Boolean Function Optimal Continuous Function Of Information Theory

To cryptanalysis for SP network iterative technology, we must let Boolean function in the SP network iterative technology be continuous. The problem that seeking for Boolean function in the SP network structure belongs to combinatorial optimization.Let Boolean function be continuous in that combinatorial optimization problem which seeking for Boolean function is transformed into the linear or nonlinear programming one of continuous function. Thus the technology and method of linear or nonlinear programming can be used for reference. Especially, if derivative of continuous function exists, we can use gradient method to solve the optimization problem.Generally speaking, the same Boolean function has different continuity functions. When the combinatorial optimization problem is transformed into the nonlinear continuous optimization one, these different functions have a great influence on ability to collect information and property of solutions of the nonlinear programming problem. We all know when the number of the levels of SP network iteration has increased, properties of object function tend to complicate, the number of local extreme value points have an exponential increase and global maximums weaken quickly. There are some requests to the continuity method of Boolean function. So we need the continuity method of Boolean function that has a strong ability to derive information.To ability in deriving information, we wonder if Boolean discrete optimal continuity function exists. Is it unique if it exists?First, we give a general definition of Boolean continuity function. Second, we analyze and prove existence and uniqueness of the optimal continuity function and give some properties of it in the article. Finally, we give the definition of relative loss of entropy of Boolean continuity function, point out the relationship between the loss of entropy and Kullback Leibler distance, and we give the necessary and sufficient condition that Boolean continuity function is optimal.We can directly extend the conclusions to continuity analysis of general discrete problem.