Power of the test based on joint application of >, within a Block>> and a generalization of Test>>

M. P. Savelov

Diskretnaya Matematika, Tome 36 (2024) no. 1, pp. 67-102

Résumé

«Approximate $\phi$-Entropy Test» is a generalization of «Approximate Entropy Test» and «Serial Test» of the NIST package. In the present paper we obtained the limit joint distribution of the statistic of «Approximate $\phi$-Entropy Test» and statistics $T_{\mathrm{mon}}$ and $T_{\mathrm{fr}}$ of the NIST package tests «Monobit Test» and «Frequency Test within a Block» respectively, in case when the tested sequence is «close» to a sequence of independent Bernoulli trials with parameter $\frac12$. The limiting value of the power of the test based on the simultaneous use of these three statistics was obtained. The limit joint distribution of the statistics $T_{\mathrm{mon}}$, $T_{\mathrm{fr}}$ and the statistics of «Approximate Entropy Test» and «Serial Test» was obtained.

Détail

Keywords: joint distribution of statistics, NIST package, power, «Approximate Entropy Test», «Approximate $\phi$-Entropy Test», «Frequency Test within a Block», «Serial Test», goodness of fit test.
Mots-clés : «Monobit Test», Markov chains

Bibliographie

[1] Rukhin A., Soto J., Nechvatal J., Smid M., Barker E., Leigh S., Levenson M., Vangel M., Banks D., Heckert A., Dray J., Vo S., A statistical test suite for random and pseudorandom number generators for cryptographic applications, NIST Special Publication 800-22 Revision 1a, 2010

[2] Serov A. A., “Formuly dlya chisel posledovatelnostei, soderzhaschikh zadannyi shablon zadannoe chislo raz”, Diskretnaya matematika, 32:4 (2020), 120–136 | DOI

[3] Zubkov A. M., Serov A. A., “A natural approach to the experimental study of dependence between statistical tests”, Matem. vopr. kriptogr., 12:1 (2021), 131–142 | DOI | MR | Zbl

[4] Zubkov A. M., Serov A. A., “Testing the NIST Statistical Test Suite on artificial pseudorandom sequences”, Matem. vopr. kriptogr., 10:2 (2019), 89–96 | DOI | MR | Zbl

[5] Zubkov A. M., Serov A. A., “Experimental study of NIST Statistical Test Suite ability to detect long repetitions in binary sequences”, Matem. vopr. kriptogr., 14:2 (2023), 137–145 | DOI | MR

[6] Zaman J. K. M. S. , Ghosh R., “Review on fifteen statistical tests proposed by NIST”, J. Theor. Phys. Cryptography, 1 (2012), 18–31

[7] Sulak F., Doǧanaksoy A., Uǧuz M., Koçak O., “Periodic template tests: A family of statistical randomness tests for a collection of binary sequences”, Discrete Applied Mathematics, 271 (2019), 191–204 | DOI | MR | Zbl

[8] Soto J., Bassham L., Randomness testing of the Advanced Encryption Standard finalist candidates, NIST IR 6483, Nat. Inst. Stand. Technol., 2000

[9] Sulak F., Uǧuz M., Koçak O., Doǧanaksoy A., “On the independence of statistical randomness tests included in the NIST test suite”, Turkish J. Electr. Eng. Comput. Sci., T., 25:5 (2017), 3673–3683 | DOI | MR

[10] Georgescu C., Simion E., “New results concerning the power of NIST randomness tests”, Proc. Romanian acad., ser. A., 18 (2017), 381–388 | MR

[11] Iwasaki A., Umeno K., “Randomness test to solve Discrete Fourier Transform Test problems”, IEICE Trans. Fundam. Electronics, Commun. Comput. Sci., E101.A:8 (2018), 1204–1214 | DOI

[12] Burciu P., Simion E., “A systematic approach of NIST statistical tests dependencies”, J. Electr. Eng., Electronics, Control and Comput. Sci., 5:15 (2019), 1–6

[13] Ryabko B. Ya., Pestunov A. I., “«Stopka knig» kak novyi statisticheskii test dlya sluchainykh chisel”, Probl. peredachi inform., 40:1 (2004), 73–78 | MR | Zbl

[14] Rukhin A. L., “Approximate entropy for testing randomness”, J. Appl. Prob., 37:1 (2000), 88–100 | DOI | MR | Zbl

[15] Karell-Albo J. A., Legón-Pérez C. M., Madarro-Capó E. J., Rojas O., Sosa-Gómez G., “Measuring independence between statistical randomness tests by mutual information”, Entropy, 22:7 (2020), 1–18 | DOI | MR

[16] Maltsev M. V., Kharin Yu. S., “O testirovanii vykhodnykh posledovatelnostei kriptograficheskikh generatorov na osnove tsepei Markova uslovnogo poryadka”, Informatika, 4 (2013), 104–111

[17] Kievets N. G., Korzun A. I., “Sravnenie statistik testov serii i approksimirovannoi entropii”, Doklady BGUIR, 3 (2014), 12–17

[18] Voloshko V. A., Trubei A. I., “O moschnosti testov mnogomernoi diskretnoi ravnomernosti, ispolzuemykh dlya statisticheskogo analiza generatorov sluchainykh posledovatelnostei”, Zhurn. Belorus. gos. un-ta. Matem. Inf., 1 (2022), 26–37 | MR

[19] Voloshko V. A., Kharin Yu. S., Trubey A. I., “On power comparison for some tests on pure randomness under Markov high-order dependencies”, Computer Data Analysis and Modeling: Stochastics and Data Science (Proc. of the XIII Intern. Conf.), 2022, 211–217

[20] Savelov M. P., “Predelnye sovmestnye raspredeleniya statistik chetyrekh kriteriev paketa NIST”, Diskretnaya matematika, 33:2 (2021), 141–154 | DOI

[21] Savelov M. P., “Predelnye sovmestnye raspredeleniya statistik trekh kriteriev paketa NIST”, Diskretnaya matematika, 33:3 (2021), 92–106 | DOI

[22] Savelov M. P., “Predelnaya teorema dlya sglazhennogo varianta spektralnogo kriteriya ravnoveroyatnosti dvoichnoi posledovatelnosti”, Diskretnaya matematika, 33:4 (2021), 132–140 | DOI | MR

[23] Savelov M. P., “Predelnoe sovmestnoe raspredelenie statistik kriteriev «Monobit test», «Frequency Test within a Block» i «Test for the Longest Run of Ones in a Block»”, Diskretnaya matematika, 34:3 (2022), 70–84 | DOI

[24] Savelov M. P., “Predelnoe sovmestnoe raspredelenie statistik kriteriev «Monobit test», «Frequency Test within a Block» i «Binary Matrix Rank Test»”, Diskretnaya matematika, 34:4 (2022), 84–98 | DOI | MR

[25] Savelov M. P., “Predelnoe sovmestnoe raspredelenie statistik kriteriev «Monobit test», «Frequency Test within a Block» i obobscheniya kriteriya «Serial Test»”, Diskretnaya matematika, 35:1 (2023), 88–106 | DOI | MR

[26] Savelov M. P., “Predelnoe sovmestnoe raspredelenie statistik kriteriev paketa NIST «Monobit Test», «Frequency Test within a Block» i obobscheniya kriteriya «Approximate Entropy Test»”, Diskretnaya matematika, 35:2 (2023), 93–108 | DOI

[27] Savelov M. P., “Predelnoe sovmestnoe raspredelenie statistik kriteriev priblizhennoi $\phi$-entropii”, Diskretnaya matematika, 35:3 (2023), 60—70 | DOI | MR

[28] Savelov M. P., “Moschnost kriteriya, osnovannogo na odnovremennom primenenii «Monobit Test», «Frequency Test within a Block» i «Serial Test»”, Diskretnaya matematika, 35:4 (2023), 79–114 | DOI | MR

[29] Pincus S., Huang W.-M., “Approximate entropy: Statistical properties and applications”, Commun. Statist. — Theory and Methods, 21:11 (1992), 3061–3077 | DOI | Zbl

[30] Pincus S., Singer B. H., “Randomness and degrees of irregularity”, Proc. Nat. Acad. Sci. USA, 93:5 (1996), 2083–2088 | DOI | MR | Zbl

[31] Pincus S., Kalman R. E., “Not all (possibly) «random» sequences are created equal”, Proc. Nat. Acad. Sci. USA, 94:8 (1997), 3513–3518 | DOI | MR | Zbl

[32] Selivanov B. I., Chistyakov V. P., “Posledovatelnyi $\chi^2$-kriterii, postroennyi po $s$-tsepochkam sostoyanii tsepi Markova”, Diskretnaya matematika, 9:4 (1997), 127–136 | DOI | MR | Zbl

[33] Rukhin A. L., “Korrelyatsionnye matritsy tsepochek dlya markovskikh posledovatelnostei i testirovanie sluchainosti”, Teoriya veroyatn. i ee primen., 51:4 (2006), 712–731 | DOI | MR