Modified Patterson–Wiedemann construction
The Patterson–Wiedemann (PW) construction, which is defined for an odd number n of variables with n being the product of two distinct prime numbers p and q , can be interpreted as idempotent functions which are represented by the ( d , r )-interleaved sequences formed by all-zero and all-one column...
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| creator | Kavut, Selçuk |
| description | The Patterson–Wiedemann (PW) construction, which is defined for an odd number
n
of variables with
n
being the product of two distinct prime numbers
p
and
q
, can be interpreted as idempotent functions which are represented by the (
d
,
r
)-interleaved sequences formed by all-zero and all-one columns, where
r
=
(
2
p
-
1
)
(
2
q
-
1
)
and
d
=
(
2
n
-
1
)
r
. We here study a modified form of the PW construction, which only requires
2
n
-
1
(
=
d
r
)
be a composite number, by relaxing the constraint on the values of
d
and
r
. We first elaborate on the case
n
=
15
and consider the functions corresponding to the (217, 151)-interleaved sequences. Taking into account those satisfying
f
(
α
)
=
f
(
α
2
k
)
for all
α
∈
F
2
n
in this scenario, where
k
is a fixed divisor of
n
, we obtain Boolean functions with nonlinearity 16268 exceeding the bent concatenation bound. Then we extend our study for the case
n
=
11
and obtain Boolean functions with nonlinearity 996 represented by the (89, 23)-interleaved sequences, which equals the best known nonlinearity result. In the process, we show that there is the possibility to exceed the best known nonlinearities using the functions corresponding to those interleaved sequences. |
| doi_str_mv | 10.1007/s10623-023-01248-y |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3015442515</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3015442515</sourcerecordid><originalsourceid>FETCH-LOGICAL-c270t-8a56789e31fdb1f1755028e8b82924101f299f41e0fdce75f6341cbed50084073</originalsourceid><addsrcrecordid>eNp9kM1KAzEUhYMoWKsv4KrgUqL33iSTzFKKf1DRheIyTGcSmWKTmkwX3fkOvqFP4gwjuHNxOHD5zrlwGDtFuEAAfZkRChIcBiFJw3d7bIJKC66VKfbZBEpSHIHokB3lvAIAFEATdv4Qm9a3rpk9VV3nUo7h-_PrtT-4dRXCrI4hd2lbd20Mx-zAV-_Znfz6lL3cXD_P7_ji8fZ-frXgNWnouKlUoU3pBPpmiR61UkDGmaWhkiQCeipLL9GBb2qnlS-ExHrpGgVgJGgxZWdj7ybFj63LnV3FbQr9SysAlZSkUPUUjVSdYs7JebtJ7bpKO4tgh1HsOIqFQcModteHxBjKPRzeXPqr_if1A9pbZIU</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3015442515</pqid></control><display><type>article</type><title>Modified Patterson–Wiedemann construction</title><source>SpringerLink Journals - AutoHoldings</source><creator>Kavut, Selçuk</creator><creatorcontrib>Kavut, Selçuk</creatorcontrib><description>The Patterson–Wiedemann (PW) construction, which is defined for an odd number
n
of variables with
n
being the product of two distinct prime numbers
p
and
q
, can be interpreted as idempotent functions which are represented by the (
d
,
r
)-interleaved sequences formed by all-zero and all-one columns, where
r
=
(
2
p
-
1
)
(
2
q
-
1
)
and
d
=
(
2
n
-
1
)
r
. We here study a modified form of the PW construction, which only requires
2
n
-
1
(
=
d
r
)
be a composite number, by relaxing the constraint on the values of
d
and
r
. We first elaborate on the case
n
=
15
and consider the functions corresponding to the (217, 151)-interleaved sequences. Taking into account those satisfying
f
(
α
)
=
f
(
α
2
k
)
for all
α
∈
F
2
n
in this scenario, where
k
is a fixed divisor of
n
, we obtain Boolean functions with nonlinearity 16268 exceeding the bent concatenation bound. Then we extend our study for the case
n
=
11
and obtain Boolean functions with nonlinearity 996 represented by the (89, 23)-interleaved sequences, which equals the best known nonlinearity result. In the process, we show that there is the possibility to exceed the best known nonlinearities using the functions corresponding to those interleaved sequences.</description><identifier>ISSN: 0925-1022</identifier><identifier>EISSN: 1573-7586</identifier><identifier>DOI: 10.1007/s10623-023-01248-y</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Boolean ; Boolean functions ; Coding and Cryptography 2022 ; Coding and Information Theory ; Computer Science ; Cryptology ; Discrete Mathematics in Computer Science ; Nonlinearity ; Prime numbers</subject><ispartof>Designs, codes, and cryptography, 2024-03, Vol.92 (3), p.653-666</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-8a56789e31fdb1f1755028e8b82924101f299f41e0fdce75f6341cbed50084073</cites><orcidid>0000-0002-9460-1418</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10623-023-01248-y$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10623-023-01248-y$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Kavut, Selçuk</creatorcontrib><title>Modified Patterson–Wiedemann construction</title><title>Designs, codes, and cryptography</title><addtitle>Des. Codes Cryptogr</addtitle><description>The Patterson–Wiedemann (PW) construction, which is defined for an odd number
n
of variables with
n
being the product of two distinct prime numbers
p
and
q
, can be interpreted as idempotent functions which are represented by the (
d
,
r
)-interleaved sequences formed by all-zero and all-one columns, where
r
=
(
2
p
-
1
)
(
2
q
-
1
)
and
d
=
(
2
n
-
1
)
r
. We here study a modified form of the PW construction, which only requires
2
n
-
1
(
=
d
r
)
be a composite number, by relaxing the constraint on the values of
d
and
r
. We first elaborate on the case
n
=
15
and consider the functions corresponding to the (217, 151)-interleaved sequences. Taking into account those satisfying
f
(
α
)
=
f
(
α
2
k
)
for all
α
∈
F
2
n
in this scenario, where
k
is a fixed divisor of
n
, we obtain Boolean functions with nonlinearity 16268 exceeding the bent concatenation bound. Then we extend our study for the case
n
=
11
and obtain Boolean functions with nonlinearity 996 represented by the (89, 23)-interleaved sequences, which equals the best known nonlinearity result. In the process, we show that there is the possibility to exceed the best known nonlinearities using the functions corresponding to those interleaved sequences.</description><subject>Boolean</subject><subject>Boolean functions</subject><subject>Coding and Cryptography 2022</subject><subject>Coding and Information Theory</subject><subject>Computer Science</subject><subject>Cryptology</subject><subject>Discrete Mathematics in Computer Science</subject><subject>Nonlinearity</subject><subject>Prime numbers</subject><issn>0925-1022</issn><issn>1573-7586</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kM1KAzEUhYMoWKsv4KrgUqL33iSTzFKKf1DRheIyTGcSmWKTmkwX3fkOvqFP4gwjuHNxOHD5zrlwGDtFuEAAfZkRChIcBiFJw3d7bIJKC66VKfbZBEpSHIHokB3lvAIAFEATdv4Qm9a3rpk9VV3nUo7h-_PrtT-4dRXCrI4hd2lbd20Mx-zAV-_Znfz6lL3cXD_P7_ji8fZ-frXgNWnouKlUoU3pBPpmiR61UkDGmaWhkiQCeipLL9GBb2qnlS-ExHrpGgVgJGgxZWdj7ybFj63LnV3FbQr9SysAlZSkUPUUjVSdYs7JebtJ7bpKO4tgh1HsOIqFQcModteHxBjKPRzeXPqr_if1A9pbZIU</recordid><startdate>20240301</startdate><enddate>20240301</enddate><creator>Kavut, Selçuk</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-9460-1418</orcidid></search><sort><creationdate>20240301</creationdate><title>Modified Patterson–Wiedemann construction</title><author>Kavut, Selçuk</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-8a56789e31fdb1f1755028e8b82924101f299f41e0fdce75f6341cbed50084073</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Boolean</topic><topic>Boolean functions</topic><topic>Coding and Cryptography 2022</topic><topic>Coding and Information Theory</topic><topic>Computer Science</topic><topic>Cryptology</topic><topic>Discrete Mathematics in Computer Science</topic><topic>Nonlinearity</topic><topic>Prime numbers</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kavut, Selçuk</creatorcontrib><collection>CrossRef</collection><jtitle>Designs, codes, and cryptography</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kavut, Selçuk</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Modified Patterson–Wiedemann construction</atitle><jtitle>Designs, codes, and cryptography</jtitle><stitle>Des. Codes Cryptogr</stitle><date>2024-03-01</date><risdate>2024</risdate><volume>92</volume><issue>3</issue><spage>653</spage><epage>666</epage><pages>653-666</pages><issn>0925-1022</issn><eissn>1573-7586</eissn><abstract>The Patterson–Wiedemann (PW) construction, which is defined for an odd number
n
of variables with
n
being the product of two distinct prime numbers
p
and
q
, can be interpreted as idempotent functions which are represented by the (
d
,
r
)-interleaved sequences formed by all-zero and all-one columns, where
r
=
(
2
p
-
1
)
(
2
q
-
1
)
and
d
=
(
2
n
-
1
)
r
. We here study a modified form of the PW construction, which only requires
2
n
-
1
(
=
d
r
)
be a composite number, by relaxing the constraint on the values of
d
and
r
. We first elaborate on the case
n
=
15
and consider the functions corresponding to the (217, 151)-interleaved sequences. Taking into account those satisfying
f
(
α
)
=
f
(
α
2
k
)
for all
α
∈
F
2
n
in this scenario, where
k
is a fixed divisor of
n
, we obtain Boolean functions with nonlinearity 16268 exceeding the bent concatenation bound. Then we extend our study for the case
n
=
11
and obtain Boolean functions with nonlinearity 996 represented by the (89, 23)-interleaved sequences, which equals the best known nonlinearity result. In the process, we show that there is the possibility to exceed the best known nonlinearities using the functions corresponding to those interleaved sequences.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10623-023-01248-y</doi><tpages>14</tpages><orcidid>https://orcid.org/0000-0002-9460-1418</orcidid></addata></record> |
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| subjects | Boolean Boolean functions Coding and Cryptography 2022 Coding and Information Theory Computer Science Cryptology Discrete Mathematics in Computer Science Nonlinearity Prime numbers |
| title | Modified Patterson–Wiedemann construction |
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