A Method to Determine the Most Suitable Initial Conditions of Chaotic Map in Statistical Randomness Applications

Abstract

The processes and systems in the real world actually contain order and symmetry. Understanding these order and symmetrical behavior has been a common effort of scientists. Chaos theory has been an interesting topic to understand these order and symmetrical behavior. One of the most important reasons for this interest is the rich dynamics of chaotic systems. A remarkable application of these systems is statistical randomness. Especially in random number generator designs based on discrete time chaotic systems, an important design parameter affecting the success of the generator (statistical randomness properties) is the initial conditions of chaotic maps. Obtaining different initial conditions that will meet the statistical requirements is important in terms of generating different random number sequences. In order to determine the initial conditions, an algorithm that updates the initial conditions depending on the number of successful statistical tests is proposed. Although the proposed design approach is similar to a back propagation neural network, it has a unique design approach. The NIST SP 800-22 test suite has been used to analyze the statistical properties of the proposed generator. It is known that the NIST SP 800-22 test suite is a hypothesis test. Therefore, in order to show the success of the proposed method in the best way, various additional analysis studies have been carried out proving that the generator outputs have a uniform distribution. Using the proposed method, six different initial conditions have been determined that provide statistical random properties for the discrete-time chaotic systems known as logistic map and tent map. 1,000,000 bits have been generated using the obtained initial conditions. These bit values are then converted to decimal values between 0-15. It is observed that the obtained numbers have a uniform distribution. These outputs are thought to be applicable in many areas such as games, simulation, modeling, determination of optimization parameters and cryptology. It is shown in a practical application for cryptographic substitution-box designs to examine the success of the outputs.