Classification of Bent Monomials, Constructions of Bent Multinomials and Upper Bounds on the Nonlinearity of Vectorial Functions
This paper is composed of two main parts related to the nonlinearity of vectorial functions. The first part is devoted to maximally nonlinear (n, m) functions (the so-called bent vectorial functions), which contribute to an optimal resistance to both linear and differential attacks on symmetric cryp...
Gespeichert in:
| Veröffentlicht in: | IEEE transactions on information theory 2018-01, Vol.64 (1), p.367-383 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext bestellen |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 383 |
|---|---|
| container_issue | 1 |
| container_start_page | 367 |
| container_title | IEEE transactions on information theory |
| container_volume | 64 |
| creator | Yuwei Xu Carlet, Claude Mesnager, Sihem Chuankun Wu |
| description | This paper is composed of two main parts related to the nonlinearity of vectorial functions. The first part is devoted to maximally nonlinear (n, m) functions (the so-called bent vectorial functions), which contribute to an optimal resistance to both linear and differential attacks on symmetric cryptosystems. They can be used in block ciphers at the cost of additional diffusion/compression/expansion layers, or as building blocks for the construction of substitution boxes (S-boxes), and they are also useful for constructing robust codes and algebraic manipulation detection codes. A main issue on bent vectorial functions is to characterize bent monomial functions Tr m n (λx d ) from F 2 n to F 2 m (where m is a divisor of n) leading to a classification of those bent monomials. We also treat the case of functions with multiple trace terms involving general results and explicit constructions. Furthermore, we investigate some open problems raised by Pasalic et al. and Muratovic-Ribic et al. in a series of papers on vectorial functions. The second part is devoted to the nonlinearity of (n, m)-functions. No tight upper bound is known when n/2 |
| doi_str_mv | 10.1109/TIT.2017.2750663 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_RIE</sourceid><recordid>TN_cdi_ieee_primary_8031071</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><ieee_id>8031071</ieee_id><sourcerecordid>2174499639</sourcerecordid><originalsourceid>FETCH-LOGICAL-c325t-508ca95a16b80c0252553ad217444fe875661581e5b7f110ba0688a3153176163</originalsourceid><addsrcrecordid>eNpFkc1LJDEQxcPiwo66d8FLwJNgj6lO5-uog18w7l5GryHTk8ZIm4xJWpibf_qmtwc9FVX5vVehHkInQOYARF2uHlbzmoCY14IRzukPNAPGRKU4aw7QjBCQlWoa-QsdpvRa2oZBPUOfi96k5DrXmuyCx6HD19Zn_Bh8eHOmTxd4EXzKcWjH9_QNDH12ewYbv8FP262N-DoMflMoj_OLxX-C7523Jrq8G5XPts0hFgm-HfxkeIx-dsXC_t7XI_R0e7Na3FfLv3cPi6tl1dKa5YoR2RrFDPC1JC2pWc0YNZsaRNM0nZWCcQ5MgmVr0ZWDrA3hUhoKjILgwOkROp98X0yvt9G9mbjTwTh9f7XU44xQxaQS9QcU9mxitzG8DzZl_RqG6Mv39P-FSnGqCkUmqo0hpWi7L1sgesxEl0z0mIneZ1Ikp5PEWWu_cEkoEAH0HwrKhuQ</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2174499639</pqid></control><display><type>article</type><title>Classification of Bent Monomials, Constructions of Bent Multinomials and Upper Bounds on the Nonlinearity of Vectorial Functions</title><source>IEEE Electronic Library (IEL)</source><creator>Yuwei Xu ; Carlet, Claude ; Mesnager, Sihem ; Chuankun Wu</creator><creatorcontrib>Yuwei Xu ; Carlet, Claude ; Mesnager, Sihem ; Chuankun Wu</creatorcontrib><description>This paper is composed of two main parts related to the nonlinearity of vectorial functions. The first part is devoted to maximally nonlinear (n, m) functions (the so-called bent vectorial functions), which contribute to an optimal resistance to both linear and differential attacks on symmetric cryptosystems. They can be used in block ciphers at the cost of additional diffusion/compression/expansion layers, or as building blocks for the construction of substitution boxes (S-boxes), and they are also useful for constructing robust codes and algebraic manipulation detection codes. A main issue on bent vectorial functions is to characterize bent monomial functions Tr m n (λx d ) from F 2 n to F 2 m (where m is a divisor of n) leading to a classification of those bent monomials. We also treat the case of functions with multiple trace terms involving general results and explicit constructions. Furthermore, we investigate some open problems raised by Pasalic et al. and Muratovic-Ribic et al. in a series of papers on vectorial functions. The second part is devoted to the nonlinearity of (n, m)-functions. No tight upper bound is known when n/2 <; m <; n. The covering radius bound is the only known upper bound in this range (the Sidelnikov- Chabaud-Vaudenay bound coincides with it when m = n - 1 and it has no sense when m <; n - 1). Finding better bounds is an open problem since the 1990s. Moreover, no bound has been found during the last 23 years, which improve upon the covering radius bound for a large part of (n, m)-functions. We derive such upper bounds for functions, which are sufficiently unbalanced or which satisfy some conditions. These upper bounds imply some necessary conditions for vectorial functions to have large nonlinearity.</description><identifier>ISSN: 0018-9448</identifier><identifier>EISSN: 1557-9654</identifier><identifier>DOI: 10.1109/TIT.2017.2750663</identifier><identifier>CODEN: IETTAW</identifier><language>eng</language><publisher>New York: IEEE</publisher><subject>Algorithms ; bent functions ; block cipher ; Boolean functions ; Boxes ; Ciphers ; Classification ; Computer systems ; covering radius bound ; Cryptography ; Diffusion barriers ; Diffusion layers ; Electronic mail ; Encryption ; Mathematics ; Nonlinearity ; Resistance ; Robustness ; s-boxes ; Symmetric cryptography ; Upper bound ; Upper bounds ; vectorial boolean functions</subject><ispartof>IEEE transactions on information theory, 2018-01, Vol.64 (1), p.367-383</ispartof><rights>Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2018</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c325t-508ca95a16b80c0252553ad217444fe875661581e5b7f110ba0688a3153176163</citedby><cites>FETCH-LOGICAL-c325t-508ca95a16b80c0252553ad217444fe875661581e5b7f110ba0688a3153176163</cites><orcidid>0000-0003-4008-2031 ; 0000-0002-6118-7927 ; 0000-0002-5044-3534</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://ieeexplore.ieee.org/document/8031071$$EHTML$$P50$$Gieee$$H</linktohtml><link.rule.ids>230,314,777,781,793,882,27905,27906,54739</link.rule.ids><linktorsrc>$$Uhttps://ieeexplore.ieee.org/document/8031071$$EView_record_in_IEEE$$FView_record_in_$$GIEEE</linktorsrc><backlink>$$Uhttps://hal.science/hal-03958972$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Yuwei Xu</creatorcontrib><creatorcontrib>Carlet, Claude</creatorcontrib><creatorcontrib>Mesnager, Sihem</creatorcontrib><creatorcontrib>Chuankun Wu</creatorcontrib><title>Classification of Bent Monomials, Constructions of Bent Multinomials and Upper Bounds on the Nonlinearity of Vectorial Functions</title><title>IEEE transactions on information theory</title><addtitle>TIT</addtitle><description>This paper is composed of two main parts related to the nonlinearity of vectorial functions. The first part is devoted to maximally nonlinear (n, m) functions (the so-called bent vectorial functions), which contribute to an optimal resistance to both linear and differential attacks on symmetric cryptosystems. They can be used in block ciphers at the cost of additional diffusion/compression/expansion layers, or as building blocks for the construction of substitution boxes (S-boxes), and they are also useful for constructing robust codes and algebraic manipulation detection codes. A main issue on bent vectorial functions is to characterize bent monomial functions Tr m n (λx d ) from F 2 n to F 2 m (where m is a divisor of n) leading to a classification of those bent monomials. We also treat the case of functions with multiple trace terms involving general results and explicit constructions. Furthermore, we investigate some open problems raised by Pasalic et al. and Muratovic-Ribic et al. in a series of papers on vectorial functions. The second part is devoted to the nonlinearity of (n, m)-functions. No tight upper bound is known when n/2 <; m <; n. The covering radius bound is the only known upper bound in this range (the Sidelnikov- Chabaud-Vaudenay bound coincides with it when m = n - 1 and it has no sense when m <; n - 1). Finding better bounds is an open problem since the 1990s. Moreover, no bound has been found during the last 23 years, which improve upon the covering radius bound for a large part of (n, m)-functions. We derive such upper bounds for functions, which are sufficiently unbalanced or which satisfy some conditions. These upper bounds imply some necessary conditions for vectorial functions to have large nonlinearity.</description><subject>Algorithms</subject><subject>bent functions</subject><subject>block cipher</subject><subject>Boolean functions</subject><subject>Boxes</subject><subject>Ciphers</subject><subject>Classification</subject><subject>Computer systems</subject><subject>covering radius bound</subject><subject>Cryptography</subject><subject>Diffusion barriers</subject><subject>Diffusion layers</subject><subject>Electronic mail</subject><subject>Encryption</subject><subject>Mathematics</subject><subject>Nonlinearity</subject><subject>Resistance</subject><subject>Robustness</subject><subject>s-boxes</subject><subject>Symmetric cryptography</subject><subject>Upper bound</subject><subject>Upper bounds</subject><subject>vectorial boolean functions</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNpFkc1LJDEQxcPiwo66d8FLwJNgj6lO5-uog18w7l5GryHTk8ZIm4xJWpibf_qmtwc9FVX5vVehHkInQOYARF2uHlbzmoCY14IRzukPNAPGRKU4aw7QjBCQlWoa-QsdpvRa2oZBPUOfi96k5DrXmuyCx6HD19Zn_Bh8eHOmTxd4EXzKcWjH9_QNDH12ewYbv8FP262N-DoMflMoj_OLxX-C7523Jrq8G5XPts0hFgm-HfxkeIx-dsXC_t7XI_R0e7Na3FfLv3cPi6tl1dKa5YoR2RrFDPC1JC2pWc0YNZsaRNM0nZWCcQ5MgmVr0ZWDrA3hUhoKjILgwOkROp98X0yvt9G9mbjTwTh9f7XU44xQxaQS9QcU9mxitzG8DzZl_RqG6Mv39P-FSnGqCkUmqo0hpWi7L1sgesxEl0z0mIneZ1Ikp5PEWWu_cEkoEAH0HwrKhuQ</recordid><startdate>201801</startdate><enddate>201801</enddate><creator>Yuwei Xu</creator><creator>Carlet, Claude</creator><creator>Mesnager, Sihem</creator><creator>Chuankun Wu</creator><general>IEEE</general><general>The Institute of Electrical and Electronics Engineers, Inc. (IEEE)</general><general>Institute of Electrical and Electronics Engineers</general><scope>97E</scope><scope>RIA</scope><scope>RIE</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7SP</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>1XC</scope><orcidid>https://orcid.org/0000-0003-4008-2031</orcidid><orcidid>https://orcid.org/0000-0002-6118-7927</orcidid><orcidid>https://orcid.org/0000-0002-5044-3534</orcidid></search><sort><creationdate>201801</creationdate><title>Classification of Bent Monomials, Constructions of Bent Multinomials and Upper Bounds on the Nonlinearity of Vectorial Functions</title><author>Yuwei Xu ; Carlet, Claude ; Mesnager, Sihem ; Chuankun Wu</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c325t-508ca95a16b80c0252553ad217444fe875661581e5b7f110ba0688a3153176163</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Algorithms</topic><topic>bent functions</topic><topic>block cipher</topic><topic>Boolean functions</topic><topic>Boxes</topic><topic>Ciphers</topic><topic>Classification</topic><topic>Computer systems</topic><topic>covering radius bound</topic><topic>Cryptography</topic><topic>Diffusion barriers</topic><topic>Diffusion layers</topic><topic>Electronic mail</topic><topic>Encryption</topic><topic>Mathematics</topic><topic>Nonlinearity</topic><topic>Resistance</topic><topic>Robustness</topic><topic>s-boxes</topic><topic>Symmetric cryptography</topic><topic>Upper bound</topic><topic>Upper bounds</topic><topic>vectorial boolean functions</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Yuwei Xu</creatorcontrib><creatorcontrib>Carlet, Claude</creatorcontrib><creatorcontrib>Mesnager, Sihem</creatorcontrib><creatorcontrib>Chuankun Wu</creatorcontrib><collection>IEEE All-Society Periodicals Package (ASPP) 2005-present</collection><collection>IEEE All-Society Periodicals Package (ASPP) 1998-Present</collection><collection>IEEE Electronic Library (IEL)</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Electronics & Communications Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>IEEE transactions on information theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Yuwei Xu</au><au>Carlet, Claude</au><au>Mesnager, Sihem</au><au>Chuankun Wu</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Classification of Bent Monomials, Constructions of Bent Multinomials and Upper Bounds on the Nonlinearity of Vectorial Functions</atitle><jtitle>IEEE transactions on information theory</jtitle><stitle>TIT</stitle><date>2018-01</date><risdate>2018</risdate><volume>64</volume><issue>1</issue><spage>367</spage><epage>383</epage><pages>367-383</pages><issn>0018-9448</issn><eissn>1557-9654</eissn><coden>IETTAW</coden><abstract>This paper is composed of two main parts related to the nonlinearity of vectorial functions. The first part is devoted to maximally nonlinear (n, m) functions (the so-called bent vectorial functions), which contribute to an optimal resistance to both linear and differential attacks on symmetric cryptosystems. They can be used in block ciphers at the cost of additional diffusion/compression/expansion layers, or as building blocks for the construction of substitution boxes (S-boxes), and they are also useful for constructing robust codes and algebraic manipulation detection codes. A main issue on bent vectorial functions is to characterize bent monomial functions Tr m n (λx d ) from F 2 n to F 2 m (where m is a divisor of n) leading to a classification of those bent monomials. We also treat the case of functions with multiple trace terms involving general results and explicit constructions. Furthermore, we investigate some open problems raised by Pasalic et al. and Muratovic-Ribic et al. in a series of papers on vectorial functions. The second part is devoted to the nonlinearity of (n, m)-functions. No tight upper bound is known when n/2 <; m <; n. The covering radius bound is the only known upper bound in this range (the Sidelnikov- Chabaud-Vaudenay bound coincides with it when m = n - 1 and it has no sense when m <; n - 1). Finding better bounds is an open problem since the 1990s. Moreover, no bound has been found during the last 23 years, which improve upon the covering radius bound for a large part of (n, m)-functions. We derive such upper bounds for functions, which are sufficiently unbalanced or which satisfy some conditions. These upper bounds imply some necessary conditions for vectorial functions to have large nonlinearity.</abstract><cop>New York</cop><pub>IEEE</pub><doi>10.1109/TIT.2017.2750663</doi><tpages>17</tpages><orcidid>https://orcid.org/0000-0003-4008-2031</orcidid><orcidid>https://orcid.org/0000-0002-6118-7927</orcidid><orcidid>https://orcid.org/0000-0002-5044-3534</orcidid></addata></record> |
| fulltext | fulltext_linktorsrc |
| identifier | ISSN: 0018-9448 |
| ispartof | IEEE transactions on information theory, 2018-01, Vol.64 (1), p.367-383 |
| issn | 0018-9448 1557-9654 |
| language | eng |
| recordid | cdi_ieee_primary_8031071 |
| source | IEEE Electronic Library (IEL) |
| subjects | Algorithms bent functions block cipher Boolean functions Boxes Ciphers Classification Computer systems covering radius bound Cryptography Diffusion barriers Diffusion layers Electronic mail Encryption Mathematics Nonlinearity Resistance Robustness s-boxes Symmetric cryptography Upper bound Upper bounds vectorial boolean functions |
| title | Classification of Bent Monomials, Constructions of Bent Multinomials and Upper Bounds on the Nonlinearity of Vectorial Functions |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-18T18%3A38%3A25IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_RIE&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Classification%20of%20Bent%20Monomials,%20Constructions%20of%20Bent%20Multinomials%20and%20Upper%20Bounds%20on%20the%20Nonlinearity%20of%20Vectorial%20Functions&rft.jtitle=IEEE%20transactions%20on%20information%20theory&rft.au=Yuwei%20Xu&rft.date=2018-01&rft.volume=64&rft.issue=1&rft.spage=367&rft.epage=383&rft.pages=367-383&rft.issn=0018-9448&rft.eissn=1557-9654&rft.coden=IETTAW&rft_id=info:doi/10.1109/TIT.2017.2750663&rft_dat=%3Cproquest_RIE%3E2174499639%3C/proquest_RIE%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2174499639&rft_id=info:pmid/&rft_ieee_id=8031071&rfr_iscdi=true |