{"pk":65002,"title":"Equivalences of biprojective almost perfect nonlinear functions","subtitle":null,"abstract":"Two important problems on almost perfect nonlinear (APN) functions are the enumeration and equivalence problems. In this paper, we solve these two problems for any biprojective APN function family by introducing a group theoretic method for those functions. Roughly half of the known APN families of functions on even dimensions are biprojective. By our method, we settle the equivalence problem for all known biprojective APN functions. Furthermore, we give a new family of such functions. Using our method, we count the number of inequivalent APN functions in all known biprojective APN families and show that the new family found in this paper gives exponentially many new inequivalent APN functions. Quite recently, the Taniguchi family of APN functions was shown to contain an exponential number of inequivalent APN functions by Kaspers and Zhou (J. Cryptol. 34(1), 2021) which improved their previous count (J. Comb. Th. A 186, 2022) for the Zhou-Pott family. Our group theoretic method substantially simplifies the work required for proving those results and provides a generic natural method for every family in the large super-class of biprojective APN functions that contains these two family along with many others.\n \nMathematics Subject Classifications: 94A60, 06E30\n \nKeywords: APN function, CCZ-equivalence, biprojective function","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"APN function"},{"word":"CCZ-equivalence"},{"word":"biprojective function"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/8h67z6mh","frozenauthors":[{"first_name":"Faruk","middle_name":"","last_name":"Göloğlu","name_suffix":"","institution":"Department of Algebra, Charles University, Prague, Czech Republic","department":""},{"first_name":"Lukas","middle_name":"","last_name":"Kölsch","name_suffix":"","institution":"Department of Mathematics, University of South Florida, St. Petersburg, FL, U.S.A.","department":""}],"date_submitted":"2025-09-12T04:37:55-07:00","date_accepted":"2025-09-12T04:37:55-07:00","date_published":"2025-09-15T00:00:00-07:00","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/65002/galley/49812/download/"}]}