Interplay between resiliency and polynomial degree — Recursive amplification, higher order sensitivity and beyond
Boolean functions are important primitives in different domains of cryptology, complexity and coding theory, and far beyond in different areas of science and technology. In this paper we connect the tools of cryptology and complexity theory in the domain of resilient Boolean functions. It is well kn...
Uloženo v:
| Vydáno v: | Discrete Applied Mathematics Ročník 365; s. 138 - 159 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier B.V
15.04.2025
|
| Témata: | |
| ISSN: | 0166-218X |
| On-line přístup: | Získat plný text |
| Tagy: |
Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
|
| Abstract | Boolean functions are important primitives in different domains of cryptology, complexity and coding theory, and far beyond in different areas of science and technology. In this paper we connect the tools of cryptology and complexity theory in the domain of resilient Boolean functions. It is well known that the resiliency of a Boolean function and its polynomial degree are directly connected. We first show that borrowing an idea from complexity theory, one can implement resilient Boolean functions on a large number of variables with little amount of circuit. Further, we also look into the search techniques used in the construction of resilient Boolean functions to show the existence and non-existence results of functions with low polynomial degree and high sensitivity on small number of variables. In the process, we settle some previously open problems. Finally, we extend the notion of sensitivity to higher order and present a construction with low polynomial degree and higher order sensitivity exploiting Maiorana-McFarland functions. The questions we raise identify novel combinatorial problems in the domain of Boolean functions. |
|---|---|
| AbstractList | Boolean functions are important primitives in different domains of cryptology, complexity and coding theory, and far beyond in different areas of science and technology. In this paper we connect the tools of cryptology and complexity theory in the domain of resilient Boolean functions. It is well known that the resiliency of a Boolean function and its polynomial degree are directly connected. We first show that borrowing an idea from complexity theory, one can implement resilient Boolean functions on a large number of variables with little amount of circuit. Further, we also look into the search techniques used in the construction of resilient Boolean functions to show the existence and non-existence results of functions with low polynomial degree and high sensitivity on small number of variables. In the process, we settle some previously open problems. Finally, we extend the notion of sensitivity to higher order and present a construction with low polynomial degree and higher order sensitivity exploiting Maiorana-McFarland functions. The questions we raise identify novel combinatorial problems in the domain of Boolean functions. |
| Author | Mukherjee, Chandra Sekhar Tang, Deng Maitra, Subhamoy Stănică, Pantelimon |
| Author_xml | – sequence: 1 givenname: Subhamoy surname: Maitra fullname: Maitra, Subhamoy email: subho@isical.ac.in organization: Applied Statistics Unit, Indian Statistical Institute, 203 B T Road, Kolkata 700108, India – sequence: 2 givenname: Chandra Sekhar surname: Mukherjee fullname: Mukherjee, Chandra Sekhar email: chandrasekhar.mukherjee07@gmail.com organization: University of Southern California, USA – sequence: 3 givenname: Pantelimon surname: Stănică fullname: Stănică, Pantelimon email: pstanica@nps.edu organization: Naval Postgraduate School, Applied Mathematics Department, Monterey, CA 93943, USA – sequence: 4 givenname: Deng orcidid: 0000-0002-8373-9200 surname: Tang fullname: Tang, Deng email: dtang@foxmail.com organization: Shanghai Jiao Tong University, Shanghai 200240, China |
| BookMark | eNp9kE1OwzAQRr0oEm3hAOx8ABrsJI0TsUIVP5UqISGQ2FmOPWmnSuzIDkXZcQhOyElwVdZsZjQjvU8zb0Ym1lkg5IqzhDNe3OwTo7okZekyYTxhLJuQadwXi5SX7-dkFsKeMcbjNCVhbQfwfatGWsPwCWCph4AtgtUjVdbQ3rWjdR2qlhrYegD68_VNX0B_-IAHoKrrW2xQqwGdvaY73O7AU-dNrAFswAEPOJyyahidNRfkrFFtgMu_PidvD_evq6fF5vlxvbrbLHSaV8PC6KYQTbYUVS2MqPMSSqFFVlaF0su0zOo0NzmrFW9ElcdnRKZTDVlRGA5C8SqbE37K1d6F4KGRvcdO-VFyJo-m5F5GU_JoSjIuo6nI3J4YiIcdELwM-ugCDHrQgzQO_6F_ARnIeIA |
| Cites_doi | 10.1007/978-3-642-22006-7_28 10.1109/TIT.2004.840902 10.1007/BF01263419 10.1090/S0002-9939-96-03732-X 10.1145/2488608.2488721 10.1145/3106234 10.1109/TIT.2007.894696 10.1007/3-540-48405-1_13 10.1007/PL00009809 10.1016/S0304-3975(01)00144-X 10.1109/ISIT.2014.6875090 10.1109/TC.2003.1190582 10.1016/j.jcss.2003.11.002 10.1109/CCC.2012.35 |
| ContentType | Journal Article |
| Copyright | 2025 Elsevier B.V. |
| Copyright_xml | – notice: 2025 Elsevier B.V. |
| DBID | AAYXX CITATION |
| DOI | 10.1016/j.dam.2025.01.003 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EndPage | 159 |
| ExternalDocumentID | 10_1016_j_dam_2025_01_003 S0166218X25000034 |
| GroupedDBID | -~X ADEZE AFTJW ALMA_UNASSIGNED_HOLDINGS FDB OAUVE AAYXX AI. CITATION FA8 VH1 WUQ |
| ID | FETCH-LOGICAL-c249t-dcf67f3579b7d7b48e87c73896ac5283b24d40ba1f79421873c2ce366d1e7a193 |
| ISICitedReferencesCount | 0 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=001399200400001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0166-218X |
| IngestDate | Sat Nov 29 08:19:22 EST 2025 Sat Mar 01 15:46:03 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | 06E30 Sensitivity 94D10 Separation Maiorana-McFarland construction 94A60 Resiliency Polynomial degree Higher order sensitivity |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c249t-dcf67f3579b7d7b48e87c73896ac5283b24d40ba1f79421873c2ce366d1e7a193 |
| ORCID | 0000-0002-8373-9200 |
| PageCount | 22 |
| ParticipantIDs | crossref_primary_10_1016_j_dam_2025_01_003 elsevier_sciencedirect_doi_10_1016_j_dam_2025_01_003 |
| PublicationCentury | 2000 |
| PublicationDate | 2025-04-15 |
| PublicationDateYYYYMMDD | 2025-04-15 |
| PublicationDate_xml | – month: 04 year: 2025 text: 2025-04-15 day: 15 |
| PublicationDecade | 2020 |
| PublicationTitle | Discrete Applied Mathematics |
| PublicationYear | 2025 |
| Publisher | Elsevier B.V |
| Publisher_xml | – name: Elsevier B.V |
| References | A. Ambainis, Superlinear advantage for exact quantum algorithms, in: Proceedings of the forty-fifth annual ACM symposium on Theory of Computing, STOC’13, 2013, pp. 891–900. R. O’Donnell, J. Wright, Y. Zhou, The Fourier entropy–influence conjecture for certain classes of Boolean functions, in: Proc. of Automata, Languages and Programming – 38th International Colloquium, 2011, pp. 330–341. S. Maitra, P. Sarkar, Highly Nonlinear Resilient Functions Optimizing Siegenthaler’s Inequality, in: Adv. Crypt. — CRYPTO’ 99. CRYPTO 1999, in: LNCS 1666, Springer, Berlin, Heidelberg, pp. 198–215 Buhrman, De Wolf (b3) 2002; 288 Qichun Wang, Chik How Tan, Cryptographic Boolean functions with a large number of variables, in: 2014 IEEE International Symposium on Information Theory, Honolulu, HI, USA, June 29 - July 4, 2014, pp. 1534–1538. Mansour (b12) 1994 Ambainis, Balodis, Belovs, Lee, Santha, Smotrovs (b2) 2017; 64 G. Kindler, R. O’Donnell, Gaussian noise sensitivity and Fourier tails, in: Proceedings of the 27th Annual Computational Complexity Conference, 2012, pp. 137–147. . Sarkar, Maitra (b18) 2003; 52 Klivans, O’Donnell, Servedio (b10) 2004; 68 Carlet, Charpin (b4) 2005; 51 Nisan, Szegedy (b13) 1994; 4 Dillon (b5) 1974 Friedgut, Kalai (b7) 1996; 124 Sarkar, Maitra (b17) 2000 Friedgut (b6) 1998; 18 Kavut, Maitra, Yucel (b8) 2007; 53 Nisan, Wigderson (b14) 1995; 15 O’Donnell (b15) 2014 Sarkar (10.1016/j.dam.2025.01.003_b17) 2000 Mansour (10.1016/j.dam.2025.01.003_b12) 1994 10.1016/j.dam.2025.01.003_b16 10.1016/j.dam.2025.01.003_b9 Buhrman (10.1016/j.dam.2025.01.003_b3) 2002; 288 Ambainis (10.1016/j.dam.2025.01.003_b2) 2017; 64 Sarkar (10.1016/j.dam.2025.01.003_b18) 2003; 52 Dillon (10.1016/j.dam.2025.01.003_b5) 1974 10.1016/j.dam.2025.01.003_b11 Klivans (10.1016/j.dam.2025.01.003_b10) 2004; 68 10.1016/j.dam.2025.01.003_b1 Nisan (10.1016/j.dam.2025.01.003_b14) 1995; 15 Friedgut (10.1016/j.dam.2025.01.003_b6) 1998; 18 Friedgut (10.1016/j.dam.2025.01.003_b7) 1996; 124 Nisan (10.1016/j.dam.2025.01.003_b13) 1994; 4 O’Donnell (10.1016/j.dam.2025.01.003_b15) 2014 10.1016/j.dam.2025.01.003_b19 Kavut (10.1016/j.dam.2025.01.003_b8) 2007; 53 Carlet (10.1016/j.dam.2025.01.003_b4) 2005; 51 |
| References_xml | – volume: 52 start-page: 410 year: 2003 end-page: 417 ident: b18 article-title: Efficient implementation of cryptographically useful “large” Boolean functions publication-title: IEEE Trans. Comp. – volume: 51 start-page: 562 year: 2005 end-page: 571 ident: b4 article-title: Cubic Boolean functions with highest resiliency publication-title: IEEE Trans. Inform. Theory – volume: 68 start-page: 808 year: 2004 end-page: 840 ident: b10 article-title: Learning intersections and thresholds of half-spaces publication-title: J. Comput. System Sci. – volume: 64 start-page: 1 year: 2017 end-page: 24 ident: b2 article-title: Separations in query complexity based on pointer functions publication-title: J. ACM – reference: Qichun Wang, Chik How Tan, Cryptographic Boolean functions with a large number of variables, in: 2014 IEEE International Symposium on Information Theory, Honolulu, HI, USA, June 29 - July 4, 2014, pp. 1534–1538. – reference: S. Maitra, P. Sarkar, Highly Nonlinear Resilient Functions Optimizing Siegenthaler’s Inequality, in: Adv. Crypt. — CRYPTO’ 99. CRYPTO 1999, in: LNCS 1666, Springer, Berlin, Heidelberg, pp. 198–215, – reference: . – year: 2014 ident: b15 article-title: Analysis of Boolean Functions – start-page: 391 year: 1994 end-page: 424 ident: b12 article-title: Learning Boolean functions via the Fourier transform publication-title: Theoretical Advances in Neural Computation and Learning – volume: 53 start-page: 1743 year: 2007 end-page: 1751 ident: b8 article-title: Search for Boolean functions with excellent profiles in the rotation symmetric class publication-title: IEEE Trans. Inform. Theory – reference: G. Kindler, R. O’Donnell, Gaussian noise sensitivity and Fourier tails, in: Proceedings of the 27th Annual Computational Complexity Conference, 2012, pp. 137–147. – reference: R. O’Donnell, J. Wright, Y. Zhou, The Fourier entropy–influence conjecture for certain classes of Boolean functions, in: Proc. of Automata, Languages and Programming – 38th International Colloquium, 2011, pp. 330–341. – start-page: 485 year: 2000 end-page: 506 ident: b17 article-title: Construction of nonlinear Boolean functions with important cryptographic properties publication-title: Adv. Crypt. - EUROCRYPT 2000 – volume: 18 start-page: 27 year: 1998 end-page: 36 ident: b6 article-title: Boolean functions with low average sensitivity depend on few coordinates publication-title: Combinatorica – volume: 124 start-page: 2293 year: 1996 end-page: 3002 ident: b7 article-title: Every monotone graph property has a sharp threshold publication-title: Proc. Amer. Math. Soc. – volume: 288 start-page: 21 year: 2002 end-page: 43 ident: b3 article-title: Complexity measures and decision tree complexity: a survey publication-title: Theoret. Comput. Sci. – volume: 4 start-page: 301 year: 1994 end-page: 313 ident: b13 article-title: On the degree of Boolean functions as real polynomials publication-title: Comput. Complexity – reference: A. Ambainis, Superlinear advantage for exact quantum algorithms, in: Proceedings of the forty-fifth annual ACM symposium on Theory of Computing, STOC’13, 2013, pp. 891–900. – year: 1974 ident: b5 article-title: Elementary Hadamard Difference Sets – volume: 15 start-page: 557 year: 1995 end-page: 565 ident: b14 article-title: On rank vs. communication complexity publication-title: Comb. – ident: 10.1016/j.dam.2025.01.003_b16 doi: 10.1007/978-3-642-22006-7_28 – volume: 51 start-page: 562 issue: 2 year: 2005 ident: 10.1016/j.dam.2025.01.003_b4 article-title: Cubic Boolean functions with highest resiliency publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.2004.840902 – volume: 4 start-page: 301 year: 1994 ident: 10.1016/j.dam.2025.01.003_b13 article-title: On the degree of Boolean functions as real polynomials publication-title: Comput. Complexity doi: 10.1007/BF01263419 – year: 2014 ident: 10.1016/j.dam.2025.01.003_b15 – volume: 124 start-page: 2293 issue: 10 year: 1996 ident: 10.1016/j.dam.2025.01.003_b7 article-title: Every monotone graph property has a sharp threshold publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-96-03732-X – ident: 10.1016/j.dam.2025.01.003_b1 doi: 10.1145/2488608.2488721 – volume: 64 start-page: 1 issue: 5 year: 2017 ident: 10.1016/j.dam.2025.01.003_b2 article-title: Separations in query complexity based on pointer functions publication-title: J. ACM doi: 10.1145/3106234 – volume: 53 start-page: 1743 issue: 5 year: 2007 ident: 10.1016/j.dam.2025.01.003_b8 article-title: Search for Boolean functions with excellent profiles in the rotation symmetric class publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.2007.894696 – ident: 10.1016/j.dam.2025.01.003_b11 doi: 10.1007/3-540-48405-1_13 – volume: 18 start-page: 27 issue: 1 year: 1998 ident: 10.1016/j.dam.2025.01.003_b6 article-title: Boolean functions with low average sensitivity depend on few coordinates publication-title: Combinatorica doi: 10.1007/PL00009809 – start-page: 485 year: 2000 ident: 10.1016/j.dam.2025.01.003_b17 article-title: Construction of nonlinear Boolean functions with important cryptographic properties – volume: 288 start-page: 21 issue: 1 year: 2002 ident: 10.1016/j.dam.2025.01.003_b3 article-title: Complexity measures and decision tree complexity: a survey publication-title: Theoret. Comput. Sci. doi: 10.1016/S0304-3975(01)00144-X – volume: 15 start-page: 557 year: 1995 ident: 10.1016/j.dam.2025.01.003_b14 article-title: On rank vs. communication complexity publication-title: Comb. – ident: 10.1016/j.dam.2025.01.003_b19 doi: 10.1109/ISIT.2014.6875090 – volume: 52 start-page: 410 issue: 4 year: 2003 ident: 10.1016/j.dam.2025.01.003_b18 article-title: Efficient implementation of cryptographically useful “large” Boolean functions publication-title: IEEE Trans. Comp. doi: 10.1109/TC.2003.1190582 – volume: 68 start-page: 808 issue: 4 year: 2004 ident: 10.1016/j.dam.2025.01.003_b10 article-title: Learning intersections and thresholds of half-spaces publication-title: J. Comput. System Sci. doi: 10.1016/j.jcss.2003.11.002 – year: 1974 ident: 10.1016/j.dam.2025.01.003_b5 – start-page: 391 year: 1994 ident: 10.1016/j.dam.2025.01.003_b12 article-title: Learning Boolean functions via the Fourier transform – ident: 10.1016/j.dam.2025.01.003_b9 doi: 10.1109/CCC.2012.35 |
| SSID | ssj0001218 ssj0000186 ssj0006644 |
| Score | 2.4132175 |
| Snippet | Boolean functions are important primitives in different domains of cryptology, complexity and coding theory, and far beyond in different areas of science and... |
| SourceID | crossref elsevier |
| SourceType | Index Database Publisher |
| StartPage | 138 |
| SubjectTerms | Higher order sensitivity Maiorana-McFarland construction Polynomial degree Resiliency Sensitivity Separation |
| Title | Interplay between resiliency and polynomial degree — Recursive amplification, higher order sensitivity and beyond |
| URI | https://dx.doi.org/10.1016/j.dam.2025.01.003 |
| Volume | 365 |
| WOSCitedRecordID | wos001399200400001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 issn: 0166-218X databaseCode: AIEXJ dateStart: 20211209 customDbUrl: isFulltext: true dateEnd: 99991231 titleUrlDefault: https://www.sciencedirect.com omitProxy: false ssIdentifier: ssj0001218 providerName: Elsevier – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 issn: 0166-218X databaseCode: AIEXJ dateStart: 20220331 customDbUrl: isFulltext: true dateEnd: 99991231 titleUrlDefault: https://www.sciencedirect.com omitProxy: false ssIdentifier: ssj0001218 providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3LbtNAFB1FKYuyQDxFeWkWrCiWMvY87GVFiwCpFRJBys4az4wVp6lTJW7U7vgIPohv4Uu48_CDQiW6YGNZVjJO5h7NPffOvWcQep1qmqQqZZGQZRxRXopIpjyLFAXnzrRWkih32IQ4OUlns-zzaPSj7YXZLkVdp5eX2fl_NTU8A2Pb1tlbmLsbFB7APRgdrmB2uP6T4X0Z4VJedTVYEFFXy8r1WDpZgNXyyjYj2-0ZA-E2sMxQ8UAti7Tpg63Zl7bUvAwZPWuIua8IcVqd-xtb9x4OnrBjFq4RZsh0DytYkICRdzz3uBOI7Wj8sawad86RXb_m8mzVlyRfnMLLFr5KyHZA6LWEde10Lrti4i-NI8JJXSl_4wmxbYupzvrigmlIiB-a4KNDiiNmdrfGN3m2WU_OI-Ais-GynXA2WHiJ14gJPpx4lfE_3IPPVCzgTVaEIPaKrZOk94Xt_v81F9kVLrY1cYschsjtEPmE5E5udicWLEvHaOfg49Hs00DCzOrz7bZJv36PC7geDcrz_s-1e-6u-vDaL_w7axowoel9dC-EMPjAQ-8BGpn6Ibo7MO8jtOlAiAMIcQ9CDNbEPQixByH--e077uCHf4PfW-zBhx348AB8biwPvsfo6_uj6bsPUTjeI1IQ8zeRViUXZcJEVggtCpqaVCgBBJpLZSWHiphqOikkKcFnwASJRMXKJJxrYoSEwOMJGter2jxFOI5JQbTJSKYURASTAj7HGHCzWBvKTLaH3rTTl597FZf8RlPuIdpOcB5oqKeXOcDo5q89u807nqPdHugv0LhZX5iX6I7aNtVm_Spg6BeSS6jH |
| linkProvider | Elsevier |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Interplay+between+resiliency+and+polynomial+degree+%E2%80%94+Recursive+amplification%2C+higher+order+sensitivity+and+beyond&rft.jtitle=Discrete+Applied+Mathematics&rft.au=Maitra%2C+Subhamoy&rft.au=Mukherjee%2C+Chandra+Sekhar&rft.au=St%C4%83nic%C4%83%2C+Pantelimon&rft.au=Tang%2C+Deng&rft.date=2025-04-15&rft.issn=0166-218X&rft.volume=365&rft.spage=138&rft.epage=159&rft_id=info:doi/10.1016%2Fj.dam.2025.01.003&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_dam_2025_01_003 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0166-218X&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0166-218X&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0166-218X&client=summon |