Equitable [[2, 10], [6, 6]]-partitions of the 12-cube
We describe the computer-aided classification of equitable partitions of the 12-cube with quotient matrix [[2, 10], [6, 6]], or, equivalently, simple orthogonal arrays OA(1536, 12, 2, 7), or order-7 correlation-immune Boolean functions in 12 arguments with 1536 ones (which completes the classificati...
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| Published in | Cryptography and communications Vol. 16; no. 5; pp. 975 - 996 |
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| Format | Journal Article |
| Language | English |
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01.09.2024
Springer Nature B.V |
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| ISSN | 1936-2447 1936-2455 |
| DOI | 10.1007/s12095-024-00716-z |
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| Abstract | We describe the computer-aided classification of equitable partitions of the 12-cube with quotient matrix [[2, 10], [6, 6]], or, equivalently, simple orthogonal arrays OA(1536, 12, 2, 7), or order-7 correlation-immune Boolean functions in 12 arguments with 1536 ones (which completes the classification of unbalanced order-7 correlation-immune Boolean functions in 12 arguments and, as derived objects, unbalanced order-6 correlation-immune Boolean functions in 11 arguments). We find that there are 103 equivalence classes of the considered objects, and there are only two almost-OA(1536, 12, 2, 8) among them. Additionally, we find that there are 40 equivalence classes of pairs of disjoint simple OA(1536, 12, 2, 7) (equivalently, equitable partitions of the 12-cube with quotient matrix [[2, 6, 4], [6, 2, 4], [6, 6, 0]]) and discuss the existence of a non-simple OA(1536, 12, 2, 7). |
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| AbstractList | We describe the computer-aided classification of equitable partitions of the 12-cube with quotient matrix [[2, 10], [6, 6]], or, equivalently, simple orthogonal arrays OA(1536, 12, 2, 7), or order-7 correlation-immune Boolean functions in 12 arguments with 1536 ones (which completes the classification of unbalanced order-7 correlation-immune Boolean functions in 12 arguments and, as derived objects, unbalanced order-6 correlation-immune Boolean functions in 11 arguments). We find that there are 103 equivalence classes of the considered objects, and there are only two almost-OA(1536, 12, 2, 8) among them. Additionally, we find that there are 40 equivalence classes of pairs of disjoint simple OA(1536, 12, 2, 7) (equivalently, equitable partitions of the 12-cube with quotient matrix [[2, 6, 4], [6, 2, 4], [6, 6, 0]]) and discuss the existence of a non-simple OA(1536, 12, 2, 7). We describe the computer-aided classification of equitable partitions of the 12-cube with quotient matrix [[2, 10], [6, 6]], or, equivalently, simple orthogonal arrays OA(1536, 12, 2, 7), or order-7 correlation-immune Boolean functions in 12 arguments with 1536 ones (which completes the classification of unbalanced order-7 correlation-immune Boolean functions in 12 arguments and, as derived objects, unbalanced order-6 correlation-immune Boolean functions in 11 arguments). We find that there are 103 equivalence classes of the considered objects, and there are only two almost-OA(1536, 12, 2, 8) among them. Additionally, we find that there are 40 equivalence classes of pairs of disjoint simple OA(1536, 12, 2, 7) (equivalently, equitable partitions of the 12-cube with quotient matrix [[2, 6, 4], [6, 2, 4], [6, 6, 0]]) and discuss the existence of a non-simple OA(1536, 12, 2, 7). |
| Author | Krotov, Denis S. |
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| Cites_doi | 10.1109/SFCS.1992.267760 10.1016/j.disc.2019.111659 10.1007/3-540-28991-7 10.1214/aoms/1177699280 10.3103/S0027132210030101 10.1002/jcd.20236 10.1007/s11202-007-0075-4 10.37236/8557 10.1016/j.jsc.2013.09.003 10.46586/tosc.v2023.i3.213-226 10.1109/TIT.1984.1056949 10.1007/3-540-44495-5_3 10.1109/ISIT.2002.1023737 10.1007/3-540-46766-1_6 10.2307/2983576 |
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| Keywords | Equitable partitions 94D10 Perfect colorings Correlation-immune boolean functions 05B15 Intriguing sets Orthogonal arrays |
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| References | Fon-Der-Flaass, D.G.: Perfect colorings of the 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$12$$\end{document}-cube that attain the bound on correlation immunity. Sib. Èlektron. Mat. Izv. 4, 292–295 (2007). In Russian. English translation: arXiv:1403.8091 McKay, B.D., Piperno, A.: Practical graph isomorphism. II. J. Symb. Comput. 60, 94–112 (2014). https://doi.org/10.1016/j.jsc.2013.09.003 Vorobev, K.V., Fon-Der-Flaass, D.G.: On perfect 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2$$\end{document}-colorings of the hypercube. Sib. Èlektron. Mat. Izv. 7, 65–75 (2010). In Russian, with English abstract Fon-Der-Flaass, D.G.: A bound on correlation immunity. Sib. Èlektron. Mat. Izv., 4, 133–135 (2007). http://mi.mathnet.ru/eng/semr149 Kaski, P., Östergård, P.R.J.: Classification Algorithms for Codes and Designs, volume 15 of Algorithms Comput. Math. Springer, Berlin (2006). https://doi.org/10.1007/3-540-28991-7 Krotov, D.S., Vorob’ev, K.V.: On unbalanced Boolean functions with best correlation immunity. Electr. J. Comb. 27(1):#P1.45(1–24) (2020). https://doi.org/10.37236/8557 KhalyavinAVEstimates of the capacity of orthogonal arrays of large strengthMosc. Univ. Math. Bull.2010653130131296197110.3103/S0027132210030101 Kirienko, D.: On new infinite family of high order correlation immune unbalanced Boolean functions. In: Proceedings 2002 IEEE International Symposium on Information Theory, Lausanne, Switzerland, June 30 – July 5, 2002, page 465. IEEE (2002). https://doi.org/10.1109/ISIT.2002.1023737 SchoenEDEendebakPTNguyenMVMComplete enumeration of pure-level and mixed-level orthogonal arraysJ. Comb. Des.2010182123140260463810.1002/jcd.20236 SeidenEZemachROn orthogonal arraysAnn. Math. Stat.19663751355137010.1214/aoms/1177699280 Kaski, P., Pottonen, O.: libexact user’s guide, version 1.0. Technical Report 2008-1, Helsinki Institute for Information Technology HIIT (2008) Fon-Der-Flaass, D.G.: Perfect 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2$$\end{document}-colorings of a hypercube. Sib. Math. J. 48(4), 740–745 (2007). translated from Sib. Mat. Zh. 48(4), 923-930 (2007) https://doi.org/10.1007/s11202-007-0075-4 Krotov, D.S.: On the OA(1536,13,2,7) and related orthogonal arrays. Discrete Math. 343(2), 111659/1–11 (2020). https://doi.org/10.1016/j.disc.2019.111659 Rao, C.R.: Factorial experiments derivable from combinatorial arrangements of arrays. J. R. Stat. Soc., Suppl. 9(1), 128–139 (1947). https://doi.org/10.2307/2983576 Rasoolzadeh, S.: Classification of all t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document}-resilient boolean functions with t+4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t+4$$\end{document} variables. IACR Trans. Symmetric Cryptology 2023(3), 213–226 (2023). https://doi.org/10.46586/tosc.v2023.i3.213-226 Camion, P., Carlet, C., Charpin, P., Sendrier, N.: On correlation-immune functions. In: Joan Feigenbaum, editor. Advances in Cryptology — CRYPTO ’91. volume 576 of Lect. Notes Comput. Sci., pages 86–100. Springer, Berlin, Heidelberg (1992). https://doi.org/10.1007/3-540-46766-1_6 Friedman, J.: On the bit extraction problem. In: Foundations of Computer Science, IEEE Annual Symposium on, pages 314–319, Los Alamitos, CA, USA. IEEE Computer Society (1992). https://doi.org/10.1109/SFCS.1992.267760 Tarannikov, Y.: On resilient Boolean functions with maximal possible nonlinearity. Cryptology ePrint Archive 2000/005 (2000). https://eprint.iacr.org/2000/005 SiegenthalerTCorrelation-immunity of nonlinear combining functions for cryptographic applicationsIEEE Trans. Inf. Theory198430577678078128310.1109/TIT.1984.1056949 716_CR11 716_CR12 AV Khalyavin (716_CR8) 2010; 65 716_CR10 T Siegenthaler (716_CR17) 1984; 30 716_CR1 716_CR13 716_CR14 716_CR19 716_CR18 ED Schoen (716_CR15) 2010; 18 E Seiden (716_CR16) 1966; 37 716_CR3 716_CR2 716_CR5 716_CR4 716_CR7 716_CR6 716_CR9 |
| References_xml | – reference: Kirienko, D.: On new infinite family of high order correlation immune unbalanced Boolean functions. In: Proceedings 2002 IEEE International Symposium on Information Theory, Lausanne, Switzerland, June 30 – July 5, 2002, page 465. IEEE (2002). https://doi.org/10.1109/ISIT.2002.1023737 – reference: Fon-Der-Flaass, D.G.: Perfect colorings of the 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$12$$\end{document}-cube that attain the bound on correlation immunity. Sib. Èlektron. Mat. Izv. 4, 292–295 (2007). In Russian. English translation: arXiv:1403.8091 – reference: KhalyavinAVEstimates of the capacity of orthogonal arrays of large strengthMosc. Univ. Math. Bull.2010653130131296197110.3103/S0027132210030101 – reference: McKay, B.D., Piperno, A.: Practical graph isomorphism. II. J. Symb. Comput. 60, 94–112 (2014). https://doi.org/10.1016/j.jsc.2013.09.003 – reference: SchoenEDEendebakPTNguyenMVMComplete enumeration of pure-level and mixed-level orthogonal arraysJ. Comb. Des.2010182123140260463810.1002/jcd.20236 – reference: SeidenEZemachROn orthogonal arraysAnn. Math. Stat.19663751355137010.1214/aoms/1177699280 – reference: Rao, C.R.: Factorial experiments derivable from combinatorial arrangements of arrays. J. R. Stat. Soc., Suppl. 9(1), 128–139 (1947). https://doi.org/10.2307/2983576 – reference: Vorobev, K.V., Fon-Der-Flaass, D.G.: On perfect 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2$$\end{document}-colorings of the hypercube. Sib. Èlektron. Mat. Izv. 7, 65–75 (2010). In Russian, with English abstract – reference: Kaski, P., Östergård, P.R.J.: Classification Algorithms for Codes and Designs, volume 15 of Algorithms Comput. Math. Springer, Berlin (2006). https://doi.org/10.1007/3-540-28991-7 – reference: SiegenthalerTCorrelation-immunity of nonlinear combining functions for cryptographic applicationsIEEE Trans. Inf. Theory198430577678078128310.1109/TIT.1984.1056949 – reference: Krotov, D.S., Vorob’ev, K.V.: On unbalanced Boolean functions with best correlation immunity. Electr. J. Comb. 27(1):#P1.45(1–24) (2020). https://doi.org/10.37236/8557 – reference: Friedman, J.: On the bit extraction problem. In: Foundations of Computer Science, IEEE Annual Symposium on, pages 314–319, Los Alamitos, CA, USA. IEEE Computer Society (1992). https://doi.org/10.1109/SFCS.1992.267760 – reference: Tarannikov, Y.: On resilient Boolean functions with maximal possible nonlinearity. Cryptology ePrint Archive 2000/005 (2000). https://eprint.iacr.org/2000/005 – reference: Fon-Der-Flaass, D.G.: A bound on correlation immunity. Sib. Èlektron. Mat. Izv., 4, 133–135 (2007). http://mi.mathnet.ru/eng/semr149 – reference: Kaski, P., Pottonen, O.: libexact user’s guide, version 1.0. Technical Report 2008-1, Helsinki Institute for Information Technology HIIT (2008) – reference: Krotov, D.S.: On the OA(1536,13,2,7) and related orthogonal arrays. Discrete Math. 343(2), 111659/1–11 (2020). https://doi.org/10.1016/j.disc.2019.111659 – reference: Camion, P., Carlet, C., Charpin, P., Sendrier, N.: On correlation-immune functions. In: Joan Feigenbaum, editor. Advances in Cryptology — CRYPTO ’91. volume 576 of Lect. Notes Comput. Sci., pages 86–100. Springer, Berlin, Heidelberg (1992). https://doi.org/10.1007/3-540-46766-1_6 – reference: Fon-Der-Flaass, D.G.: Perfect 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2$$\end{document}-colorings of a hypercube. Sib. Math. J. 48(4), 740–745 (2007). translated from Sib. Mat. Zh. 48(4), 923-930 (2007) https://doi.org/10.1007/s11202-007-0075-4 – reference: Rasoolzadeh, S.: Classification of all t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document}-resilient boolean functions with t+4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t+4$$\end{document} variables. IACR Trans. Symmetric Cryptology 2023(3), 213–226 (2023). https://doi.org/10.46586/tosc.v2023.i3.213-226 – ident: 716_CR19 – ident: 716_CR5 doi: 10.1109/SFCS.1992.267760 – ident: 716_CR10 doi: 10.1016/j.disc.2019.111659 – ident: 716_CR6 doi: 10.1007/3-540-28991-7 – volume: 37 start-page: 1355 issue: 5 year: 1966 ident: 716_CR16 publication-title: Ann. Math. Stat. doi: 10.1214/aoms/1177699280 – volume: 65 start-page: 130 issue: 3 year: 2010 ident: 716_CR8 publication-title: Mosc. Univ. Math. Bull. doi: 10.3103/S0027132210030101 – volume: 18 start-page: 123 issue: 2 year: 2010 ident: 716_CR15 publication-title: J. Comb. Des. doi: 10.1002/jcd.20236 – ident: 716_CR2 – ident: 716_CR3 doi: 10.1007/s11202-007-0075-4 – ident: 716_CR11 doi: 10.37236/8557 – ident: 716_CR12 doi: 10.1016/j.jsc.2013.09.003 – ident: 716_CR14 doi: 10.46586/tosc.v2023.i3.213-226 – volume: 30 start-page: 776 issue: 5 year: 1984 ident: 716_CR17 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.1984.1056949 – ident: 716_CR18 doi: 10.1007/3-540-44495-5_3 – ident: 716_CR4 – ident: 716_CR7 – ident: 716_CR9 doi: 10.1109/ISIT.2002.1023737 – ident: 716_CR1 doi: 10.1007/3-540-46766-1_6 – ident: 716_CR13 doi: 10.2307/2983576 |
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| Title | Equitable [[2, 10], [6, 6]]-partitions of the 12-cube |
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