{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T08:14:57Z","timestamp":1776327297049,"version":"3.50.1"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2012,10,1]],"date-time":"2012-10-01T00:00:00Z","timestamp":1349049600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012,10,1]]},"abstract":"<jats:title>Abstract.<\/jats:title>\n                  <jats:p>\n                    This paper studies degree 3 Boolean functions in\n                    <jats:italic>n<\/jats:italic>\n                    variables\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0020_d2116c999557bef931786c54169d7a4b.png\"\/>\n                    which are rotation symmetric, that is, invariant under any cyclic shift of the indices of the variables. These rotation symmetric functions have\nbeen extensively studied in the last dozen years or so because of their importance in cryptography. Let\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0020_af1eb97decb097c4f506d2b014175bd7.png\"\/>\n                    denote the (Hamming) weight of the Boolean function\n                    <jats:italic>f<\/jats:italic>\n                    . We extend and simplify results\nof Bileschi, Cusick and Padgett, who gave an algorithm for finding a recursion for the sequence\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0020_1ad517e29f75398f77bbd965d7f0550f.png\"\/>\n                    , where\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0020_a83c30855320fc6b430feae8c864ac2a.png\"\/>\n                    is the cubic rotation symmetric function generated by the monomial\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0020_3f719a0c969346895bce32d0162395b8.png\"\/>\n                    We show that the weights for the function\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0020_30fda1bb4d1c72db561df75305e7861d.png\"\/>\n                    (note this is just the sum of the first\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0020_cff2979f89210f2c15b888c893f5bd66.png\"\/>\n                    terms in the definition of\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0020_3521354d6cdb3d9d13a2020dbceeb008.png\"\/>\n                    )\nsatisfy the same recursion as the weights for\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0020_3521354d6cdb3d9d13a2020dbceeb008.png\"\/>\n                    . We prove the recursion for the weights of\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0020_df03865dc9741688903c894da712cb13.png\"\/>\n                    by a method based on the work by Bileschi\u2013Cusick\u2013Padgett, but\nwe are able to avoid a lot of the complications in their proofs. In particular, we do not need the technical notion of \u201c\n                    <jats:italic>g<\/jats:italic>\n                    terms\u201d or the elaborate proof of similarity of certain matrices.\n                  <\/jats:p>","DOI":"10.1515\/jmc-2011-0020","type":"journal-article","created":{"date-parts":[[2012,7,5]],"date-time":"2012-07-05T02:50:25Z","timestamp":1341456625000},"page":"105-135","source":"Crossref","is-referenced-by-count":16,"title":["Recursive weights for some Boolean functions"],"prefix":"10.1515","volume":"6","author":[{"given":"Alyssa","family":"Brown","sequence":"first","affiliation":[{"name":"Department of Mathematics, University at Buffalo, 244 Math Building, Buffalo, NY 14260, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Thomas W.","family":"Cusick","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University at Buffalo, 244 Math Building, Buffalo, NY 14260, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2012,10,5]]},"container-title":["Journal of Mathematical Cryptology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2011-0020\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2011-0020\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:30:55Z","timestamp":1764981055000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2011-0020\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,1]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2012,10,5]]},"published-print":{"date-parts":[[2012,10,1]]}},"alternative-id":["10.1515\/jmc-2011-0020"],"URL":"https:\/\/doi.org\/10.1515\/jmc-2011-0020","relation":{},"ISSN":["1862-2984","1862-2976"],"issn-type":[{"value":"1862-2984","type":"electronic"},{"value":"1862-2976","type":"print"}],"subject":[],"published":{"date-parts":[[2012,10,1]]}}}