On Independence and Sensitivity of Statistical Randomness Tests - Sequences and Their Applications-SETA 2008 - page 26

Table 3. Relation of tests for all sequences of length n =30for α =0 . 01

Test Frequency Overlapping Longest Run Runs RW RW LC k-error MOC Lempel-Ziv

Template of Ones Height Excursion LC

Frequency - 0.3853 0.1231 0.0409 0.5591 0 0.0339 0.0207 0.1035 0.3399

Overlapping 0.3500 - 0.3096 0.0733 0.2490 0 0.0309 0.0185 0.0875 0.1631

Template
Longest 0.1851 0.5124 - 0.0869 0.1602 0 0.0303 0.0172 0.1026 0.0991

Run of Ones
Runs 0.0409 0.0807 0.0578 - 0.0441 0.0876 0.0343 0.0212 0.1016 0.1308

RW 0.8630 0.4231 0.1645 0.0681 - 0 0.0355 0.0204 0.1498 0.3520

Height
RW 0 0 0 0.1195 0 - 0.0320 0.0192 0.0328 0.0395

Excursion
LC

0.0175 0.0175 0.0104 0.0177 0.0119 0.0121 - 0.1398 0.0250 0.0328

1-error LC 0.0172 0.0170 0.0095 0.0176 0.0110 0.0117 0.2251 - 0.0184 0.0330

MOC 0.1025 0.0953 0.0676 0.1005 0.0960 0.0238 0.0480 0.0219 - 0.0917

Lempel-Ziv 0.1747 0.0922 0.0339 0.0672 0.1172 0.0149 0.0327 0.0204 0.0476 -

results of linear complexity and 1-error linear complexity tests. Moreover, a sig-
nificant relation is observed between Lempel-Ziv and frequency test. As also
mentioned in the theoretical results part in Sect. 3.1, no correlation between the
weight and the linear complexity is observed.
Another interesting result is that none of the sequences that fail the random
walk excursion test, fail any of the (i) frequency; (ii) overlapping template; (iii)
longest run of ones or (iv) random walk height tests for sequences of length
n = 20 and 30. This means that including the random walk excursion test
increases the coverage of test suites significantly. To measure the effect of each
test to the coverage, we present the number of 20 bit sequences that only fail the
given test in Table 4. The tests based on ordering of k-tuples seem to increase
the coverage of the selection more compared to tests based on k -tuple pattern
frequency and this is mainly due to the correlation of these tests presented in
above tables. Also, it is observed that all sequences that fail frequency test also
fail any of the other tests in our scheme. So, there is no contribution of frequency
tests to the coverage of selected tests, for sequences of length 20.
We also calculated the coverage, that is

10

i =1 R i n |

|∪ ,for n = 20 and 30 as 0 . 122948

and 0 . 134930, respectively. Whenever R T n

sets are disjoint, coverage takes its

Table 4. Number of sequences that only fail the given test (but pass all other tests)

Tests Number of Sequences
Linear Complexity 22436
Lempel-Ziv Complexity 19680
Random Walk Excursion 19428
Maximum Order Complexity 8419
k =1-errorLinearComplexity7895
Runs

6454

Longest Run of Ones 6196

Overlapping Template 4765

Random Walk Height 1163

Frequency 0