Li­ne­ar­ly Self-Equi­va­lent APN Per­mu­ta­ti­ons in Small Di­men­si­on

Chris­tof Bei­er­le, Mar­cus Brink­mann, Gre­gor Le­an­der

IEEE Tran­sac­tions on In­for­ma­ti­on Theo­ry


Ab­stract

All al­most per­fect non­line­ar (APN) per­mu­ta­ti­ons that we know to date admit a spe­cial kind of li­ne­ar self-equi­va­lence, i.e., there exists a per­mu­ta­ti­on G in their CCZ-equi­va­lence class and two li­ne­ar per­mu­ta­ti­ons A and B, such that G⃘A = B⃘G. After pro­vi­ding a sur­vey on the known APN func­tions with a focus on the exis­tence of self-equi­va­len­ces, we se­arch for APN per­mu­ta­ti­ons in di­men­si­on 6, 7, and 8 that admit such a li­ne­ar self-equi­va­lence. In di­men­si­on six, we were able to con­duct an ex­haus­ti­ve se­arch and ob­tain that there is only one such APN per­mu­ta­ti­on up to CCZ-equi­va­lence. In di­men­si­ons 7 and 8, we per­for­med an ex­haus­ti­ve se­arch for all but a few clas­ses of li­ne­ar self-equi­va­len­ces and we did not find any new APN per­mu­ta­ti­on. As one in­te­res­ting re­sult in di­men­si­on 7, we ob­tain that all APN per­mu­ta­ti­on po­ly­no­mi­als with co­ef­fi­ci­ents in F2 must be (up to CCZ-equi­va­lence) mo­no­mi­al func­tions.

[DOI] [PDF]

Tags: APN per­mu­ta­ti­ons, au­to­mor­phism, CCZ-equi­va­lence, dif­fe­ren­ti­al cryp­t­ana­ly­sis, ex­haus­ti­ve se­arch, self-equi­va­lence