The next step before testing a 106 bits number (Section 7.1) was running the statistical tests with a bigger number. Thus, it was requested to the Luna SA HSM a generation of a n = 109 bits number. The focus of this analysis was to test whether the statistical data previously obtained are consistent. It was not possible to run the Discrete Fourier Transform (Spectral) Test, the Random Excursions Test and the Random Excursions Variant Test with such a large number, because of an insufficient memory error. It took about 12 hours to run all the tests.
7.2.1 Frequency Test
Since comprises 109 bits, the input size recommendation is fulfilled. The calculated statistics are presented in Table 7.17.
7.2. Quality analysis of a Hardware Security Module (HSM) generated number of 109 bits 87
Computational Information P-value Result x =−9 Total visits 1134 0.822167 Success x =−8 Total visits 1078 0.591242 Success x =−7 Total visits 986 0.270316 Success x =−6 Total visits 1015 0.308536 Success x =−5 Total visits 1042 0.346996 Success x =−4 Total visits 1004 0.173158 Success x =−3 Total visits 1019 0.140604 Success x =−2 Total visits 1041 0.100846 Success x =−1 Total visits 1112 0.167661 Success x = 1 Total visits 1103 0.117560 Success x = 2 Total visits 985 0.021078 Success x = 3 Total visits 996 0.091918 Success x = 4 Total visits 1014 0.199041 Success x = 5 Total visits 944 0.106712 Success x = 6 Total visits 937 0.132939 Success x = 7 Total visits 960 0.210994 Success x = 8 Total visits 954 0.231553 Success x = 9 Total visits 933 0.219192 Success
Table 7.16: Statistics table of the Random Excursions Variant Test for a 1000000 bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
Computational Information Result The nth partial sum (Sn) 8364
Sn/n 0.000008
P-value 0.791400 Success
Table 7.17: Statistics table of the Frequency Test for a 109bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
P-value = 0.791400 > 0.01 and thus, the tested sequence is considered to be random.
7.2.2 Frequency Test Within a Block
The block length of this test is M = 128.
comprises 109 bits and thus the input size recommendation is fulfilled. The condition n ≥ MN is satisfied, with N = jn
M k
= 7812500 and M× N = 109. Since n is a multiple of
128, there wouldn’t be any discarded bits in the realization of this test. The calculated statistics are presented in Table 7.18.
P-value = 0.320670 > 0.01 and thus, the tested sequence is considered to be random. This result reveals that each one of the M-bits block of have approximately M/2 ones and M/2
88 Chapter 7. Quality analysis of generated numbers Computational Information Result
χ2 7814340.812500 Number of substrings 7812500
Block length (M) 128 Bits discarded 0
P-value 0.320670 Success
Table 7.18: Statistics table of the Frequency Test Within a Block for a 109bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
zeros.
7.2.3 Runs Test
comprises 109bits and thus the input size recommendation is fulfilled. The calculated statistics are presented in Table 7.19.
Computational Information Result
π 0.500004
vn(obs) (Total number of runs) 499962850
|vn(obs)− 2nπ(1 − π)|
2√2nπ(1− π) 1.661397
P-value 0.018795 Success
Table 7.19: Statistics table of the Runs Test for a 109bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
P-value = 0.018795 > 0.01 and thus, the tested sequence is considered to be random.
7.2.4 Test for the Longest Run of Ones in a Block
Since comprises 109 bits, the input size recommendation is fulfilled and, depending on the chosen value of n, the value of M will be assigned according to Table 6.3.
The calculated statistics are presented in Table 7.20.
P-value = 0.480659 > 0.01 and thus, the tested sequence is considered to be random.
7.2.5 Binary Matrix Rank Test
Since comprises 109 bits, the input size recommendation is fulfilled. The calculated statistics are presented in Table 7.21.
7.2. Quality analysis of a Hardware Security Module (HSM) generated number of 109 bits 89 Computational Information Result
N (number of substrings) 100000 M (Substring Length) 10000 χ2 5.506601
P-value 0.480659 Success
Table 7.20: Statistics table of the Test for the Longest Run of Ones in a Block for a 109bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
Computational Information Result P32 0.288788 P31 0.577576 Probability P30 0.133636 F32 281676 F31 564029 Frequency F30 130857 Number of matrices 976562 χ2 1.375720 Bits discarded 512 P-value 0.502651 Success
Table 7.21: Statistics table of the Binary Matrix Rank Test for a 109bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
P-value = 0.907914 > 0.01 and thus, the tested sequence is considered to be random.
7.2.6 Non-overlapping Template Matching Test
In this evaluation m = 9, as recommended. Also, the conditions M > 0.01· n = 109 and N =bn/Mc = 8 (with M = 125000) are met. All the recommendations are fulfilled.
The calculated statistics are presented in Table 7.22. In this test m = 9 and thus there will be 29 different patterns (except the all ones and all zeros patterns) to be tested. This means that there will be 29− 2 P-values. The sequence is split into N independent blocks and the frequency of the pattern occurrences in each block is logged in Table 7.22.
There are more than one calculated P-value, presented in Table 7.22, that have values less than 0.01. For the patterns 000001001, 011101111 and 101000100 the P-values are 0.007990, 0.008124 and 0.001514, respectively. This indicates that, for the tested sequence, the patterns mentioned before occur a different number of times from what will be expected in a truly random sequence.
By convention, if the sequence fails the test in at least one pattern, then the sequence fails the test. Thus, this sequence fails this test and it is not considered to be random.
90 Chapter 7. Quality analysis of generated numbers
Computational Information
M = 125000000 N = 8 m = 9 Frequency
Template W1 W2 W3 W4 W5 W6 W7 W8 χ2 P-value Result
000000001 244453 243866 244528 243860 243632 244292 244701 244669 5.408738 0.713129 Success 000000011 244246 244208 244181 244114 243820 243561 244738 243642 4.500244 0.809409 Success 000000101 243851 244593 244777 243523 244060 244104 243564 244391 6.261703 0.617943 Success 000000111 243702 243982 244577 243827 244504 244555 244255 244321 3.625408 0.889242 Success 000001001 244032 243022 244004 244622 244548 243187 245637 244381 20.699494 0.007990 FAILURE .. . 011010111 244024 244098 243871 243788 244538 244715 244831 243603 6.210717 0.623641 Success 011011111 244361 243475 244224 244784 244656 244316 243641 244265 6.244790 0.619833 Success 011101111 244355 244085 244510 243922 243593 244657 246004 243364 20.654521 0.008124 FAILURE 011111111 243889 243960 244026 244166 244974 244945 245145 244106 10.427673 0.236282 Success .. . 101000000 243727 243152 244648 243802 244232 243868 244753 243636 9.459824 0.305001 Success 101000100 244833 243380 243710 244680 245369 243892 245194 242836 25.067049 0.001514 FAILURE 101001000 243399 244164 244159 244215 243963 244509 243992 244634 4.190828 0.839509 Success .. . 111111100 243873 244625 244474 244260 244530 244650 243662 244193 4.552591 0.804149 Success 111111110 243889 243960 244026 244166 244974 244945 245145 244106 10.427673 0.236282 Success
Table 7.22: Statistics table of the Non-Overlapping Template Matching Test for a 109bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random). The complete table is in C, table C.2
7.2.7 Overlapping Template Matching Test
comprises 109 bits and thus the input size recommendation is fulfilled. In this evaluation, m = 9.
The calculated statistics are presented in Table 7.23.
P-value = 0.00000 < 0.01 and thus, the tested sequence is not considered to be random.
7.2.8 Maurer’s "Universal Statistical" Test
Since comprises 109 bits, the input size recommendation is fulfilled. Q and L are chosen according to Table 6.4. The sequence was partitioned into two segments: an initialization segment consisting of Q L-bit non-overlapping blocks, and a test segment consisting of K L-bit non-overlapping blocks. After this partition, 10 bits remained in the end of the sequence, not belonging to any block. Those 10 bits were discard.
7.2. Quality analysis of a Hardware Security Module (HSM) generated number of 109 bits 91
Computational Information Result m (block length of ones) 9
M (length of substring) 1032 N (number of substrings) 968992 λ[(M− m + 1)/2m] 2 η 1 χ2 116.476980 { P-value} 0.000000 FAILURE Frequency 0 1 2 3 4 ≥ 5 351982 180536 135143 97572 68021 135738
Table 7.23: Statistics table of the Overlapping Template Matching Test for a 109 bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
Computational Information Result
L 15 Q 327680 K 66338986 sum 939865144.454580 σ 0.000149 v ar i anc e 3.419000 ex pec ted V al ue 14.167488 Bits discarded 10 P-value 0.398955 Success
Table 7.24: Statistics table of the Maurer’s "Universal Statistical" Test for a 109 bits number. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
P-value = 0.398955 > 0.01 and thus, the tested sequence is considered to be random.
7.2.9 Linear Complexity Test
comprises 109 bits and thus, the input size recommendation is fulfilled. In this evaluation, M was chosen to be equal to 500 and N was processed by the test source code. The values Ti = (−1)M· (Li − µ) +
2
9 are recorded in v0, v1, . . . , v6, as presented in Table 6.5. The calculated statistics are presented in Table 7.25.
P-value = 0.520163 > 0.01 and thus, the tested sequence is considered to be random.
7.2.10 Serial Test
comprises 109 bits and m was chosen to be equal to 16 (and 16 <blog2nc − 2 = 27) and
92 Chapter 7. Quality analysis of generated numbers
Computational Information Result M (substring length) 500 N (number of substrings) 2000000 χ2 5.186203 Bits Discarded 0 P-value 0.520163 Success Frequency v0 v1 v2 v3 v4 v5 v6 20888 62344 249283 1001018 499452 125349 41666
Table 7.25: Statistics table of the Linear Complexity Test for a 109 bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
7.2. Quality analysis of a Hardware Security Module (HSM) generated number of 109 bits 93
Computational Information Result Block length (m) 16 ψm 66755.999433 ψm−1 33531.165344 ψm−2 16705.072693 P-value1 0.037629 Success P-value2 0.466093 Success
Table 7.26: Statistics table of the Serial Test for a 109bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
P-value1 = 0.037629 > 0.01 and P-value2 = 0.466093 > 0.01. Thus, the tested sequence is considered to be random.
7.2.11 Approximate Entropy Test
For this test comprises 109bits and m was chosen to be equal to 10 (and 10 <blog2nc−5 =
24). So, the input size recommendation is fulfilled. The calculated statistics are presented in Table 7.27.
Computational Information Result Block length (m) 10 χ2 1002.410764 φm -6.931471 φm+1 -7.624618 ApEn(m) = ϕ(m)− ϕ(m+1) 0.693147 P-value 0.679283 Success
Table 7.27: Statistics table of the Approximate Entropy Test for a 109bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
P-value = 0.679283 > 0.01. Thus, it can be concluded that the tested sequence is consid- ered to be random. ApEn(m) has a small value, and this implies a strong regularity in .
7.2.12 Cumulative Sums Test
comprises 109bits and thus, the input size recommendation is fulfilled.
The calculated statistics for the Cumulative Sums Test Mode = 0 are presented in Table 7.28 and the calculated statistics for the Cumulative Sums Test Mode = 1 are presented in Table 7.29.
From table 7.28, P-value = 0.842729 > 0.01. Thus, the tested sequence is considered to be random.
94 Chapter 7. Quality analysis of generated numbers
Computational Information Result Maximum partial sum 24288
P-value 0.842729 Success
Table 7.28: Statistics table of the Cumulative Sums (forward) Test for a 109bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
Computational Information Result Maximum partial sum 24606
P-value 0.834051 Success
Table 7.29: Statistics table of the Cumulative Sums (reverse) Test for a 109bits number generated by a Luna SA HSM. Success means the acceptance of the null hypothesis (the hypothesis that states the sequence is random).
From table 7.29, P-value = 0.834051 > 0.01. Thus, the tested sequence is considered to be random.