{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T10:30:57Z","timestamp":1723545057816},"reference-count":26,"publisher":"World Scientific Pub Co Pte Lt","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Found. Comput. Sci."],"published-print":{"date-parts":[[2011,9]]},"abstract":"A function F from Fpn<\/jats:sup><\/jats:sub>to itself is planar if for any [Formula: see text] the function F(x+a)-F(x) is a permutation. CCZ-equivalence is the most general known equivalence relation of functions preserving planar property. This paper considers two possible extensions of CCZ-equivalence for functions over fields of odd characteristics, one proposed by Coulter and Henderson and the other by Budaghyan and Carlet. We show that the second one in fact coincides with CCZ-equivalence, while using the first one we generalize one of the known families of PN functions. In particular, we prove that, for any odd prime p and any positive integers n and m, the indicators of the graphs of functions F and F' from Fpn<\/jats:sup><\/jats:sub>to Fpm<\/jats:sup><\/jats:sub>are CCZ-equivalent if and only if F and F\u2032 are CCZ-equivalent.<\/jats:p>We also prove that, for any odd prime p, CCZ-equivalence of functions from Fpn<\/jats:sup><\/jats:sub>to Fpm<\/jats:sup><\/jats:sub>, is strictly more general than EA-equivalence when n \u2265 3 and m is greater or equal to the smallest positive divisor of n different from 1.<\/jats:p>","DOI":"10.1142\/s0129054111008684","type":"journal-article","created":{"date-parts":[[2011,9,12]],"date-time":"2011-09-12T13:56:16Z","timestamp":1315835776000},"page":"1243-1258","source":"Crossref","is-referenced-by-count":11,"title":["ON ISOTOPISMS OF COMMUTATIVE PRESEMIFIELDS AND CCZ-EQUIVALENCE OF FUNCTIONS"],"prefix":"10.1142","volume":"22","author":[{"given":"LILYA","family":"BUDAGHYAN","sequence":"first","affiliation":[{"name":"Department of Informatics, University of Bergen, PB 7803, 5020 Bergen, Norway"}]},{"given":"TOR","family":"HELLESETH","sequence":"additional","affiliation":[{"name":"Department of Informatics, University of Bergen, PB 7803, 5020 Bergen, Norway"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1952-0047027-4"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1961.11.1"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/s10623-009-9318-7"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/BF00630563"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2007.11.002"},{"key":"rf8","author":"Budaghyan L.","journal-title":"\"Contemporary Mathematics\" of American Mathematical Society"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2008.928275"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2005.864481"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008344232130"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(82)90045-X"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008292303803"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2007.07.007"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1007\/s10623-007-9097-y"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01111042"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1906-1500755-5"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1906-1500764-6"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-35-00112-0"},{"key":"rf24","first-page":"1526","volume":"133","author":"Ding C.","journal-title":"J. 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