Statistical assessment of two rekeying mechanisms applied to the generation of random numbers

 range 
 goodness score 
 Statistical test Number of testing 
 [0.985 ; 0.995] Desirably good 0 
 frequency 1 
 [0.980561; 0.985) Almost not good 
 2 
 block-frequency 1 (0.995 ; 0.999439] but still good 
 cumulative-sums 2 [0 ; 0.980561) 
 Not good 4 
 runs 1 (0.999439 ; 1] 
40 No 2.CS (12) 2020 
 Khoa học và Công nghệ trong lĩnh vực An toàn thông tin 
 Only for uniformity at the check of that We recommend the use of Fortuna, and so of 
criteria, we excluded of the NIST suite: Random Fortuna with CTR-ACPKM, in real and practical 
Excursions Test and Random Excursions Variant applications. Nevertheless, it should be taken into 
Test, since both analyze only an arbitrary amount account that each generated sequence must be 
of sequences which satisfy some conditions. A carefully analyzed for randomness before being 
similar approach is adopted in [33], [34]. used although similar results are expected. 
 This way, with the remaining tests, we IV. CONCLUSION 
evaluate 162 proportion values per file, and the 
 Generation of keys, initialization vectors, 
accumulated score gives us another idea of 
 seeds, and other kinds of sequences with random 
goodness. The possible best score is 0 and the 
 behavior play a fundamental role in 
possible worst score is 648 for every file. 
 cryptography. Thus, a pseudo-random number 
 A comparative analysis between the twenty generator cryptographically strong with proven 
files generated with Fortuna and the twenty files pseudo-random behavior is a powerful tool in 
generated with the proposed variant is shown this area. 
below. All results have been tabulated and are 
 The used simple technique is the 
shown in the Appendix D, E. These results do not 
 implementation of one block cipher algorithm 
throw significant differences between both 
 using one mode of operation. This way, good 
variants, and we appreciate a random behavior 
 random behavior is achieved if the underlying 
since the final score of every file is very low. 
 encryption method has good diffusion properties. 
 Generator Score The pseudo-random number generator 
 12 – best score Fortuna is based on this principle, using the 
 Fortuna (original) 
 38 – worst score standard AES and the mode of operation CTR as 
 underlying primitives, and one internal rekeying 
 8 – best score 
 Fortuna-CTR-ACPKM mechanism proposed by their own designers. 
 40 – worst score 
 In this paper, we have analyzed the random 
[Aspect 3.] The other approach according to [15] behavior of the sequences generated by Fortuna 
is to examine the distribution of the p-values in as well as a variant that uses another internal 
the range [0 ; 1], so this interval is divided into rekeying mechanism that included in the 
ten equal sub-intervals and then the number of p- Russian Standardization System and named 
values that lie within each sub-interval are CTR-ACPKM. In both cases, twenty files of 120 
counted. The optimal case for randomness is that MB were generated, all of them show a good 
100 p-values fall in each sub-interval. random behavior. 
 As reference we used the NIST test suite, 
 The NIST test suite returns a p-value of p-
 although we have removed from it two tests and 
values in every testing, corresponding to the chi 
 have included an extra analysis technique 
square test for the same ones. If that p-value is 
 performed previously, but using the Diehard 
greater than 0.0001, then the sequence can be 
 statistical test suite. In terms of this approach, the 
considered to be uniformly distributed, in all the 
 sequences generated by Fortuna seem as random 
checks carried out this was satisfactorily fulfilled. 
 as the sequences generated by the variant of 
B. Final considerations Fortuna, although the difference observed is 
 really small to make a comparison between the 
 Both the original Fortuna and Fortuna with the two kinds of sequences. 
internal rekeying mechanism CTR-ACPKM, 
have shown that the generated sequences have a To conclude this paper, we propose the 
random behavior. The performing tests do not application of the two analyzed pseudo-random 
offer sufficient reasons to establish a significant number generators, Fortuna and Fortuna with 
comparison between both types of generated internally rekeying mechanism CTR-ACPKM, in 
sequence actually, using the criteria established real and practical applications, as long as the 
in [16] the sequences generated by Fortuna have generated sequences in both cases are previously 
similar randomness than the other kind of analyzed and show a random behavior similar to 
generated sequences. the examples shown here. 
 Số 2.CS (12) 2020 41 
Journal of Science and Technology on Information Security 
 V. APPENDIX C. Proportion analysis of two samples generated 
 by Fortuna with CTR-ACPKM 
A. Pseudo-code of Fortuna with both, original 
 and new rekeying mechanisms 
 Function: Pseudo-random generator Fortuna 
 Input: Counter with 128 bits (random) 
 Key 퐾 with 256 bits (random) 
 Number of bytes 푛 to generate 
 Procedure: 1. Define an empty string 
 2. Define = ⌈푛/16⌉ 
 3. For 푖 from 1 to − 1 do 
 4. = || 퐾( ) 
 5. = + 1 
 6. 푛 = 푛 − 16 
 7. If 푖 = 216 then 
 8. Change the key 퐾 using the 
 internal rekeying method of the 
 original Fortuna or the ACPKM 
 method 
 9. = + 1 
 10. End If 
 D. Score analysis of the 20 samples generated by 
 11. 푖 = 푖 + 1 Fortuna 
 12. End For 
 24 14 32 12 26 26 38 30 22 20 
 13. = ||first-푛-bytes-( ( )) 
 퐾 22 32 34 26 24 30 24 32 26 22 
 Return: Pseudo-random string of 푛 bytes 
 E. Score analysis of the 20 samples generated by 
B. Proportion analysis of two samples generated Fortuna with CTR-ACPKM 
 by the original Fortuna 
 18 40 34 18 36 28 34 12 34 30 
 26 24 24 32 30 8 18 28 8 32 
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