Variant Map System of Random Sequences

Abstract

Sequences of random variables play a key role in probability theory, stochastic processes, and statistics to analyze dynamic behavior. Speckle patterns have emerged as useful tools to explore space–time variations of random sequences in various measurement applications of comprehensive properties in complex space–time variation events. In this chapter, a variant map system is proposed to analyze statistical properties of random sequences in visual representations. An input 0–1 sequence will be divided into multiple segments and each segment of a fixed length will be transformed into a 2-tuple pair of measures. Five measuring sets are identified and rearranged in a 1D or 2D numerical array as a histogram representing a visual map. These five types of maps consist of two types in 1D format as classical maps and three types in 2D format as variant maps. Properties are analyzed on all five types of maps. A cryptographic sequence of the AES cipher is selected as a sample stream. The five types of visual maps are generated and refined clustering characteristics are organized into four groups on changes of segmented and shifted lengths for visual comparisons on enlarged 2DP maps. Speckle patterns of various distributions are observed. Three variant maps with distinct statistic distributions could be useful to provide new visual tools to explore comprehensive cryptographic sequences on complex nonlinear dynamic behavior in global network environments.