Some Research On Resilient Functions In Cryptosystem

Resilient functions play more important roles in both the design and analysis of stream ciphers, block ciphers and hash functions. Areas where resilient functions find their applications widely include fault-tolerant distributed computing, quantum cryptographic key distribution and resilient network coding. Besides, the study of resilient functions clearly improve the research on combinatorial theory, graph theory and sequence theory. Inherently resilient functions have close relation to correlation-immune functions, which is early from the coloring problem in combinatorial theory. Some miscellaneous important problems on resilient functions are mainly studied. main results are as follows:1. Maiorana-Mafarland construction used in cryptographic functions is carefully studied. We analysis the advantages and disadvantages of the functions of super-class of Maiorana-Mafarland's class introduced by Carlet and Maiorana-Mafarland's subclass with degree optimization proposed by Pasalic. By concatenating nonlinear resilient functions, a class of new resilient functions with more nonlinear terms is constructed, which also satisfy a variety of desirable criteria: high nonlinearity, degree optimization, reasonable high resilient order and nonzero linear structures,and so on.2. We prove partial results about how to drive a general explicit formula for the cardinality of determine the maximum cardinality of # S kt(T ) as a function of k , t , and | T|, which is Open Problem proposed by Pasalic. and put forward a combinatorial conjecture.3. We have deeply studied the design idea of constructing highly nonlinear multi-output resilient functions with linear codes, and then analyzed the design principles using Jahanson and Pasalic's disjoint-code sets. A new class of multi-output resilient functions correspond to linear codes meeting Griesmer's bound is presented.4. We have deeply discussed the constructing principles of Reed-Muller codes and analyzed the relation between resilient functions and Reed-Muller codes from a cryptographic point of view. We also have made a study of the relation between RM codes'covering radius and algebraic immunity order in the resilient functions.