Experiment Results of Key Space Testing of Robust Hyper-Chaotic System

Experiment Results of Key Space Testing of

Robust Hyper-Chaotic System

Experiment Results of Key Space Testing of

Robust Hyper-Chaotic System

Content:

I. Introduction…………..………... 2

II.

System with 2 Coupled Maps (n=2)….……….……... 3

III. System with 3 Coupled Maps (n=3)……… 7

IV. System with 4 Coupled Maps (n=4)….……….……... 11

V

System with 10 Coupled Maps (n=10)……….……... 15

To test the randomness of the output sequence of Robust Hyper-Chaotic System (RHCS), the

toolbox [1] is used in analysis process. The masking sequence of the RHCS is generated with each

own parameters. Since, the selections of parameters of RHCS are growing with the precision of the

system and the number of coupled maps. To reduce the experiment time, the system precision is set

to 48-bits, and the various parameter “r1” will be tested in each experiment configuration. For each

“r1”, 40 sequences of 10

6

bits are generated. Each test of the sequence will produce a “p-values”

from SP800-22 testing package. The higher p-value (a minimal default value is recommended by

0.01), the more random the test case. As suggested by SP800-22, for each statistical test item, the

minimum pass rate of a well random source is 0.9428 out of 40 sequences.

II. System with 2 Coupled Maps (n=2)

 Case 1

System Configuration:

Precision (m) = 12

Length (bits) = 48

Parameters:

r

1

=

20.ab56cd56

r

2

=

15.102fe487

c

11

=

0.e93455000000

c

12

=

0.16cbab000000

c

21

=

0.325438000000

c

22

=

0.cdabc8000000

Initial Values:

x

1

=

0.123456789abc

x

2

=

0.cda987654321

Scan Target:

r

1

Range:

20.ab56cd56 ~ 1af.ab56cd56

Average 1.0000 1.0000 1.0000 0.9625 1.0000 1.0000 0.9875 0.9750 0.9750 0.9875 1.0000 0.9625 1.0000 1.0000 1.0000 1.0000 0.9750 0.9875 0.9625 0.9875 0.9881 Serial 1.0000 1.0000 1.0000 0.9625 1.0000 1.0000 0.9875 0.9750 0.9750 0.9875 1.0000 0.9625 1.0000 1.0000 1.0000 1.0000 0.9750 0.9875 0.9625 0.9875 0.9881 L.Z. 0.9750 0.9750 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 0.9750 0.9500 0.9875 Rand om Excu-Va riant 0.9889 0.9960 0.9911 0.9847 0.9850 0.9976 0.9911 0.9957 0.9872 0.9924 1.0000 0.9879 0.9784 0.9921 0.9662 0.9952 0.9931 0.9949 0.9976 1.0000 0.9908 Rand om Excu. _{0.9917 0.9955 0.9950 1.0000 0.9856 0.9946 0.9800 0.9760 0.9952 0.9943 0.9943 0.9891 1.0000 0.9941 0.9783 0.9891 0.9896 0.9943 0.9891 0.9822 0.9904} Apen _{1.0000 1.0000 0.9500 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9500 0.9750 0.9900} Univ-er s al 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9500 1.0000 1.0000 1.0000 0.9750 0.9913 Over lappi ng Tem p . 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 0.9750 0.9750 0.9750 1.0000 1.0000 0.9925 Non periodi c Tem p . 0.9910 0.9875 0.9892 0.9907 0.9890 0.9890 0.9905 0.9905 0.9900 0.9907 0.9882 0.9899 0.9912 0.9902 0.9894 0.9921 0.9873 0.9899 0.9916 0.9889 0.9898 S.DFT 0.9750 0.9750 1.0000 1.0000 0.9500 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 0.9750 0.9500 0.9750 0.9750 1.0000 1.0000 0.9850 Rank _{1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9500 0.9750 0.9750 1.0000 0.9913} Lon gest Run 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 0.9938 Runs _{0.9250 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 0.9500 1.0000 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9863} Cum u . Sums 1.0000 1.0000 0.9375 0.9625 0.9875 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9625 1.0000 0.9750 0.9750 1.0000 0.9750 0.9750 1.0000 0.9850 Block freq. 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 0.9750 0.9950 Freq. _{1.0000 1.0000 0.9500 0.9500 0.9750 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 0.9750 0.9750 1.0000 0.9863}

Case 1

Scan Value 00 20.AB5 6CD56 00 35.AB5 6CD56 00 4A.AB5 6CD56 00 5F.AB5 6CD56 00 74.AB5 6CD56 00 89.AB5 6CD56 00 9E.AB5 6CD56 00 B3.AB5 6CD56 00 C8.AB5 6CD56 00 DD.AB5 6CD56 00 F2.AB5 6CD56 01 07.AB5 6CD56 01 1C.AB5 6CD56 01 31.AB5 6CD56 01 46.AB5 6CD56 01 5B.AB5 6CD56 01 70.AB5 6CD56 01 85.AB5 6CD56 01 9A.AB5 6CD56 01 AF.AB5 6CD56 Average

 Case 2

System Configuration:

Precision (m) = 12

Length (bits) = 48

Parameters:

r

1

=

0F.0141712721

r

2

=

25.ABD43143C1

c

11

= 0. EE6CD1607E20

c

12

= 0. 11932E9F81E0

c

21

= 0. 2C366F84797E

c

22

= 0. D3C9907B8682

Initial Values:

x

1

= 0. 778F25E2FF4B

x

2

= 0. 9EDEC745E89E

Scan Target:

r

1

Range:

0F.0141712721 ~ 28.2F799F135D

Average 1.0000 1.0000 0.9750 1.0000 0.9875 0.9750 0.9750 0.9875 1.0000 0.9875 0.9875 0.9625 0.9500 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 0.9868 Serial 1.0000 1.0000 0.9750 1.0000 0.9875 0.9750 0.9750 0.9875 1.0000 0.9875 0.9875 0.9625 0.9500 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 0.9868 L.Z. 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 0.9500 0.9750 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 0.9500 0.9889 Rand om Excu-Var iant 0.9778 0.9841 0.9829 0.9967 0.9874 0.9883 1.0000 0.9942 0.9806 0.9757 1.0000 1.0000 0.9907 0.9946 0.9899 1.0000 0.9903 0.9931 1.0000 0.9868 0.9903 Rand om Excu. 0.9917 0.9866 1.0000 0.9927 1.0000 0.9737 1.0000 0.9934 0.9875 0.9766 0.9943 1.0000 0.9861 0.9960 0.9716 0.9891 0.9946 1.0000 0.9750 1.0000 0.9908 Apen _{1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9500 1.0000 0.9500 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9889} Univ-er s al 0.9750 0.9500 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9500 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9750 0.9875 Over lappi ng Tem p . 0.9500 1.0000 1.0000 0.9750 1.0000 0.9750 0.9750 1.0000 0.9750 0.9750 1.0000 0.9500 1.0000 0.9500 1.0000 0.9750 1.0000 0.9750 0.9750 0.9750 0.9819 Non periodi c Tem p . 0.9912 0.9890 0.9909 0.9865 0.9887 0.9912 0.9894 0.9905 0.9919 0.9897 0.9885 0.9877 0.9904 0.9917 0.9878 0.9922 0.9887 0.9883 0.9873 0.9900 0.9897 S.DFT 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9500 0.9500 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 0.9750 0.9500 1.0000 0.9750 0.9750 0.9875 Rank _{1.0000 1.0000 1.0000 0.9500 0.9750 1.0000 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 0.9889} Lon gest Run _{1.0000 1.0000 0.9750 0.9750 1.0000 0.9750 0.9750 0.9500 0.9750 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9500 0.9500 1.0000 1.0000 0.9819} Runs _{0.9500 1.0000 0.9750 0.9750 0.9750 1.0000 0.8750 0.9250 1.0000 0.9500 1.0000 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9500 1.0000 0.9764} Cum u . Sums _{0.8750 0.9500 0.8250 0.7625 1.0000 0.9000 0.7625 0.9625 0.9500 0.6750 0.9500 0.9750 0.9125 1.0000 0.9750 0.9750 1.0000 0.9125 0.8875 0.9500 0.9090} Block freq. 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 0.9500 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 0.9875 Freq. _{0.8250 0.9250 0.8000 0.7000 0.9750 0.8750 0.7250 0.9750 0.9250 0.6750 0.9250 0.9750 0.9250 0.9750 0.9750 1.0000 1.0000 0.9000 0.8750 0.9500 0.8931}

Case 2

Scan Value 0F .01417 12721 10 .5487C 46975 11 .A7CE1 7ABC9 12 .FB146 AEE1D 14 .4E5AB E3071 15 .A1A11 172C5 16 .F4E76 4B519 18 .482DB 7F76D 19 .9B740 B39C1 1A .EEBA5 E7C15 1C .4200B 1BE69 1D .95470 500BD 1E .E88D5 84311 20 .3BD3A B8565 21 .8F19F EC7B9 22 .E2605 20A0D 24 .35A6A 54C61 25 .88ECF 88EB5 26 .DC334 BD109 28 .2F799 F135D Average

III. System with 3 Coupled Maps (n=3)

 Case 3

System Configuration:

Precision (m) = 12

Length (bits) = 48

Parameters:

r

1

=

10.f234ad23

r

2

=

12.2467920c

r

3

=

55.4098bcef

c

11

= 0. 236400000000

c

12

=

0.120100000000

c

13

= 0. ca9b00000000

c

21

= 0. c2e400000000

c

21

=

0.147000000000

c

23

= 0. 28ac00000000

c

31

= 0. 134100000000

c

22

=

0.947b00000000

c

33

= 0. 584400000000

Initial Values:

x

1

= 0. 123456789abc

x

2

= 0. cda987654321

x

3

= 0. 42143463df24

Scan Target:

r

1

Range:

0010.f234ad23 ~ 0153.f234ad23

Average 0.9875 0.9875 1.0000 0.9500 0.9875 0.9750 1.0000 1.0000 1.0000 0.9875 1.0000 0.9750 0.9750 0.9875 1.0000 1.0000 1.0000 1.0000 1.0000 0.9875 0.9900 Serial 0.9875 0.9875 1.0000 0.9500 0.9875 0.9750 1.0000 1.0000 1.0000 0.9875 1.0000 0.9750 0.9750 0.9875 1.0000 1.0000 1.0000 1.0000 1.0000 0.9875 0.9900 L.Z. 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 0.9750 1.0000 0.9913 Rand om Excu-Va riant 0.9876 0.9822 1.0000 0.9848 1.0000 0.9701 0.9889 0.9911 1.0000 0.9944 0.9734 0.9943 0.9903 1.0000 0.9928 0.9911 1.0000 0.9866 0.9978 0.9978 0.9912 Rand om Excu. 1.0000 0.9911 0.9886 0.9943 0.9948 0.9904 0.9700 1.0000 0.9800 0.9958 0.9511 0.9957 0.9891 0.9856 0.9783 0.9800 0.9891 0.9957 1.0000 1.0000 0.9885 Apen _{0.9000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9500 0.9750 1.0000 0.9750 1.0000 1.0000 0.9750 0.9500 0.9850} Univ-er s al 1.0000 1.0000 0.9750 1.0000 0.9500 0.9500 1.0000 0.9500 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 0.9863 Over lappi ng Tem p . 1.0000 0.9750 0.9750 0.9750 1.0000 0.9500 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 0.9750 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 0.9875 Non periodi c Tem p . 0.9899 0.9895 0.9897 0.9894 0.9921 0.9877 0.9873 0.9916 0.9914 0.9885 0.9877 0.9868 0.9909 0.9912 0.9887 0.9883 0.9902 0.9889 0.9936 0.9912 0.9897 S.DFT 0.9750 0.9500 1.0000 1.0000 0.9750 0.9750 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9750 1.0000 0.9750 1.0000 0.9750 0.9750 0.9838 Rank _{1.0000 1.0000 1.0000 1.0000 1.0000 0.9500 0.9750 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 0.9900} Lon gest Run 1.0000 0.9750 0.9500 1.0000 1.0000 0.9750 0.9750 0.9750 0.9500 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9875 Runs _{1.0000 1.0000 1.0000 0.9750 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9925} Cum u . Sums 1.0000 0.9750 0.9750 0.9750 1.0000 1.0000 1.0000 0.9750 0.9875 1.0000 1.0000 0.9750 1.0000 1.0000 0.9875 1.0000 1.0000 0.9875 0.9750 1.0000 0.9906 Block freq. 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9975 Freq. _{1.0000 0.9750 0.9750 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9938}

Case 3

Scan Value 00 10.F23 4AD23 00 21.F23 4AD23 00 32.F23 4AD23 00 43.F23 4AD23 00 54.F23 4AD23 00 65.F23 4AD23 00 76.F23 4AD23 00 87.F23 4AD23 00 98.F23 4AD23 00 A9.F23 4AD23 00 BA.F23 4AD23 00 CB.F23 4AD23 00 DC.F23 4AD23 00 ED.F23 4AD23 00 FE.F23 4AD23 01 0F.F23 4AD23 01 20.F23 4AD23 01 31.F23 4AD23 01 42.F23 4AD23 01 53.F23 4AD23 Average

 Case 4

System Configuration:

Precision (m) = 12

Length (bits) = 48

Parameters:

r

1

=

34.3b620cecA0

r

2

=

51.246745920c

r

3

=

a6.4098bc23ef

c

11

= 0. 2AB8903D8652 c

12

= 0. 5541497F2083 c

13

= 0. 80062643592B

c

21

= 0.7868B9793908 c

21

= 0. 4390381F4386 c

23

= 0. 44070E678372

c

31

= 0.B727A0783234 c

22

= 0. 118926208464

c

33

= 0. 374F39674968

Initial Values:

x

1

= 0. 123456789abc

x

2

= 0. cda987654321

x

3

= 0. 42143463df24

Scan Target:

r

1

Range:

34.3B620CECA0 ~ 88.A4A0E52C1D

Average 1.0000 0.9750 0.9750 1.0000 1.0000 0.9875 0.9750 0.9750 1.0000 0.9750 0.9750 0.9875 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 0.9875 Serial 1.0000 0.9750 0.9750 1.0000 1.0000 0.9875 0.9750 0.9750 1.0000 0.9750 0.9750 0.9875 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 0.9875 L.Z. 0.9750 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 0.9500 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9903 Rand om Excu-Var iant 0.9815 1.0000 0.9962 0.9756 1.0000 0.9823 0.9850 0.9838 0.9856 0.9893 0.9918 0.9907 0.9874 1.0000 0.9944 1.0000 0.9856 0.9833 0.9896 Rand om Excu. 0.9896 0.9931 0.9871 0.9600 1.0000 0.9943 0.9808 0.9896 0.9676 0.9952 0.9861 0.9948 1.0000 0.9938 0.9938 0.9938 1.0000 0.9938 0.9896 Apen _{0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9944} Univ-er s al 0.9750 1.0000 0.9750 0.9500 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 0.9750 0.9750 0.9500 1.0000 1.0000 0.9847 Over lappi ng Tem p . 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9944 Non periodi c Tem p . 0.9905 0.9931 0.9894 0.9905 0.9885 0.9890 0.9892 0.9895 0.9899 0.9895 0.9873 0.9914 0.9887 0.9885 0.9904 0.9916 0.9902 0.9897 0.9898 S.DFT 0.9750 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 0.9750 0.9917 Rank _{1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9750 0.9500 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9500 0.9750 0.9861} Lon gest Run 1.0000 1.0000 0.9750 1.0000 1.0000 0.9500 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 0.9500 1.0000 0.9750 1.0000 1.0000 1.0000 0.9889 Runs _{0.9750 1.0000 0.9750 0.9500 1.0000 0.9750 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 0.9875} Cum u . Sums 1.0000 0.9625 0.9750 1.0000 0.9750 0.9875 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 0.9375 0.9875 1.0000 1.0000 0.9500 0.9847 Block freq. 1.0000 1.0000 1.0000 0.9750 0.9750 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9944 Freq. _{1.0000 0.9750 0.9750 1.0000 0.9750 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9500 0.9750 0.9750 1.0000 0.9750 0.9861}

Case 4

Scan Value 34 .3B620 CECA0 38 .ACB63 340CF 3D .1E0A5 994FE 41 .8F5E7 FE92D 46 .00B2A 63D5C 4A .7206C C918B 4E .E35AF 2E5BA 53 .54AF1 939E9 57 .C6033 F8E18 5C .37576 5E247 60 .A8AB8 C3676 65 .19FFB 28AA5 69 .8B53D 8DED4 6D .FCA7F F3303 72 .6DFC2 58732 76 .DF504 BDB61 7B .50A47 22F90 7F .C1F89 883BF Average

IV. System with 4 Coupled Maps (n=4)

 Case 5

System Configuration:

Precision (m) = 12

Length (bits) = 48

Parameters:

r

1

= 0010. 52564735

r

2

= 00c1. 87692765

r

3

= 0022. ad685943

r

4

= 00e3. 86fd7678

c

11

=

0.236400000000

c

12

= 0.120100000000

c

13

=

0.944700000000

c

14

=

0.365400000000

c

21

=

0.2a7400000000

c

22

= 0.147000000000

c

23

=

0.28ac00000000

c

24

=

0.987000000000

c

31

=

0.134100000000

c

32

=

0.947b00000000

c

33

=

0.473300000000

c

34

=

0.111100000000

c

41

=

0.234500000000

c

42

=

0.a23400000000

c

43

=

0.098400000000

c

44

=

0.310300000000

Initial Values:

x

1

= 0. f4dc982e6c51

x

2

= 0. 123456789abc

x

3

= 0. cda987654321

x

4

= 0. 42143463df24

Scan Target:

r

1

Range:

0010.52564735 ~ 02cf.52564735

Average 0.9875 1.0000 1.0000 0.9750 1.0000 1.0000 0.9875 0.9875 0.9500 1.0000 1.0000 1.0000 1.0000 0.9875 0.9750 1.0000 0.9875 1.0000 1.0000 0.9625 0.9910 Serial 0.9875 1.0000 1.0000 0.9750 1.0000 1.0000 0.9875 0.9875 0.9500 1.0000 1.0000 1.0000 1.0000 0.9875 0.9750 1.0000 0.9875 1.0000 1.0000 0.9625 0.9910 L.Z. 0.9750 1.0000 1.0000 0.9250 1.0000 0.9500 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 0.9750 0.9750 0.9750 0.9861 Rand om Excu-Va riant 0.9957 0.9924 0.9959 0.9773 0.9921 0.9933 0.9903 1.0000 0.9976 1.0000 1.0000 0.9979 0.9884 1.0000 1.0000 0.9861 1.0000 1.0000 0.9952 1.0000 0.9948 Rand om Excu. _{0.9712 0.9886 0.9861 0.9886 0.9822 0.9950 0.9783 0.9908 1.0000 0.9946 0.9875 0.9904 0.9844 1.0000 0.9773 1.0000 0.9850 1.0000 1.0000 0.9957 0.9889} Apen _{0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9944} Univ-er s al 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9000 0.9750 1.0000 0.9500 0.9500 1.0000 0.9861 Over lappi ng Tem p . 0.9750 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9917 Non periodi c Tem p . 0.9889 0.9865 0.9890 0.9910 0.9922 0.9900 0.9914 0.9897 0.9921 0.9899 0.9892 0.9883 0.9902 0.9890 0.9919 0.9914 0.9904 0.9880 0.9897 0.9907 0.9899 S.DFT 0.9500 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9500 0.9500 1.0000 1.0000 0.9750 0.9250 0.9750 0.9889 Rank _{1.0000 0.9750 0.9750 1.0000 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9944} Lon gest Run _{1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9500 0.9903} Runs _{0.9500 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9903} Cum u . Sums 0.9750 1.0000 0.9625 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9875 1.0000 0.9750 1.0000 0.9750 0.9375 1.0000 0.9625 0.9875 0.9625 1.0000 0.9854 Block freq. 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9944 Freq. _{0.9500 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 0.9500 1.0000 0.9750 0.9750 0.9750 1.0000 0.9861}

Case 5

Scan Value 10.52564 735 35 .52564 735 00 5A.525 64735 00 7F.525 64735 00 A4.525 64735 00 C9.525 64735 00 EE.525 64735 11 3.5256 474 13 8.5256 474 01 5D.525 64735 18 2.5256 474 01 A7.525 64735 01 CC.525 64735 01 F1.525 64735 21 6.5256 474 02 3B.525 64735 26 0.5256 474 28 5.5256 474 02 AA.525 64735 02 CF.525 64735 Average

 Case 6

System Configuration:

Precision (m) = 12

Length (bits) = 48

Parameters:

r

1

= 0010. 52564735

r

2

= 0043. 87692765

r

3

= 0022. ad685943

r

4

= 0011. 86fd7678

c

11

=

0.236400000000

c

12

= 0.120100000000

c

13

=

0.944700000000

c

14

=

0.365400000000

c

21

=

0.2a7400000000

c

22

= 0.147000000000

c

23

=

0.28ac00000000

c

24

=

0.987000000000

c

31

=

0.134100000000

c

32

=

0.947b00000000

c

33

=

0.473300000000

c

34

=

0.111100000000

c

41

=

0.234500000000

c

42

=

0.a23400000000

c

43

=

0.098400000000

c

44

=

0.310300000000

Initial Values:

x

1

= 0. f4dc982e6c51

x

2

= 0. 123456789abc

x

3

= 0. cda987654321

x

4

= 0. 42143463df24

Scan Target:

r

1

Range:

0010.52564735 ~ 01dd.5d844397

Average 1.0000 0.9625 1.0000 0.9875 0.9750 0.9750 0.9875 0.9875 1.0000 0.9750 1.0000 0.9625 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9882 Serial 1.0000 0.9625 1.0000 0.9875 0.9750 0.9750 0.9875 0.9875 1.0000 0.9750 1.0000 0.9625 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9882 L.Z. 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 0.9944 Rand om Excu-Vari ant 0.9792 0.9874 0.9745 0.9960 0.9960 0.9978 0.9850 0.9847 0.9855 0.9979 0.9959 0.9975 0.9782 0.9959 0.9936 1.0000 0.9978 1.0000 0.9844 0.9928 0.9913 Rand om Excu. 0.9844 1.0000 0.9896 0.9866 0.9866 0.9950 0.9856 0.9914 1.0000 0.9676 0.9861 0.9943 0.9911 0.9769 0.9952 0.9950 0.9950 0.9917 0.9950 0.9837 0.9896 Apen _{1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9986} Univ-er sal 0.9750 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 0.9750 1.0000 0.9917 Over lappi ng Tem p . 0.9750 0.9750 1.0000 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 0.9903 Non periodi c Tem p . 0.9870 0.9919 0.9883 0.9897 0.9900 0.9919 0.9889 0.9904 0.9924 0.9894 0.9877 0.9889 0.9885 0.9900 0.9922 0.9916 0.9882 0.9916 0.9882 0.9929 0.9899 S.DFT 0.9500 1.0000 1.0000 1.0000 1.0000 0.9250 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9500 0.9750 1.0000 0.9750 0.9750 0.9750 0.9750 0.9750 0.9833 Rank _{0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 0.9750 0.9931} Lon gest Run 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9750 1.0000 1.0000 0.9500 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9903 Runs _{0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9931} Cum u . Sums 1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 0.9875 1.0000 1.0000 1.0000 1.0000 0.9875 0.9875 0.9875 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9944 Block freq. 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 1.0000 0.9750 0.9500 1.0000 1.0000 1.0000 0.9917 Freq. _{1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9944}

Case 6

Scan Value 10.52564 735 00 28.964 B398B 00 40.DA4 02BE1 00 59.1E3 51E37 00 71.622 A108D 00 89.A61 F02E3 00 A1.EA1 3F539 00 BA.2E0 8E78F 00 D2.71F DD9E5 00 EA.B5F 2CC3B 01 02.F9E 7BE91 01 1B.3DD CB0E7 01 33.81D 1A33D 01 4B.C5C 69593 01 64.09B B87E9 01 7C.4DB 07A3F 01 94.91A 56C95 01 AC.D59 A5EEB 01 C5.198 F5141 01 DD.5D8 44397 Average

V. System with 10 Coupled Maps (n=10)

 Case 7

System Configuration:

Precision (m) = 12

Length (bits) = 48

Parameters:

r

1

= 3F.26B3575797

r

2

= 11.23434CD1F0

r

3

= 10.25CA67FD2E

r

4

= 15.398CDBAEF5 r

5

= 1E.F9478E2945

r

6

= 26.398ED0A20F

r

7

= 2F.10171B1702

r

8

= 3D.3C26280529

r

7

= 43.0F18301E03

r

10

= 57.1523141712

(Note, parameter c are not shown here)

Initial Values:

x

1

= 0.1B0335031128

x

2

= 0.0111132F180A x

3

= 0.2B042E34180C

x

4

= 0.09081B081A28 x

5

= 0.180C10251C33 x

6

= 0.10171B17022B

x

7

= 0.06201F331F05

x

8

= 0.03132C07112B x

9

= 0.1B270408181A

x

10

= 0.1A1017161322

Scan Target:

r

1

Range:

05.2E152C2A1D ~ AA.CFD9153E55

Average 1.0000 0.9875 1.0000 0.9625 1.0000 1.0000 1.0000 0.9875 0.9875 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 0.9750 0.9750 0.9750 0.9625 0.9875 0.9889 Serial 1.0000 0.9875 1.0000 0.9625 1.0000 1.0000 1.0000 0.9875 0.9875 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 0.9750 0.9750 0.9750 0.9625 0.9875 0.9889 L.Z. 1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 0.9750 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 0.9903 Rand om Excu-Va riant 0.9935 0.9899 0.9883 1.0000 0.9899 0.9967 1.0000 1.0000 0.9972 0.9974 1.0000 0.9784 0.9889 1.0000 1.0000 0.9854 1.0000 1.0000 0.9942 1.0000 0.9948 Rand o m Excu. 0.9853 0.9886 0.9934 0.9904 0.9886 0.9853 1.0000 0.9822 1.0000 0.9881 0.9943 0.9861 1.0000 1.0000 0.9943 1.0000 0.9922 0.9927 0.9737 1.0000 0.9923 Apen _{1.0000 0.9750 1.0000 0.9750 1.0000 0.9750 0.9750 0.9500 0.9250 1.0000 1.0000 1.0000 0.9500 0.9500 0.9750 1.0000 0.9750 1.0000 0.9500 1.0000 0.9792} Univ-er s al 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 0.9750 0.9750 0.9750 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 0.9875 Over lappi ng Tem p . 1.0000 1.0000 0.9500 0.9750 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 0.9500 1.0000 0.9750 0.9750 1.0000 1.0000 0.9861 Non periodi c Tem p . 0.9889 0.9892 0.9878 0.9900 0.9887 0.9880 0.9894 0.9895 0.9887 0.9887 0.9900 0.9905 0.9921 0.9882 0.9914 0.9902 0.9900 0.9914 0.9894 0.9895 0.9896 S.DFT 0.9750 1.0000 0.9750 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9917 Rank _{0.9750 0.9750 0.9750 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 1.0000 0.9750 1.0000 0.9750 0.9889} Lon ges t Ru n 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 0.9750 0.9750 0.9500 1.0000 0.9500 1.0000 1.0000 0.9500 1.0000 1.0000 1.0000 1.0000 1.0000 0.9875 Runs _{0.9250 0.9500 0.9250 0.9750 0.9500 0.9500 0.9500 0.9750 0.9250 0.9250 1.0000 1.0000 1.0000 1.0000 0.9750 0.9500 0.9750 0.9750 0.9750 0.9750 0.9625} Cum u . Sums 0.7125 0.6375 0.7625 0.6375 0.6625 0.8000 0.8125 0.7500 0.7875 0.6625 0.9750 0.9625 0.9625 0.9250 0.9000 0.9250 0.8500 0.9250 0.8750 0.9125 0.8139 Block freq. 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9750 1.0000 0.9750 1.0000 0.9750 1.0000 0.9958 Freq. _{0.7000 0.6250 0.7750 0.6750 0.6750 0.7750 0.8000 0.7000 0.7000 0.7000 0.9750 0.9750 0.9750 0.9000 0.9000 0.9250 0.8500 0.9250 0.8500 0.9250 0.8083}

Case 7

Scan Value 0D .762BC 49E53 15 .BE425 D1289 1E .0658F 586BF 26 .4E6F8 DFAF5 2E .96862 66F2B 36 .DE9CB EE361 3F .26B35 75797 47 .6EC9E FCBCD 4F .B6E08 84003 57 .FEF72 0B439 60 .470DB 9286F 68 .8F245 19CA5 70 .D73AE A10DB 79 .1F518 28511 81 .67681 AF947 89 .AF7EB 36D7D 91 .F7954 BE1B3 9A .3FABE 455E9 A2 .87C27 CCA1F AA .CFD91 53E55 Average

VI. Conclusions

Even the true random sequence can 100% pass the testing. From n=2, 3, 4 and 10, the results

show generated output sequences of RHCS with different parameters are indeed holding better

statistical properties.

[1] S. L. Chen, S. M. Chang, T. T. Hwang and W. W. Lin, “Toolbox for Key Space Testing of

Robust Hyper-Chaotic System”, NCTS Preprints Series 2006-7-016 in Taiwan, 2006.

(http://math.cts.nthu.edu.tw/Mathematics/preprints/content.html\#2006)

[2] “Random Number Generation and Testing”,

http://csrc.nist.gov/rng/

[3] S. L. Chen, S. M. Chang, T. T. Hwang and W. W. Lin, “Digital Secure-Communication Using

Robust Hyper-Chaotic Systems”, NCTS Preprints Series 2006-7-015 in Taiwan, 2006.

History

Date Version

Description

2006.07.18 1.0 First

Release.