2D Spatial Distributions for Measures of Random Sequences Using Conjugate Maps

Advanced visual tools are useful to provide additional information for modern information warfare. 2D spatial distributions of random sequences play an important role to understand properties of complex sequences. This paper proposes time-sequences from a given logical function of 1D Cellular Automata in both Poincare map and conjugate map. Multiple measure sequences of Markov chains can be used to display spatial distributions using conjugate maps. Measure sequences recursively produced by different logical functions generating maps. Possible complementary feature exits between pair functions, Conjugate symmetry relationships between a pair of logical functions in conjugate maps can be observed.


INTRODUCTION
Random sequences are widely used in many security based applications such as security communication, cryptology coding and information security systems (Key, 1976). To make proper analysis, Markov chain methodologies and technologies provide a series of important methods and tools to help analysers decoding process (James & Michael, 1972;Haccoun, 1980;Sheynin, 1989). In modern information warfare, it is essential for analysers to detect and decrypt the opponent's communications using information acquisition toolkits from real coding sequences (Widnall & Fogelman, 1997).
Information Warfare describes terms of 'actions' executed to achieve a sought outcome -denial, exploitation, corruption and destruction of an opponent's 'information' and related functions, and prevention of such 'actions' executed by an opponent (Borden, 1999).
The battle between the obscurers and those who sought to break the codes has been a continual one, but it reached a new level of stature and importance during World War II with its decryption of Germany's Enigma messages. Historic events are approved that statistical and probability tools are extremely important in Information Warfare applications. This battle of wits fought by British mathematicians and statisticians shortened World War II and ushered in the age of information warfare (Budiansky & Stephen, 2000).
Prerequisite of executing these attack actions is thorough understood the mechanism of information encryption that opponent uses (Carlo & Bruce, 2002). In information warfare, secured-communications among opposite parts may use public networks. It is feasible to capture relevant information for further analysis. Different quantitative tools and methods are useful to provide additional information in decoding process. Variant features play an important role for measurement and analysis of random sequences (Denning, 1999).
Because of the implicated expression of functions that generate random sequences, it is hard to get the characteristic of random sequences from the function and coding sequences themselves (Li & Tian, 2006). Traditionally, time sequence Map and Poincare Map are two most popular methods to take the measure features of a random sequence in 2 dimensions (Graeme & Matthew, 2009). From a visual viewpoint, current Markov chain schemes do not provide efficient visual mechanism to display multiple measurement sequences from the spatial characteristic of complex random sequences.
To extract further information from random sequences, this paper establishes a visual system to illustrate multiparameter measurement sequences of Markov chains as conjugate maps. For a given set of measurement sequences, the conjugate map proposed in this paper can provide refined information of distributed structure than present map technologies (Li & Zheng, 2010).
In the second section, respective characteristics of traditional methods and conjugate method are discussed. The measurement mechanism of logical function's spatial characteristics: disposal model, measuring model, and visualizing model are described in the third section. The results of maps and analysis of the results are discussed in the fourth and fifth sections, and then, conclusion remarks are provided in the last section.

TRADITIONAL METHODS AND CONJUGATE METHOD
In this section, two typically traditional methods: Time Sequence Map and Poincare Map are discussed for comparison.
Time Sequence Map generates a 2D coordinate, X axis is determined by the time scale t and Y axis is determined by the value of measured parameter ( ) f t , as shown in Figure 1(a).
The measure sequence f t with length of T can form Poincare Map according to the matching pattern considering data correlation. Poincare method maps one group of measures of time sequence to a 2D map. It detects spatial distribution of sequence through the distribution of point cluster. In Poincare Map, X axis is determined by the value of ( ) f t while Y is ( ) f t l . It's vicinity related patterns map when 1 l , as shown in Figure 1(b). Different from Poincare method based on one group of measures, new map proposed in this paper chooses 2 groups of measures from relevant parallel measures sequences. As two different groups of measures are acted simultaneously, the value of each axis is determined by these two groups of measurements. It is convenient to name new map as Conjugate Map to present this kind of multiple parameter measurement map.

GENERATE AND MEASURE MECHANISM OF TIME SEQUENCE
In this section, the CA (Cellular Automata) method is applied to generate time sequence and then to make concomitant measurement sequence. First, the initial sequence inputted, and the output sequence is generated by a given logical function using 1D cellular automata; Using this data sequence, measurements are formed by probability measurement according to pairs of Input and Output sequences; Finally, the generated measure sequences can be used to construct a 2D conjugate map showing 2D spatial distribution of the time sequence. The processing flow of the mechanism is shown in Figure 2.  Table 1. A total of 2 N states of N-length initial input sequence are exhaustively generated and the corresponding sequence under the logical function : f X Y o can be generated. The input and the output sequences are in the same group corresponded to each other; there are 2 N groups of corresponding relationship (Wolfram, 1986). Exhaustion of all the initial input sequences is shown in Table 2.

Measure Model
The basic model of measurement can be confirmed to establish the transformation relation between the input sequence In the transformation of : there are a total of 4 types of transformations, each type determines a number, and corresponding relationships are shown in Table 3. This type of measurement structure has a directly corresponding relationship to the Markov chain mechanism (Sheynin, 1989).
as the serial number of different initial input sequence. There are 4 measurements that can be identified by the measurement parameters above shown in Table 4 with Markov chain properties respectively.

Visualization Model
Based on the probability measurements presented above, two measurements are chosen to construct 2D map. As two different groups of measurements are used simultaneously, to name this kind of map conjugate map, of which the value of each axis is determined by these two groups of measurements.
According to the construction pattern introduced above, there are 2 On the same group of sequences, construct 2D conjugate maps respectively by using the combinations above as shown in Figure 3. This paper chooses the typical combination 01 10 { ( ), ( )} P j P j constructing 2D conjugate map to detect the special distribution of time sequences for N=13 condition.

VISUALIZATION RESULT
Because of the restriction of the structural complexity of the logical function, 16 functions of 2 variables are used to describe them in the way of exhaustion (Wan & Zheng, 2010). Output sequences are generated by different initial input sequences under the given logical function and then obtaining various measure data from the corresponding I/O sequence based on probability method. Then the map is constructed using these measurement data.
This paper choose No. 1, 5, 6 and 13 functions which are typical functions as an example, observing the characteristic of three kinds of maps which are given in Figure 4. According to the variable-value logic theory, 3 kinds of encoding model can be distinguished: W, F and C .
The visualization information that can be acquired from a single function's map is rather limited. In order to compare the spatial property of different logical function, an 4 4 u array is constructed using the maps that are generated from 16 logical functions in different encoding pattern as shown in Figure 5. To compare conveniently, combinations of 16 recursive images which generated from 16 functions are given in this paper under different codes. Recursive images in W-code, F-code and C-code from a given initial sequence are shown in Figure 6-Figure 8 respectively.    Under C-Code structure, Poincare maps and conjugate maps are shown in Figure 14 and Figure 15.
In the above maps, 2D conjugate maps not only show spatial distributions of different logical functions, but also have special holistic symmetries under the F and C code conditions.

ANALYZE
Through three types of different maps, three different coding schemes can be observed.
Time sequence map can show the simple trend of single measurement series with time variations, but it has difficult for the scheme to describe spatial distributions of time sequence.
Poincare map can apply a single measurement sequence, although the map can be generated under different length in a correlation, information of distribution is naturally limited by the selected measurement sequence.
A 2D conjugate map uses 2 groups of independent measurements simultaneously; this scheme can show differences and connections between spatial distributions of logical functions, furthermore, through different coding model, it can illustrate holistic relationships among different functions. i.e. function pairs of 0:15, 1:7, 2:11, 4:13 and 8:14 have clear conjugated symmetry in conjugate maps. In addition, for C code condition, the points of 4 functions on each edge of maps are located on the same side of edge. For example, points clusters of (0, 4, 1, 5), (0, 2, 8, 10), (10, 14, 11, 15) and (5, 7, 13, 15) functions are separately located on four side of the 2D map space.

CONCLUSION
Refined property of various time sequences can be identified from 2D conjugate maps to illustrate multiple measurement sequences under Markov chain mechanism. Spatial property of time sequence plays an important role in the study of dynamic sequence's behaviour. The stable distribution under visualization method can help people understand relevant issues.
In compared with Poincare maps and conjugate maps, there are additional properties in the complex dynamic sequences. Conjugate map method uses multiple parameters of Markov chains to make independent measurements simultaneously.
Proposed technology can provide further structural information among multiple measurements, refined relationship via spatial distributions can be established. It is possible for the scheme to use statistical and probability methodologies to enhance visual tools of Markov chain mechanisms to resolve real problems and requirements for modern information warfare and information security applications in near future.