
Abstract:Higher order differential cryptanalysis is based on the property of higher order derivatives of Boolean functions that the degree of a Boolean function can be reduced by at least 1 by taking a derivative on the function at any point. We define \emph{fast point} as the point at which the degree can be reduced by at least 2. In this paper, we show that the fast points of a n-variable Boolean function form a linear subspace and its dimension plus the algebraic degree of the function is at most n. We also show that non-trivial fast point exists in every n-variable Boolean function of degree n-1, every symmetric Boolean function of degree d where n \not\equiv d \pmod{2} and every quadratic Boolean function of odd number variables. Moreover we show the property of fast points for n-variable Boolean functions of degree n-2.
ePrint: https://eprint.iacr.org/2010/417
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