{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T18:40:12Z","timestamp":1772908812955,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,4]],"date-time":"2020-07-04T00:00:00Z","timestamp":1593820800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The analysis of independence between statistical randomness tests has had great attention in the literature recently. Dependency detection between statistical randomness tests allows one to discriminate statistical randomness tests that measure similar characteristics, and thus minimize the amount of statistical randomness tests that need to be used. In this work, a method for detecting statistical dependency by using mutual information is proposed. The main advantage of using mutual information is its ability to detect nonlinear correlations, which cannot be detected by the linear correlation coefficient used in previous work. This method analyzes the correlation between the battery tests of the National Institute of Standards and Technology, used as a standard in the evaluation of randomness. The results of the experiments show the existence of statistical dependencies between the tests that have not been previously detected.<\/jats:p>","DOI":"10.3390\/e22070741","type":"journal-article","created":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T03:19:27Z","timestamp":1594005567000},"page":"741","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Measuring Independence between Statistical Randomness Tests by Mutual Information"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7260-2444","authenticated-orcid":false,"given":"Jorge Augusto","family":"Karell-Albo","sequence":"first","affiliation":[{"name":"Instituto de Criptograf\u00eda, Universidad de La Habana, Havana 10400, Cuba"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6104-9671","authenticated-orcid":false,"given":"Carlos Miguel","family":"Leg\u00f3n-P\u00e9rez\u00a0","sequence":"additional","affiliation":[{"name":"Instituto de Criptograf\u00eda, Universidad de La Habana, Havana 10400, Cuba"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5004-2960","authenticated-orcid":false,"given":"Evaristo Jos\u00e9","family":"Madarro-Cap\u00f3\u00a0","sequence":"additional","affiliation":[{"name":"Instituto de Criptograf\u00eda, Universidad de La Habana, Havana 10400, Cuba"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0681-3833","authenticated-orcid":false,"given":"Omar","family":"Rojas","sequence":"additional","affiliation":[{"name":"Facultad de Ciencias Econ\u00f3micas y Empresariales, Universidad Panamericana, \u00c1lvaro del Portillo 49, Zapopan, Jalisco 45010, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7793-896X","authenticated-orcid":false,"given":"Guillermo","family":"Sosa-G\u00f3mez","sequence":"additional","affiliation":[{"name":"Facultad de Ciencias Econ\u00f3micas y Empresariales, Universidad Panamericana, \u00c1lvaro del Portillo 49, Zapopan, Jalisco 45010, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,4]]},"reference":[{"key":"ref_1","unstructured":"Sulak, F. (2011). Statistical analysis of block ciphers and hash functions. Publication of the Middle East Technical University Ph.D. Examinations, METU. Supervisor Assoc. Prof. Dr. Ali Doganaksoy."},{"key":"ref_2","first-page":"1","article-title":"Statistical testing of cryptographic randomness","volume":"9","author":"Demirhan","year":"2016","journal-title":"J. Stat. Stat. Actuar. Sci."},{"key":"ref_3","unstructured":"Knuth, D.E. (1969). The Art of Computer Programming, Addison-Wesley. [1st ed.]."},{"key":"ref_4","unstructured":"Ko\u00e7ak, O. (2016). A Unified Evaluation of Statistical Randomness Tests and Experimental Analysis of their Relations. [Ph.D. Thesis, Middle East Technical University]."},{"key":"ref_5","unstructured":"Bassham, L.E., Rukhin, A.L., Soto, J., Nechvatal, J.R., Smid, M.E., Barker, E.B., Leigh, S.D., Levenson, M., Vangel, M., and Banks, D.L. (2001). A Statistical Test Suite for Random And Pseudorandom Number Generators for Cryptographic Applications, BoozAllen and Hamilton."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.jspi.2004.02.010","article-title":"Using information theory approach to randomness testing","volume":"133","author":"Ryabko","year":"2005","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ryabko, B. (2020). Time-Adaptive Statistical Test for Random Number Generators. Entropy, 22.","DOI":"10.3390\/e22060630"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1177\/0037549717726145","article-title":"A simulation study on the accuracy of cryptographic randomness tests","volume":"93","author":"Demirhan","year":"2017","journal-title":"Simulation"},{"key":"ref_9","unstructured":"Do\u011fnaksoy, A., Ege, B., and Mu\u015f, K. (2008, January 25\u201327). Extended results for independence and sensitivity of NIST randomness tests. Proceedings of the Information Security and Cryptography Conference, Istanbul, Turkey."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"655","DOI":"10.3906\/elk-1503-214","article-title":"Mutual correlation of NIST statistical randomness tests and comparison of their sensitivities on transformed sequences","volume":"25","author":"Sulak","year":"2017","journal-title":"Turk. J. Electr. Eng. Comput. Sci."},{"key":"ref_11","unstructured":"Marsaglia, G. (2020, April 03). The Marsaglia Random Number CDROM, with the DIEHARD Battery of Tests of Randomness. Available online: http:\/\/www.cs.hku.hk\/diehard."},{"key":"ref_12","unstructured":"Knuth, D.E. (1998). The Art of Computer Programming, Addison-Wesley. [3rd ed.]."},{"key":"ref_13","unstructured":"Caelli, W., Dawson, E., Nielsen, L., and Gustafson, H. (1992). CRYPT-X Stastical Package Manual, Measuring the Strength of Stream and Block Ciphers, Information Security Research Centre, Queensland University of Technology."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"L\u2019Ecuyer, P., and Simard, R. (2007). TESTU01: A Software Library in ANSI C for Empirical Testing of Random Number Generators. TOMS.","DOI":"10.1145\/1268776.1268777"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Fan, L., Chen, H., and Gao, S. (2013). A general method to evaluate the correlation of randomness tests. International Workshop on Information Security Applications, Springer.","DOI":"10.1007\/978-3-319-05149-9_4"},{"key":"ref_16","unstructured":"Cover, T.M., and Thomas, J.A. (2012). Elements of Information Theory, John Wiley & Sons."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1002\/j.1538-7305.1948.tb00917.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Verd\u00fa, S. (2019). Empirical Estimation of Information Measures: A Literature Guide. Entropy, 21.","DOI":"10.3390\/e21080720"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Hern\u00e1ndez, D.G., and Samengo, I. (2019). Estimating the Mutual Information between Two Discrete, Asymmetric Variables with Limited Samples. Entropy, 21.","DOI":"10.3390\/e21060623"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Carrara, N., and Ernst, J. (2020). On the Estimation of Mutual Information. Multidiscip. Digit. Publ. Inst. Proc., 33.","DOI":"10.3390\/proceedings2019033031"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Soto, J., and Bassham, L. (2000). Randomness Testing of the Advanced Encryption Standard Finalist Candidates, BoozAllen and Hamilton.","DOI":"10.6028\/NIST.IR.6483"},{"key":"ref_22","unstructured":"Turan, M.S., Do\u011fanaksoy, A., and Bozta\u015f, S. (2008). On independence and sensitivity of statistical randomness tests. International Conference on Sequences and Their Applications, Springer."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Georgescu, C., Simion, E., Nita, A.P., and Toma, A. (July, January 29). A view on NIST randomness tests (in) dependence. Proceedings of the IEEE 2017 9th International Conference on Electronics, Computers and Artificial Intelligence (ECAI), Targoviste, Romania.","DOI":"10.1109\/ECAI.2017.8166460"},{"key":"ref_24","first-page":"1","article-title":"A Systematic Approach of NIST Statistical Tests Dependencies","volume":"5","author":"Burciu","year":"2019","journal-title":"J. Electr. Eng. Electron. Control Comput. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kv\u00e5lseth, T.O. (2017). On normalized mutual information: Measure derivations and properties. Entropy, 19.","DOI":"10.3390\/e19110631"},{"key":"ref_26","unstructured":"Good, P.I. (2006). Permutation, Parametric, and Bootstrap Tests of Hypotheses, Springer Science & Business Media."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Berry, K.J., Johnston, J.E., and Mielke Jr, P.W. (2014). A Chronicle of Permutation Statistical Methods, Springer.","DOI":"10.1007\/978-3-319-02744-9"},{"key":"ref_28","unstructured":"Meyer, P.E., and Meyer, M.P.E. (2009). Package \u2018infotheo\u2019. R Package Version, Citeseer."},{"key":"ref_29","first-page":"1469","article-title":"Entropy inference and the James-Stein estimator, with application to nonlinear gene association networks","volume":"10","author":"Hausser","year":"2009","journal-title":"J. Mach. Learn. Res."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1261","DOI":"10.1007\/s10994-019-05795-1","article-title":"Efficient feature selection using shrinkage estimators","volume":"108","author":"Sechidis","year":"2019","journal-title":"Mach. Learn."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"175","DOI":"10.3233\/IDA-2005-9204","article-title":"Optimal bin number for equal frequency discretizations in supervized learning","volume":"9","author":"Boulle","year":"2005","journal-title":"Intell. Data Anal."},{"key":"ref_32","unstructured":"Hernandez, J., and Barrero, D.F. (2017, January 5\u20138). Evolutionary generation and degeneration of randomness to assess the indepedence of the Ent test battery. Proceedings of the 2017 IEEE Congress on Evolutionary Computation (CEC), San Sebastian, Spain."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/7\/741\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:47:18Z","timestamp":1760176038000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/7\/741"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,4]]},"references-count":32,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["e22070741"],"URL":"https:\/\/doi.org\/10.3390\/e22070741","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,4]]}}}