By Chuan-Kun Wu, Dengguo Feng
This ebook makes a speciality of the several representations and cryptographic houses of Booleans capabilities, offers structures of Boolean services with a few solid cryptographic houses. extra particularly, Walsh spectrum description of the conventional cryptographic houses of Boolean features, together with linear constitution, propagation criterion, nonlinearity, and correlation immunity are provided. structures of symmetric Boolean capabilities and of Boolean variations with stable cryptographic houses are particularly studied. This ebook isn't really intended to be complete, yet with its personal specialize in a few unique examine of the authors some time past. To be self content material, a few simple innovations and homes are brought. This booklet can function a reference for cryptographic set of rules designers, relatively the designers of move ciphers and of block ciphers, and for lecturers with curiosity within the cryptographic houses of Boolean functions.
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Extra resources for Boolean Functions and Their Applications in Cryptography
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.
2/ in the direction ei . More specifically, when c D 0, Hi is called a linear hyperplane, and when c D 1, Hi is called an affine hyperplane. 1/ Hi as the affine hyperplane. Otherwise, Hi denotes either a linear hyperplane or an affine hyperplane where these two cases do not make difference. x/ under the condition of xi D c. x/. 2/. x/ is independent of h˛; xi. , the image of a Boolean function in n variables on a hyperplane is degenerate to a Boolean function in n 1 variables. x1 ; x2 ; : : : ; xi 1 ; xiC1 ; : : : ; xn / which is a Boolean function in Fn 1 .
In the design of stream ciphers, Boolean functions are often required to be balanced, as this would give outputs with balanced number of 0’s and 1’s and looks more random. When a Boolean function is balanced, the requirement of algebraic degree has to have some sacrifice. f /, and when the minterms are expanded to convert to the algebraic normal form, every such minterm contributes a product of all the variables x1 x2 xn . 2. x/ 2 Fn . f / Ä n 1. f / Ä n 1. 3 Nonlinearity For security reasons, pseudorandom sequences generated by linear shift registers are not supposed to be used in encryptions directly.
Boolean Functions and Their Applications in Cryptography by Chuan-Kun Wu, Dengguo Feng