Boolean functions, Hadamard matrices, orthogonal designs applicable to security and communication

This thesis is about Boolean functions and their cryptographic properties, Hadamard matrices, orthogonal designs, and their applications. An original definition of Boolean functions and their properties are given. After an introduction to the background of Boolean functions, different cryptographic properties of Boolean functions are described. Bent functions play a most important role in cryptography. Several classes of bent functions are introduced. Some methods of construction high nonlinearity and balanced Boolean functions base on bent functions are described. In the thesis, the author shows some applications on Boolean functions. The author also introduced the relationship between coding theory and Boolean functions. Hadamard matrices are introduced after Boolean functions. In the thesis the author shows the definitions and relationship between Hadamard matrices and difference sets (DS), supplementary difference sets (SDS), orthogonal designs (OD) and symmetric balanced incomplete block design (SBIBD). The author also re-states some methods of constructing Hadamard matrices. Since Hadamard matrices be sparked the interest in 1970s, Hadamard matrices can be implemented as a base in error correcting, data communication, cryptography, etc. The thesis includes a preliminary section orthogonal designs, where the author gives some known results about orthogonal designs. Amicable orthogonal designs are also reminded in the thesis. Some applications of orthogonal designs in code theory are introduced. New results about construction cubic homogeneous bent functions on V2n for all n ≥ 3, n ≠ 4 are given. There is no homogeneous bent function of degree n exists on V2n for n ≥ 4. New results about homogeneous functions with high nonlinearity in odd space are found. Some new families of C—partitions and T—matrices are given, which can construct new Hadamard matrices. New results about T—matrices, difference sets, SBIBD and Hadamard matrices are described. The author also gives some new infinite families of orthogonal designs using Kharaghani arrays.