On Properties And Constructions Of Boolean Functions In Cryptography

Boolean functions play a central role in symmetric key cryptosystems. In thisthesis, some important properties of Boolean functions are investigated. The maincontents and fruits of this thesis are outlined as follows:1. The relationship among cross-correlation functions of arbitrary four Booleanfunctions is presented. Some properties of cross-correlation functions andauto-correlation functions are given, and several known cross-correlationproperties of Boolean functions are generalized. Based on them, a lower boundfor the maximal cross-correlation functions of two Boolean functions isobtained.2. By using Convolution Theorem, the relationship between the spectrums of twoarbitrary Boolean functions and the spectrum of their derivatives is presented.Also, the relationship between the Walsh spectrums and the decompositions oftwo arbitrary Boolean functions is deduced. Based on them, some propertiesbetween the Walsh spectrum and the cross-correlation function are generalized.3. Based on the relationship between cheating immune secret sharing function andcryptographic criteria of Boolean functions, a technique on constructing acheating immune secret sharing function is studied. By usingMariorana-McFarland construction technique, two new methods to constructcheating secret sharing functions are proposed.4. The duality of Bent functions is studied. By using the definition of the dualfunctions, we prove that the Walsh spectrum of any derivative of two Bentfunctions is linked with the values of the Walsh spectrums of the derivatives oftheir dual functions. Also, it is shown that the sum of two Boolean functionsand the sum of their dual functions have the same Hamming weight.5. The special Boolean function f₁||f₂||f₃||f₄by concatenation is presented. Itscryptographic properties are investigated, such as correlation immunity,resiliency, propagation characteristics, and algebraic immunity and so on. Weobtain its cryptographic properties of f₁||f₂||f₃||f₄are good when the cryptographicproperties of the Boolean functions f₁, f₂, f₃and f₄are good.