Research On Several Classes Of Multi-valued Logical Functions With Excellent Cryptological Properties

The investigation of logical functions with excellent cryptological properties is very important in cryptological system designs and analyses. Applying the theories of probability, algebra and number theory, and the knowledge of spectrum theory, this thesis is devoted to the research of the k-th order strict avalanche criterion of multi-output Boolean functions,the relation between generalized partially Bent functions and generalized Bent functions over finite fields, the multi-output perfect nonlinear vector functions and multi-output generalized Bent functions over ring Zpl, the cryptographic properties of the multi-output rotation symmetric functions and p -valued rotation symmetric functions. The main results are as follows:1. The k-th order strict avalanche criterion of multi-output Boolean function is researched. The necessary and sufficient conditions of multi-output Boolean function fulfilling the strict avalanche criterion and the propagation criterion are also presented independently. The rule that multi-output Boolean function fulfils the lower order strict avalanche criterion if it fulfils the higher order strict avalanche criterion is proved. Two necessary and sufficient conditions of the multi-output Boolean functions satisfying k-th order strict avalanche criterion are presented. Combing the properties of symmetric functions, the strict avalanche criterion and propagation criterion of the multi-output symmetric functions are researched. Especially, it gets two combination discriminants of the multi-output symmetric functions satisfying k-th order strict avalanche criterion. It provides the theory to construct this class of function with good cryptological properties.2. The concept of partially Bent functions is extends to finite fields, which is still named generalized partially Bent functions. By applying Chrestenson cyclic spectrum characteristic of generalized partially Bent functions, and using the relation between the logical functions over finite fields and normal basis decomposition functions over corresponding prime fields, the relation of generalized partially Bent functions and generalized Bent functions over finite fields is discussed. Then the function relation expression and the spectrum relation expression of the two classes of functions are presented. According to the relation expressions, it can construct more generalized partially Bent functions from generalized Bent functions.3. Using p-adic decomposition of the variables over ring Zpl, the multi-output perfect nonlinear functions and multi-output generalized Bent functions are researched. Firstly, it presents the p-adic vector functional expression and operational properties of pl-valued multi-output functions. Then pl-valued multi-output perfect nonlinear functions are discussed, a necessary and sufficient condition is given by using the p-adic decomposition. At last, the multi-output generalized Bent functions over residue class ring is discussed. The relation expression between two classes of functions is presented under the p-adic decomposition. And this result shows that the multi-output perfect nonlinear vector functions and multi-output generalized Bent functions are consistent over residue ring Zpl. It's answers partially Mr. Nyberg's question that"whether these two classes of functions are equivalent over residue class ring".4. The cryptographic properties of multi-output rotation symmetric functions and multi-valued functions are studied. On the one hand, the notion of multi-output rotation symmetric functions is firstly introduced. The properties of generalized walsh spectrum and the generalized autocorrelation functions are presented. Then by constructing matrixes, it gets necessary and sufficient conditions for this class of functions to satisfying cryptographic properties such as balancedness, correlation immunity, strict avalanche criterion and so on. When n is odd number, the matrixes of n variables multi-output plateaued functions have special properties, so it gives a method to find the odd number variables multi-output plateaued rotation symmetric functions. And to decide the functions are multi-output plateaued rotation symmetric functions or not, this method only need calculate part of the generalized walsh cycle spectrum and the calculation could be reduced by almost half of the amount. On the other hand, the multi-valued rotation symmetric functions over Fp are researched. The Chrestenson spectrum and the autocorrelation functions of rotation symmetric functions are studied. The polynomials of rotation symmetric functions have special properties, by constructing matrixes, it establishes the relation of the truth table, short algebric normal form and the Chrestenson spectrum. By studying these matrixes, we get some sufficient and necessary conditions for rotation symmetric functions to satisfying cryptographic properties of balancedness, correlation immunity and so on.