By Chuan-Kun Wu, Dengguo Feng
This publication makes a speciality of the several representations and cryptographic houses of Booleans features, provides buildings of Boolean services with a few stable cryptographic homes. extra in particular, Walsh spectrum description of the normal cryptographic homes of Boolean capabilities, together with linear constitution, propagation criterion, nonlinearity, and correlation immunity are provided. buildings of symmetric Boolean services and of Boolean diversifications with solid cryptographic homes are in particular studied. This e-book isn't really intended to be complete, yet with its personal concentrate on a few unique learn of the authors some time past. To be self content material, a few uncomplicated thoughts and houses are brought. This ebook can function a reference for cryptographic set of rules designers, quite the designers of circulate ciphers and of block ciphers, and for lecturers with curiosity within the cryptographic houses of Boolean functions.
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Extra info for Boolean Functions and Their Applications in Cryptography (Advances in Computer Science and Technology)
X/ is said to be independent of variables xi1 ; xi2 ; : : : ; xik . For simplicity, we simply call the function to be independent of xi1 ; xi2 ; : : : ; xik . 1. x/ which only depends on the value of x1 . Mitchell  called the functions defined above as degenerate functions and recommended that in cryptographic applications, degenerate functions should be avoided. We will generalize this concept to a more general case in the next section. 1. x/. x/. So we only need to prove the necessity. x/ ¤ 0.
X/. There is a Walsh spectrum description of the invariants of Boolean functions as described below. 5. Let D ei1 ˚ ei2 ˚ ˚ eik . 2/ with hw; i D 1. w/ D n 1 2X D ei1 ˚ ei2 ˚ ˚ eik . x/. x ˚ /. x/. x˚ /i xD0 D . 1/ hw; i D . x/. w/ D 0 must hold. w/. 7), we have Rf . w/. x/. x/ is independent of variables xi1 ; xi2 ; : : : ; xik , the function can be treated as a function in only n k variables. x/. x/. y/. 2/ is the vector generated from w after removing all the ij -th coordinates of w, j D 1; 2; : : : ; k.
X/ 2 Fn . w/ D 2 n n 1 2X Sf . 20) D0 On the other hand, if Eq. x/ must be a Boolean function. Proof. x/ must satisfy that f 2 D f , then Eq. 20 holds. 2, the right-hand side of Eq. x/ is a Boolean function. 3 can be used to judge whether a real-valued function is a Boolean function when only the Walsh spectrum is given. In this case, the Walsh spectrum uniquely determines a Boolean function. We do not usually treat the Walsh spectrum as a representation of Boolean functions, because to check whether the Walsh spectrum is from a Boolean function requires much computation.