Relationbetweeno-equivalenceandEA-equivalenceforNihobentfunctionsD - PDF document

Relationbetweeno-equivalenceandEA-equiva
Relationbetweeno-equivalenceandEA-equivalenceforNihobentfunctionsDianaDavidovay,LilyaBudaghyan;yClaudeCarletz,yTorHelleseth;yFerdinandIhringerx,TimPenttila{AbstractBooleanfunctions,andbentfunctionsinparticular,areconsidereduptoso-calledEA-equivalence,whichisthemostgeneralknownequiv-alencerelationpreservingbentnessoffunctions.However,foraspecialtypeofbentfunctions,so-calledNihobentfunctionsthereisamoregen-eralequivalencerelationcalledo-equivalencewhichisinducedfromtheequivalenceofo-polynomials.Inthepresentworkwestudy,foragiveno-polynomial,ageneralconstructionwhichprovidesallpossibleo-equivalentNihobentfunctions,andweconsiderablysimplifyittoaformwhichexcludesEA-equivalentcases.Thatis,weidentifyallcaseswhichcanpotentiallyleadtopairwiseEA-inequivalentNihobentfunctionsderivedfromo-equivalenceofanygivenNihobentfunction.Furthermore,wede-termineallpairwiseEA-inequivalentNihobentfunctionsarisingfromallknowno-polynomialsviao-equivalence.Keywords:Bentfunction,Booleanfunction,EA-equivalence,maxi-mumnonlinearity,Magicaction,modiedMagicaction,Nihobentfunc-tion,o-equivalence,o-polynomials,ovals,hyperovals,Walshtransform.1IntroductionBooleanfunctionsofnvariablesarebinaryfunctionsoverthevectorspaceFn2ofallbinaryvectorsoflengthn,andcanbeviewedasfunctionsovertheGaloiseldF2n,thankstothechoiceofabasisofF2noverF2.Inthispaper,weshallalwayshavethislastviewpoint.Booleanfunctionsareusedinthepseudo-randomgeneratorsofstreamciphersandplayacentralroleintheirsecurity.Bentfunctions,introducedbyRothaus[35]in1976,areBooleanfunctionshavinganevennumberofvariablesn,thataremaximallynonlinearinthesensethattheirnonlinearity,theminimumHammingdistancetoallanefunctions,SomeresultsofthispaperwerepresentedatIrsee2014conference,BFA2018andBFA2019workshops.yDepartmentofInformatics,UniversityofBergen,PB7803,N-5020Bergen,Norway,e-mail:fDiana.Davidova,Lilya.Budaghyan,Tor.Hellesethg@uib.nozDepartmentofMathematics,UniversitiesofParis8andParis13,2ruedelaliberte,93526Saint-DenisCedex,France,e-mail:claude.carlet@univ-paris8.frxDepartmentofMathematics:Analysis,LogicandDiscreteMathematics,GhentUniver-sity,Belgium,e-mail:ferdinand.ihringer@ugent.be{DepartmentofMath

ematics,ColoradoStateUniversity,FortColl
ematics,ColoradoStateUniversity,FortCollins,CO80523-1874,e-mail:penttila@math.colostate.edu1isoptimal(formoreinformationonbentfunctionssee,forinstance,[11]).ThiscorrespondstothefactthattheirWalshtransformtakesthevalues2n=2,only.Bentfunctionshaveattractedalotofresearchinterestinmathematicsbecauseoftheirrelationtodierencesetsandtodesigns,andintheapplicationsofmathematicstocomputersciencebecauseoftheirrelationstocodingtheoryandcryptography.Despitetheirsimpleandnaturaldenition,bentfunctionsadmitaverycomplicatedstructureingeneral.Animportantfocusofresearchistondconstructionsofbentfunctions.Manymethodsareknownandsomeofthemallowexplicitconstructions.Wedistinguishbetweenprimaryconstruc-tionsgivingbentfunctionsfromscratchandsecondaryconstructionsbuildingnewbentfunctionsfromoneorseveralgivenbentfunctions(inthesamenumberofvariablesorindierentones).Booleanfunctions,andbentfunctionsinparticular,areconsidereduptoso-calledEA-equivalence,whichisthemostgeneralknownequivalencerelationpreservingbentnessoffunctions[3,4].Bentfunctionsareoftenbetterviewedintheirbivariaterepresentation,intheformf(x;y),wherexandybelongtoFm2ortoF2m,wherem=n=2.ThisrepresentationhasledtothetwogeneralfamiliesofexplicitbentfunctionswhicharetheoriginalMaiorana-McFarland[29]andthePartialSpreads(PSap)classes(thislatterclassisincludedinthemoregeneralbutlessexplicitPSclass).Bentfunctionscanalsobeviewedintheirunivariateform,expressedbymeansofthetracefunctionoverF2n.Findingexplicitbentfunctionsinthistracerepresen-tationisusuallymoredicultthaninthebivariaterepresentation.Referencescontaininginformationonexplicitprimaryconstructionsofbentfunctionsintheirbivariateandunivariateformsare[9,25].ItiswellknownthatsomeoftheseexplicitconstructionsbelongtotheMaiorana-McFarlandclassandtothePSapclass.When,intheearly1970s,Dillonintroducedinhisthesis[17]thetwoabovementionedclasses,healsointroducedanotheronedenotedbyH,wherebentnesswasprovenundersomeconditionswhichwerenotobvioustoachieve.ThismadeclassHanexampleofanon-explicitconstruction:atthattime,Dillonwasabletoexhibitonlyfunctionsbelonging,uptotheaneequivalence(whichisaparticularcaseofEA-equivalence),totheMaiorana-McF

arlandclass.Itwasobservedin[10]thatthecl
arlandclass.Itwasobservedin[10]thattheclassofthe,socalled,Nihobentfunctions(introducedin[18]byDobbertinetal)is,uptoEA-equivalence,equaltotheDillon'sclassH.NotethatfunctionsinclassHaredenedintheirbivariaterepresentationandNihobentfunctionshadoriginallyaunivariateformonly.ThreeinnitefamiliesofNihobinomialbentfunctionswereconstructedin[18]andoneoftheseconstructionswaslatergeneralizedbyLeanderandKholosha[26]intoafunctionwith2rNihoexponents.Anotherclasswasalsoextendedin[20].In[6]itwasproventhatsomeoftheseinnitefamiliesofNihobentfunc-tionsareEA-inequivalenttoanyMaiorana-McFarlandfunctionwhichimpliedthatclassesHandMaiorana-McFarlandaredierentuptoEA-equivalence.Inthesamepaper[10],itwasshownthatNihobentfunctionsdeneo-polynomialsand,conversely,everyo-polynomialdenesaNihobentfunction.Theyalsodiscoveredthatagiveno-polynomialFcanproducetwodierent(uptoEA-equivalence)Nihobentfunctions,namely,theonesderivedfromFanditsinverseF�1.Sincetakingtheinverseofano-polynomialisaparticularcaseoftheequivalenceofo-polynomials,anaturalquestionwastoexplorethisequivalenceforconstructionoffurtherEA-inequivalentcasesofNihobentfunctions.Therstworkinthisdirectionwasdonein[7]wherethegroup2oftransformations(introducedin[14])oforder24preservingtheequivalenceofo-polynomialswasstudiedforrelationtoEA-equivalence.ItwasshownthatthesetransformationscanleadtouptofourEA-inequivalentfunctionsincludingthosederivedfromano-polynomialanditsinverse.Thatis,twonewtransformationswhichcanpotentiallyprovideEA-inequivalentfunctionsfromagiveno-polynomialwerediscovered.Hence,applicationoftheequivalenceofo-polynomialscanbeconsideredasaconstructionmethodfornew(uptoEA-equivalence)Nihobentfunctionsfromtheknownones.Notethatthegroupoftransformationsfrom[14]doesnotcoverallpossi-bletransformationswithinequivalenceofo-polynomials.Amoregeneralgroupoftransformations,so-calledtheMagicaction,waspresentedin[21],whichisanactionofagroupoftransformationsactingonprojectivelineonthesetofo-permutations.InthispaperwestudythemodiedMagicaction,atrans-formationofo-polynomialspreservingprojectiveequivalence.Weshowthato-polynomialsareprojectivelyequivalentifandonlyiftheylie

onthesameorbitunderthemodiedMagicac
onthesameorbitunderthemodiedMagicactionandtheinversemap.Furtherweprovethat,foragiveno-polynomial,EA-inequivalentNihobentfunctionscanariseonlyfromaspecicformulainvolvingparticularcompositionsoftransforma-tionsofthemodiedmagicactionandtheinversemap.Weshowthateacho-monomialcandeneuptofourEA-inequivalentbentfunctions.Weprove,forinstance,thatthePynehyperovalcangiverisetoEA-inequivalentNihobentfunctionsdenedbyo-polynomialswhichlieon3dierentorbitsofthemodiedMagicaction.Foreveryknowno-polynomialweprovideanexplicitnumberofpairwiseEA-inequivalentNihobentfunctionswhichcanbederivedviao-equivalence.Moreover,wegiveanexplicitdescription(involvingtransfor-mationsofthemodiedmagicactionandtheinversemap)ofallo-polynomialsprovidingpairwiseEA-inequivalentNihobentfunctions.Thepaperisorganizedasfollows.InSection2werecallnecessaryback-ground,inSection3wedeneNihobentfunctionsviao-polynomialsandviceversa.InSection4weprovethataneequivalenceofo-polynomialsinsomecasesyieldsEA-equivalenceofthecorrespondingNihobentfunctions.Theknownfactthateveryo-polynomialonF2mnecessarilydenesavectorialNihobentfunctionfromF22mtoF2mcanbeseenasacorollary.InSection5themod-iedmagicactionisintroducedanditisproventhatpotentiallyEA-inequivalentNihobentfunctionscanarisefromo-polynomialswhichlieonthesameorbitunderthemodiedMagicactionandtheinversemap.ThemainresultsofthepaperarecontainedinSections6and7,whereweobtainanexactformoftheorbitonwhicho-polynomialsshouldlietoproducepotentiallyEA-inequivalentNihobentfunctions.Foreachoftheknowno-polynomialsweprovidetheex-plicitnumberandrepresentationsforallequivalento-polynomialswhichprovidepairwiseEA-inequivalentNihobentfunctions.32NotationandPreliminaries2.1TraceRepresentation,BooleanFunctionsinUnivari-ateandBivariateFormsForanypositiveintegerkandanyrdividingk,thetracefunctionTrkristhemappingfromF2ktoF2rdenedbyTrkr(x):=kr�1Xi=0x2ir=x+x2r+x22r+


+x2k�r:Inparticular,theabsolutetraceoverF2kisthefunctionTrk1(x)=Pk�1i=0x2i(inwhatfollows,wejustuseTrktodenotetheabsolutetrace).RecallthatthetracefunctionsatisesthetransitivitypropertyTrk=TrrTrkr.Theunivariaterep

resentationofaBooleanfunctionisdene
resentationofaBooleanfunctionisdenedasfollows:weidentifyFn2(then-dimensionalvectorspaceoverF2)withF2nandconsidertheargumentsoffaselementsinF2n.AninnerproductinF2nisx
y=Trn(xy).ThereexistsauniqueunivariatepolynomialP2n�1i=0aixioverF2nthatrepresentsf(thisistrueforanyvectorialfunctionfromF2ntoitselfandthereforeforanyBooleanfunctionsinceF2isasubeldofF2n).Thealgebraicdegreeoffisequaltothemaximum2-weightoftheexponentsofthosemonomialswithnonzerocoecientsintheunivariaterepresentation,wherethe2-weightw2(i)ofanintegeriisthenumberofonesinitsbinaryexpansion.Moreover,fbeingBoolean,itsunivariaterepresentationcanbewrittenuniquelyintheformoff(x)=Xj2�nTro(j)(ajxj)+a2n�1x2n�1;where�nisthesetofintegersobtainedbychoosingthesmallestelementineachcyclotomiccosetmodulo2n�1(withrespectto2),o(j)isthesizeofthecyclotomiccosetcontainingj,aj2F2o(j)anda2n�12F2.Thefunctionfcanalsobewritteninanon-uniquewayasTrn(P(x))whereP(x)isapolynomialoverF2n.ThebivariaterepresentationofaBooleanfunctionisdenedinthispaperasfollows:weidentifyFn2withF2mF2m(wheren=2m)andconsidertheargumentoffasanorderedpair(x;y)ofelementsinF2m.ThereexistsauniquebivariatepolynomialP0i;j2m�1ai;jxiyjoverF2mthatrepresentsf.Thealgebraicdegreeoffisequaltomax(i;j)jai;j6=0(w2(i)+w2(j)).AndfbeingBoolean,itsbivariaterepresentationcanbewrittenintheformf(x;y)=Trm(P(x;y)),whereP(x;y)issomepolynomialoftwovariablesoverF2m.Remark1.Letg(x;y)beaBooleanfunctionoverF2mF2m.Thenonecangetaunivariaterepresentationofgmakingthefollowingsubstitutions:x=t+t2mandy=t+(t)2m;whereisaprimitiveelementofF22m.2.2WalshTransformandBentFunctionsLetfbeann-variableBooleanfunction.Its\sign"functionistheinteger-valuedfunctionf:=(�1)f.TheWalshtransformoffisthediscreteFourier4transformoffwhosevalueatpointw2F2nisdenedbybf(w)=Xx2F2n(�1)f(x)+Trn(wx):Forevenn,aBooleanfunctionfinnvariablesissaidtobebentifforanyw2F2nwehavebf(w)=2n2.Itiswellknown(see,forinstance,[9])thatthealgebraicdegreeofabentBooleanfunctioninn�2variablesisatmostn2.Bentnessandalgebraicdegree(whenlargerthan1)arepreservedbyextended-ane(EA-)equivalence.TwoBool

eanfunctionsfandginnvariablesarecalledEA
eanfunctionsfandginnvariablesarecalledEA-equivalentifthereexistsananepermutationAofF2nandananeBooleanfunction`suchthatf=gA+`.Ifl=0thenfandgarecalledaneequivalent.Inthecaseofvectorialfunctionsthereexistsamoregeneralnotionofequivalence,calledCCZ-equivalence,butforBooleanfunctions,itreducestoEA-equivalence,see[3](aswellasforbentvectorialfunctions[4]).TwofunctionsFandF0fromF2ntoitselfarecalledEA-equivalentifA1FA2+AforsomeanepermutationsA1andA2andforsomeanefunctionA.IfA=0thenFandF0arecalledaneequivalent.Forpositiveintegersnandt,avectorialBooleanfunctionFfromFn2toFt2iscalledbentifforanya2Fn2nf0gtheBooleanfunctiona
F(x)isbent.Bentfunctionsexistifandonlyifnisevenandtn=2(see[30]).2.3Projectiveplane,Ovals,HyperovalsInthefollowingwegiveashortintroductiontotheprojectiveplane.Wereferto[16]foradetailedintroductiontoprojectivegeometry.AprojectiveplaneconsistsofasetofpointsP,asetoflinesL,andanincidencerelationIbetweenPandL.TheclassicalprojectiveplanePG(2;q)overF3qhasthe1-spacesofF3qaspointsandthe2-spacesofF3qaslines.Apointpiscontainedinaline`ifp`inF3q.Asetofpointsiscalledcollineariftheyalllieonthesameline.NotethatPG(2;q)hasq2+q+1points,q2+q+1lines,eachlinecontainsq+1points,andeachpointliesinq+1lines.ThegroupP�L(3;q)actsnaturallyonPG(2;q).Inparticular,itpreservesincidence.LetObeasetofpointsinPG(2;q)suchthatnothreepointsarecollinear.Itiswell-knownthatjOjq+1ifqisoddandjOjq+2isqiseven.Onecanseethisasfollows:ConsiderapointP2O.Eachoftheq+1linesonPcontainsatmostonemorepoints,sojOjq+2.Supposethatequalityholds.Theneachlinecontainseither0or2points.ConsiderapointR2O.ThenthereareslinesthroughRwith2pointsandq+1�slinesthroughRwith0points.Hence,q+2=2s,soqiseven.Callaline`passant,tangent,respectively,secantifj`\Oj=0,j`\Oj=1,respectively,j`\Oj=2.IfjOj=q+1,thenOiscalledanoval.FromtheargumentaboveitfollowsthatinthiscaseeachpointofOliesonexactlyonetangentandqsecants.ForqeventhesesecantsallmeetinonepointN,thenucleusofO.IfjOj=q+2,thenOiscalledahyperovalandweusuallywriteHinsteadofO.IfjOj=q+1andqeven,thenO[fNgisahyperoval.Inthefollowingwelimitourselvestoq=2meven.AframeofPG(2;q)isasetoffourpointsP=

fP1;P2;P3;P4gsuchthatany3-subsetofPspans
fP1;P2;P3;P4gsuchthatany3-subsetofPspansF3q.Thefundamentaltheoremofprojectivegeometry(for5projectiveplanes)statesthatP�L(3;q)actstransitiveonframes.AsanyfourpointsofahyperovalHareaframe,wecanassumethatanovalOcontainsh(1;0;0)i;h(0;0;1)i;h(1;1;1)i2Oandhash(0;1;0)iasitsnucleus.Inthefollowingweusuallyleaveoutthebracketsh
iforthesakeofreadibility.Hence,wecanwriteOasO=f(x;F(x);1):x2F2mg[f(1;0;0)g;wherethepolynomialFsatisesthefollowing:(a)FisapermutationpolynomialoverF2mofdegreeatmostq�2satisfyingF(0)=0andF(1)=1.(b)Foranys2F2mthefunctionFs(x):=(F(x+s)+F(x)xifx6=0;0otherwise.isapermutationpolynomial.HereandfurtherinthepaperwedenoteF2m=F2mnf0g.SuchapolynomialFiscalledano-polynomialand,conversely,eacho-polynomialdenesanoval.IfwedonotrequireF(1)=1,thenFiscalledano-permutation.WewriteO(F)fortheovaldenedbytheo-polynomialF,andwewriteH(F)forthehyperovaldenedbyF.NotethatthroughoutthispaperOconsistsofpointsoftheform(x;F(x);1),whileinthehyperplaneliterature,usuallytheform(1;x;f(x))isused.ForahyperovalHwehave2m+2choicesforthenucleusN2Htoob-tainanovalHnfNg.Hence,eachhyperovalHdenes2m+2o-polynomials.Twoo-polynomialsarecalled(projectively)equivalent,iftheydeneequivalenthyperovals(underthenaturalactionofP�L(3;q)).2.4NihoBentFunctionsApositiveintegerd(alwaysunderstoodmodulo2n�1withn=2m)isaNihoexponentandt!tdisaNihopowerfunctioniftherestrictionoftdtoF2mislinearor,equivalently,ifd2j(mod2m�1)forsomejn.AsweconsiderTrn(atd)witha2F2n,withoutlossofgenerality,wecanassumethatdisinthenormalizedform,i.e.,withj=0.Thenwehaveauniquerepresentationd=(2m�1)s+1with2s2m.Ifsomesiswrittenasafraction,thishastobeinterpretedmodulo2m+1(e.g.,1=2=2m�1+1).FollowingareexamplesofbentfunctionsconsistingofoneormoreNihoexponents:1.QuadraticfunctionTrm(at2m+1)witha2F2m(heres=2m�1+1).2.Binomialsoftheformf(t)=Trn(1td1+2td2),where2d12m+1(mod2n�1)and1;22F2naresuchthat(1+2m1)2=2m+12.Equivalently,denotinga=(1+2m1)2andb=2wehavea=b2m+12F2mandf(t)=Trm(at2m+1)+Trn(btd2):6Wenotethatifb=0anda6=0thenfisabentfunctionlistedundernumber1.Thepossiblevaluesofd2

are[18,20]:d2=(2m�1)3+1;6d2=(2m�1)
are[18,20]:d2=(2m�1)3+1;6d2=(2m�1)+6(takingmeven):Thesefunctionshavealgebraicdegreemanddonotbelongtothecom-pletedMaiorana-McFarlandclass[6].3.Take1rmwithgcd(r;m)=1anddenef(t)=Trn a2t2m+1+(a+a2m)2r�1�1Xi=1tdi!;(1)where2rdi=(2m�1)i+2randa2F2nissuchthata+a2m6=0[26,27].Thisfunctionhasalgebraicdegreer+1(see[5])andbelongstothecompletedMaiorana-McFarlandclass[12].4.Bentfunctionsinabivariaterepresentationobtainedfromtheknowno-polynomials.Considerthelistedabovetwobinomialbentfunctions.Ifgcd(d2;2n�1)=dandb=dforsome2F2nthenbcanbe\absorbed"inthepowertermtd2byalinearsubstitutionofvariablet.Inthiscase,uptoEA-equivalence,b=a=1.Inparticular,thisappliestoanybwhengcd(d2;2n�1)=1thatholdsinbothcasesexceptwhend2=(2m�1)3+1withm2(mod4)whered=5.Inthisexceptionalcase,wecangetupto5dierentclassesbuttheexactsituationhastobefurtherinvestigated.3ClassHofBentFunctionsando-polynomialsHerewerestrictourselfwitheldsF2nwithneven,n=2m.Inhisthesis[17],DillonintroducedtheclassofbentfunctionsdenotedbyH.Thefunctionsinthisclassaredenedintheirbivariateformasf(x;y)=Trm(y+xF(yx2m�2));wherex;y2F2m,andFisapermutationofF2ms.t.F(x)+xdoesn'tvanish,forany2F2mthefunctionF(x)+xis2-to-1.DillonwasabletoexhibitbentfunctionsinHthatalsobelongtothecompletedMaiorana-McFarlandclass.Dillon'sclassHwasmodiedin[10]intoaclassHofthefunctions:g(x;y)=8:TrmxGyx;ifx6=0Trm(y);otherwise(2)7where2F2m;G:F2m7!F2msatisfyingthefollowingconditions:F:z7!G(z)+zisapermutationoverF2m;(3)z7!F(z)+zis2-to-1onF2mforany2F2m:(4)Herecondition(4)impliescondition(3)anditisnecessaryandsucientforgbeingbent.FunctionsinHandtheDillonclassarethesameuptoadditionofalineartermTrm((+1)y)to(2).NihobentfunctionsarefunctionsinHintheirunivariantrepresentation.Theorem1([10]).ApolynomialFonF2msatisfyingF(0)=0andF(1)=1isano-polynomialifandonlyifz7!F(z)+zis2-to-1onF2mforany2F2m:(5)Hence,obviouslyeveryo-polynomialdenesaNihobentfunction.Andviceversa,everyNihobentfunctiondenesano-polynomialsinceitdenesapolynomialFsatisfyingcondition(5)ofTheorem1,andwecanderiveano-polyn

omialF0(x)=F(x)+F(0)F(1)+F(0)which
omialF0(x)=F(x)+F(0)F(1)+F(0)whichxestherequirementsF0(0)=0andF0(1)=1.NotethattogetaNihobentfunctionfromapolynomialFitissucientthatFsatisesonlycondition(5)whiletheconditionsF(0)=0andF(1)=1arenotnecessary.InSection2.3wesawthateacho-polynomialcorrespondstoahyperovalandviceversa,eachhyperovalcorrespondstoano-polynomial.WesaythatNihobentfunctionsareo-equivalentiftheydeneprojectivelyequivalenthyperovals.Asshownin[7,10],o-equivalentNihobentfunctionsmaybeEA-inequivalent.Forexample,Nihobentfunctionsdenedbyo-polynomialsFandF�1areo-equivalentbuttheyare,ingeneral,EA-inequivalent.Hereisthelistofallknowno-polynomials(wealsogivenamesofthecorre-spondinghyperovals):1.F(x)=x2,regularhyperoval;2.F(x)=x2i,iandmarecoprime,i�1,irregulartranslationhyperoval;3.F(x)=x6,misodd,Segrehyperoval;4.F(x)=x3
2k+4,m=2k�1,GlynnI;5.F(x)=x2k+22k,m=4k�1,GlynnII;6.F(x)=x22k+1+23k+1,m=4k+1,GlynnII;7.F(x)=x2k+x2k+2+x3
2k+4,m=2k�1,Cherowitzohyperoval;8.F(x)=x16+x12+x56,misodd,Paynehyperoval;9.F(x)=2(x4+x)+2(1++2)(x3+x2)x4+2x2+1+x12,whereTrm(1)=1(ifm2(mod4),then=2F4),Subiacohyperoval(form=4alsoknownasLunelli-Scehyperoval);810.F(x)=1Trnm(v)Trnm(vr)(x+1)+(x+Trnm(v)x12+1)1�rTrnm(vx+v2m)r+x12,wheremiseven,r=2m�13,v2F22m;v2m+16=1;v6=1,Adelaidehyperoval.11.F(x)=x4+x16+x28+!11(x6+x10+x14+x18+x22+x26)+!20(x8+x20)+!6(x12+x24)with!5=!2+1andm=5,O'Keefe-Penttilahyperoval.Notethatano-polynomialFdenedonF2mhasthefollowingform[16]:F(x)=2m�22Xk=1b2kx2k:4VectorialNihobentfunctionsfromo-polynomialsItisknownsince2011thateveryo-polynomialdenesaBooleanNihobentfunc-tion[10].Inthissection,werevisitthefactthat,actually,everyo-polynomialonF2mdenesavectorialNihobentfunctionfromF2mF2mtoF2m.Thisconnectionhasbeenoriginallyobservedin[28].Inthepresentpaper,wederivethisresultbystudyingsomesimpletransformationsofo-polynomials.Belowweshowthatinsomecases,aneequivalenceofo-polynomialsyieldsEA-equivalenceofthecorrespondingNihobentfunctions.NotethatingeneralifafunctionF0isaneequivalenttoano-polynomialFthenF0isnotnecessarilyano-polynomial.Lemma1.LetFbeano-polynomi

aldenedonF2manda;b2F2m.ThenG(x)
aldenedonF2manda;b2F2m.ThenG(x)=aF(bx)isano-polynomialonF2mifandonlyifa=1F(b)(or,whatisthesame,b=F�1(a�1)).TheNihobentfunctionsdenedbytheo-polynomialsFandG=1F(b)F(bx)areaneequivalent.Proof.SupposeG(x)=aF(bx)isano-polynomial,thenG(0)=aF(0)=0foranya;b2F2mand1=G(1)=aF(b),henceGisano-polynomialifandonlyifa=1F(b).TheNihobentfunctioncorrespondingtotheo-polynomialFisf(x;y)=Trm(xF(yx)),andtheonecorrespondingtoGisg(x;y)=Trm(xG(yx))=Trm(xaF(byx))=Trm(xaF(abyax))=Trm(vF(uv));wherev=ax,u=aby.Hence,g=fAwithA(x;y)=(ax;aby),and,therefore,fandgareaneequivalent.Corollary1.Foreveryo-polynomialFdenedonF2mthefunctionxF(yx)fromF2mF2mtoF2misbent.Thatis,everyo-polynomialonF2mdenesavectorialNihobentfunctionxF(yx)fromF2mF2mtoF2m.Proof.FromLemma1wehavethatforagiveno-polynomialFandanya2F2mthefunctiong(x;y)=Trm(axF(byx))isNihobentwhereb=F�1(a�1).Thenthefunctiong(x;y)=Trm(axF(yx))isalsobentsincegandgareaneequiva-lent,thatis,g=gAwithA(x;y)=(x;by),andclearly,suchatransformationAkeepsgasaNihofunction.9Lemma2.LetFbeano-polynomialonF2mandA(x)=x2jbeanautomor-phismoverF2m.ThentheNihobentfunctionsdenedbyo-polynomialsFandG=AFA�1areaneequivalent.Proof.ObviouslyifFisano-polynomial,thenG(x)=(F(x2�j))2jisalsoano-polynomial.ConsidertheNihobentfunctiondenedbyG:g(x;y)=TrmxGyx=TrmxAFA�1yx=TrmxFyx2�j2j=Trmx2�jFyx2�j=TrmuFvu;whereu=x2�jandv=y2�j.Thus,fandgareaneequivalent(g=fAwithA(x;y)=(x;y)2�j).Lemma3.LetFbeano-polynomialonF2mandA1(x)=x+aandA2(x)=x+bfora;b2F2m.ThenG=A1FA2isano-polynomialonF2mifandonlyifb=F(a)andF(a+1)+F(a)=1.Furthermore,theNihobentfunctionsdenedbyo-polynomialsFandGareEA-equivalent.Proof.SupposeG(x)=A1FA2(x)=F(x+a)+bisano-polynomial.Then0=G(0)=F(a)+band,therefore,F(a)=band1=G(1)=F(1+a)+b=F(1+a)+F(a).Furtherwehaveg(x;y)=TrmxA1FA2yx=TrmxFyx+a+b=TrmxFy+axx&

#17;+Trm(bx)=TrmxFux
#17;+Trm(bx)=TrmxFux+Trm(bx),whereu=y+ax.Thus,gandfareEA-equivalent(g=fA+lwithA(x;y)=(x;y+ax)andl(x;y)=Trm(bx)).5ThemodiedMagicactionLetFbethecollectionofallfunctionsF:F2m7!F2msuchthatF(0)=0.ThefollowingsetP�L(2;2m)=fx7!Ax2jjA2GL(2;F2m);0jm�1gisagroupoftransformationsactingontheprojectivelines,i.e.onthesetwiththeelementsoftheform:f(a
x;a
y)j(x;y)6=(0;0);x;y2F2m;a6=0g.AnactionofthegroupP�L(2;2m)onFwasintroducedanddescribedin[21].DenetheimageofF2Funderthetransformation 2P�L(2;2m), :x7!Ax2j,A=abcd2GL(2;2m),0jm�1,asafunction F:F2m7!F2msuchthat F(x)=jAj�12h(bx+d)F2jax+cbx+d+bxF2jab+dF2jcdi:ThisyieldsanactionofP�L(2;2m)onF,whichiscalledthemagicaction.Themagicactiontakeso-permutationstoo-permutationsanditisasemi-linear10transformation,i.e. (F+G)= F+ G;foranyF;G2F, aF=a2j F,foranya2F2m,F2F,0jm�1.Letusrecalltwotheorems(Theorem4andTheorem6)from[21].Foragiveno-polynomialFdenoteO(F)theovaldenedbyF.Theorem2.[21]LetFbeano-permutationonF2mandlet 2P�L(2;2m)be :x7!Ax2jforA=abcd2GL(2;F2m)and0jm�1.ThenG= Fisalsoano-permutationonF2m.Infact,O(G)= (O(F)),where 2P�L(3;2m)isdenedby :x7!Ax2j,whereA=0@d0cb F(db)jAj12a F(ca)b0a1A.Notethattheformulationofthetheoremabovediersfromtheonein[21]becauseinthecurrentpaper(followingnotationsof[7])thepointsoftheoval(orthehyperoval)denedbyano-polynomialFareconsideredas(x;F(x);1),meanwhilein[21]theform(1;x;F(x))isused.Theorem3.[21]LetFandGbeo-permutationsonF2m,andsupposefurtherthattheovalsdenedbyFandG,i.e.O(F)andO(G)areequivalentunderP�L(3;2m).Thenthereexists 2P�L(2;2m)suchthatG= F.ThemagicactioncanbealsodescribedbyacollectionofgeneratorsofP�L(2;2m)[21]:a:x7!a001x;aF(x)=a�12F(ax);a2F2m;c:x7!10c1x;cF(x)=F(x+c)+F(c);c2F2m;':x7!0110x;'F(x)=xF(x�1);2j:x7!x2j;2jF(x)=(F(x2�j))2j;0jm�1:(6)Weslightlymodifythemagicactiongeneratorsaandcmultiplyingthembyappropriateconstantstopreservetheimageof1at1:~aF(x)=a1

2F(a)aF(x)=1F(a)F(ax);a2F2m;
2F(a)aF(x)=1F(a)F(ax);a2F2m;~cF(x)=1F(1+c)+F(c)cF(x)=1F(1+c)+F(c)(F(x+c)+F(c));c2F2m:(7)ThenewsetofgeneratorsH=f~a;~c;';2jj0jm�1;c2F2m;a2F2mpreservesthepropertyF(1)=1ofthefunctionF.TheactionofthegroupwiththenewsetofgeneratorsHonthesetofallfunctionsFdenedonF2mwiththepropertiesF(0)=0andF(1)=1willbecalledthemodiedmagicaction.11Proposition1.Twoo-polynomialsarisefromequivalenthyperovalsifandonlyiftheylieonthesameorbitofthegroupgeneratedbyHandtheinversemap.Proof.AccordingtotherstpartofTheorem2,themagicactiontakeso-permutationstoo-permutations.Sincethegeneratorsofthemodiedmagicactiondierfromtheoriginalmagicactiongeneratorsonlybyconstantcoe-cient(whatallowsastopreservethepropertyofF(1)=1foranyo-polynomialF),thenthemodiedmagicactiontakeso-polynomialstoo-polynomials.AccordingtothesecondpartofTheorem2,iftwoo-permutationslieonthesameorbitunderthemagicaction,thenthecorrespondingovalsareequivalentandhavexednucleus(0;1;0).Nowsupposethattwoo-polynomialslieonthesameorbitunderthemodiedmagicactionandtheinversemap.Sinceeacho-polynomialisano-permutation,thenthecorrespondingovalsdenedbyo-polynomialsareequivalentandhavenucleus(0;1;0).Asweknow,eachovaliscontainedinauniquehyperoval,whichisobtainedbyaddingnucleustothepointsofoval.So,hyperovalsdenedbytheo-polynomialsonthesameorbitunderthemodiedmagicactionareequivalent.Alsoitiswellknownthato-polynomialsFandF�1deneequivalenthyperovals.Thus,weconcludethathyperovalsdenedbytheo-polynomialsonthesameorbitunderthemodiedmagicactionandtheinversemapareequivalent.Let'sshowtheconversestatement.SupposethathyperovalsH(F)andH(G)denedbyo-polynomialsFandGareequivalent.ItmeansthatthereisacollineationwhichmapsH(F)toH(G).Considerthepreimageof(0;1;0)underthiscollineation,thereare3possiblecases:1.Thepreimageof(0;1;0)is(0;1;0).Itmeansthatthiscollineationxespoint(0;1;0).SodeletingthispointfromhyperovalsH(F)andH(G),wewillgetequivalentovalswithxednucleus,hencebyTheorem3,theirgeneratoro-polynomialsareonthesameorbitunderthemagicaction,henceunderthemodiedmagicaction.2.Thepreimageof(0;1;0)i

s(1;0;0).Sincehyperovalsdenedbyo-po
s(1;0;0).Sincehyperovalsdenedbyo-polynomialanditsinverseo-polynomialareequivalent,thenhyperovalH(F)isequivalenttoahyperovalH(F�1)andbythecorrespondingcollineationthepoint(1;0;0)haspreimage(0;1;0).So,attheendwehavethathyperovalsH(F�1)andH(G)areequivalentandthepreimageof(0;1;0)is(0;1;0).Hencebythepreviouscase1(andthefactthatano-polynomialanditsinversebelongtothesameorbitundermodiedactionandtheinverse)o-polynomialsFandGareonthesameorbitundermodiedmagicactionandtheinversemap.Thefollowingdiagramillustratesthepreviousdecisions.H(F�1)=H(F)=H(G)222(0;1;0)7!(1;0;0)7!(0;1;0)3.Thepreimageof(0;1;0)is(t;f(t);1).Chooseanelement'ofP�L(2;2m)taking(1;t)to(0;1)(suchauthomorphismalwaysexist,forexampleitcanbedenedbymatrixA=0010).Applying'toFwewillgetahyperovalH('F)equivalenttoH(G)wherethepreimageof(0;1;0)is(1;0;0).Becauseofthecase2,wegetthat'FandGbelongtothesameorbitunderthemodiedmagicactionandtheinversemapandsodoFandG.12Weformulatethenexttheoremwithoutproof.Firstthisresultwasan-nouncedinSeptember2014attheForthIsreeConference"FiniteGeometries"[8]bytheauthorsofthispaper,thecompleteproofcanbefoundin[1].Theorem4.TwoNihobentfunctionsareEA-equivalentifandonlyifthecorrespondingovalsareequivalent.Hence,thenumberofEA-equivalenceclassesofNihobentfunctionsarisingfromahyperovalofPG(2;2m)isthenumberoforbitsofthecollineationstabiliserofthehyperovalonthepointsofthehyperoval.6NihobentfunctionsandthemodiedmagicactionAgroupoftransformationsoforder24with3generatorspreservingo-polynomialswasconsideredin[7].Thisgroupoftransformationsisasubgroupofthegroupwiththe(modied)magicactiongeneratorsandtheinversemap.Precisely,theyarethetransformationsgeneratedby',~1=1andtheinversemap.Only4ofthesetransformationscanleadtoEA-inequivalentNihobentfunctions[7].Asacontinuationoftheworkof[7],let'sconsiderthemodiedmagicactiongenerators,andtheinversemapandseewhichofthemgiverisetoEA-inequivalentNihobentfunctions.FromProposition1itisclearthato-polynomialsonthesameorbitunderthemodiedmagicactionandthein-versemapandonlytheyareprojectivelyequivalent.SinceweareinterestedinEA-inequivalentNihobentfunctionsarising

fromprojectivelyequivalento-polynomials,
fromprojectivelyequivalento-polynomials,wefocusonorbitsofthemodiedmagicactiontogetherwiththeinversemap.WeprovebelowthattogetEA-inequivalentNihobentfunctionsfromagiveno-polynomialitissucienttouseonly~and'generatorstogetherwithinversemapwhileand~donotplayanyroleinit.Moreover,weshowthatallEA-inequivalentNihobentfunctionscanbeobtainedfromaspecialformula.6.1PreliminaryresultsFollowingnotationsof[7]thegenerator'willbedenotedby0whenneeded.Let'srecallthesetofgeneratorsH=f~c;~a;0;2jjc2F2m;a2F2m;0jm�1g;where~aF(x)=1F(a)F(ax);a2F2m;~cF(x)=cFcF(x)=cF(F(x+c)+F(c));c2F2m;wherecF=1cF(1);F0(x)='F(x)=xF(x�1);2jF(x)=(F(x2�j))2j;0jm�1;andproveafewstatementsaboutthegeneratorsofmagicactionandtheinversemap.13Lemma4.LetFbeano-polynomialonF2m.Thenthefollowingidentitieshold:~c~dF=~c+dF;(8)~a~bF=~abF;(9)2j2iF=2j+iF;(10)wherea;b2F2m,c;d2F2m;0i;jm�1.Proof.ToprovetherstequalitynotethatcdF(x)=dF(x+c)+dF(c)=F(x+c+d)+F(d)+F(c+d)+F(d)=F(x+c+d)+F(c+d)=c+dF:Sincemagicactionisasemilineartransformationweget:~c~dF(x)=1F(1+d)+F(d)1~dF(1+c)+~d(c)c(d(F(x))=1F(1+d)+F(d)F(1+d)+F(d)F(1+d+c)+F(d+c)c+dF(x)=1F(1+d+c)+F(d+c)c+dF(x)=~c+dF(x):Theothertwoequalitiesarestraightforwardtoprove:~a~bF=1~bF(a)~bF(ax)=11F(b)F(ab)1F(b)F(abx)=1F(ab)F(abx)=~abF(x);2i2jF(x)=2i(F(x12j))2j=F(x12j+i)2j+i=2i+jF(x):Corollary2.LetFbeano-polynomialonF2mandkapositiveinteger.Then(~a1~a2:::~ak)F=~a1
a2
:::
akF;(~c1~c2:::~ck)F=~c1+c2+:::ckF;(2i12i2:::2ik)F=2i1+i2+:::+ikF;wherea1;:::;ak2F2m;c1;:::;ck2F2m,0ijm�1forallj2f1;:::;kg.Proof.TheprooffollowsbyinductionusingLemma4.Lemma5.LetFbeano-polynomialonFm2.Thenthefollowingidentitieshold:(~cF)�1(x)=~F(c)F�11cFx;(11)(~aF)�1(x)=~F(a)F�1(x);(1

2)(2jF)�1(x)=2jF�1(x);(1
2)(2jF)�1(x)=2jF�1(x);(13)wherea2F2m,c2F2mand0jm�1.14Proof.Itiseasytoseethat~F(c)F�11cF=1,therefore(~cF)�1(x)=(cF(F(x+c)+F(c)))�1=F�11cFx+F(c)+c=F�11cFx+F(c)+F�1(F(c))=~F(c)F�11cFx:Equalities(12)and(13)arestraightforwaredtoprove:(~aF)�1(x)=1F(a)F(ax)�1=1aF�1(F(a)x)=~F(a)F�1(x);(2jF)�1(x)=((F(x2�j))2j)�1=(F(x2�j)�1)2j=2jF�1(x):Lemma6.LetFbeano-polynomialonF2m.Thenthefollowingidentitieshold:~c2jF=2j~c2�jF;(14)~c~aF=~a~acF;(15)(2jF)0=2jF0(16)(~aF)0=~1aF0;(17)wherea2F2m,c2F2m;0jm�1.Proof.Toprovetherstequality,transformitsleftandrightsides.~c2jF(x)=c2jF(2jF(x+c)+2jF(c))=c2jF((F((x+c)2�j))2j+(F(c2�j))2j)=c2jF((F(x2�j+c2�j))2j+(F(c2�j))2j)=c2jF(F(x2�j+c2�j)+F(c2�j))2jOntheotherhand,2j~c2�jF(x)=(~c2�jF(x2�j))2j=(c2�jF(F(x2�j+c2�j)+F(c2�j))2j:So,itislefttocheckthat(c2�jF)2j=c2jF.Indeed,c2jF=12jF(1+c)+2jF(c)=1(F((1+c)2�j))2j+(F(c2�j))2j=1F(1+c2�j)+F(c2�j)2j=(c2�jF)2j:Thusweprovedthat~c2jF=2j~c2�jF.Computingtheleftandtherightsidesofequality(15)weget~c~aF(x)=c~aF(~aF(x+c)+~aF(c))=c~aF(1F(a)F(a(x+c))+1F(a)F(ac));~a~acF(x)=1~acF(a)acF(F(ax+ac)+F(ac)):15Notethatthecoecients1F(a)c~aFand1~acF(a)acFareequalwhichmeansthat~c~aF=~a~acF:Indeed,1F(a)c~aF=1F(a)1~aF(1+c)+~aF(c)=1F(a)F(a)F(a(1+c))+F(ac)=1F(a+ac)+F(ac);1~acF(a)acF=F(1+ac)+F(ac)F(a+ac)+F(ac)1F(1+ac)+F(ac)=1F(a+ac)+F(ac):Theremainingtwoequalitiesareprovedsimilarly.For(16)weget2jF0(x)=(F0(x2�j))2j=(x2�jF(1x2�j))2j=x(F(1x2�j))2j=x2jF(1x)=(2jF)0(x):Transformingbothsi

desofEquality(17)weget(~aF)0(x)=x~&
desofEquality(17)weget(~aF)0(x)=x~aF1x=xF(a)Fax:~1aF0(x)=1F0(1a)F0xa=aF(a)xaFax=xF(a)Fax:6.2EA-inequivalentNihobentfunctionsandorbitsFurtherweneedthefollowingequalityfrom[7]((F0)�1)0=((F�1)0)�1(18)Let'sintroduceafewnotations.DenotebygFtheNihobentfunctiondenedbyano-polynomialF.WhenNihobentfunctionsgFandgFareEA-equivalent(EA-inequivalent),wewillwritegFEAgF(respectively,gFEAgF).Wewillusenotation"A(p)=B",whentheexpressionBisobtainedfromtheexpressionAusingequalitynumberp.Theorem5.LetFbeano-polynomial.Thenano-polynomialFobtainedfromFusingonegeneratorofthemodiedmagicactionandtheinversemapcanproduceaNihobentfunctionEA-inequivalenttothosedenedbyFandF�1onlyifF=(F0)�1.Proof.AssumeFisano-polynomialwhichisobtainedfromo-polynomialFusingonegeneratorofthemodiedmagicactionandtheinversemap,i.e.Fhasoneofthefollowingforms:hF;hF�1;(hF)�1;(hF�1)�1,whereh2H.Asweshowbelow,whenhis~a,~cor2j,FdenesaNihobentfunctionEA-equivalenttothosedenedbyForF�1.a)Lethbe~a;a2F2m.ThenhF(x)=~aF(x)=1F(a)F(ax)andbyLemma1,thecorrespondingNihobentfunctionisEA-equivalenttothosedenedbyF.BythesamereasonhF�1=~aF�1andF�1deneEA-equivalentNihobentfunctions.Furthernotethat(hF)�1(x)=(~aF)�1(x)(12)=~F(a)F�1(x):16Hence,g(~aF)�1EAgF�1and(hF�1)�1(x)=(~aF�1)�1(x)(12)=~F�1(a)(F�1)�1(x)=~F�1(a)F(x);andthereforeg(~aF�1)�1EAgF.b)Supposehis~cwithc2F2m.ThenhF(x)=~cF(x)=cF(F(x+c)+F(c))andhF�1(x)=~cF�1deneNihobentfunctionsEA-equivalenttothosedenedbyFandF�1respectively(byLemma3).Hence,(hF)�1(x)=(~cF(x))�1(x)(11)=F(c)F�1((cF)�1x)yieldsthatg(hF)�1EAgFandfrom(hF�1)�1(x)=(~cF�1)�1(x)(11)=F�1(c)(F�1)�11cF�1x=F�1(c)F1cF�1xfollowsg(hF�1)�1EAgF.c)Takenowh=2jwith0jm�1.ThenhF(x)=2jF(x)=(F(x2�i))2iandhF&#

0;1(x)=2jF�1=(F�1(x2�i))2i
0;1(x)=2jF�1=(F�1(x2�i))2i,andbyLemma2wegetthatg2jFandg2jF�1areEA-equivalenttogFandgF�1,respectively.Therefore,from(hF)�1(x)=(2jF)�1(x)(13)=2jF�1and(hF�1)�1(x)=(2jF�1)�1(13)=2jFitfollowsthatg(2jF)�1EAgF�1andg(2jF�1)�1EAgF.d)Considerh=0.TheNihobentfunctiondenedbyano-polynomialhF(x)=F0(x)=xF(x�1)isgF0(x;y)=Trm(x(F0(yx)))=Trm(xyxF((yx)�1))=Trm(yF(xy))=gF(y;x);i.e.gF0EAgF.Similarly,g(F�1)0EAgF�1.Thefunction(hF)�1(x)=(F0)�1(x)=(xF(x�1))�1candeneaNihobentfunctionEA-inequivalenttothosedenedbyFandF�1.Forexample,ano-monomialx2idenesthreesurelyEA-inequivalentNihobentfunctionscorrespondingtoo-polynomialsF,F�1and(F0)�1[7].Usingequality(18),weimmediatelygetthataNihobentfunctiondenedbytheo-polynomial(hF�1)�1(x)=((F�1)0)�1(x)isEA-equivalenttoonedenedby(F0)�1.WerewritetheequalitiesofLemmas4,5and6inamorecompactway.Equalities(8)-(10)ashb1hb2F=hb3F;(19)wherehb1;hb2;hb3arethesamegeneratorsfromthesetHnf0gwithdierentparametersb1;b2;b32F2m.Equalities(11)-(13)as(hb1F)�1=hb2F�1;(20)wherehb1;hb2arethesamegeneratorsfromthesetHnf0gwithdierentpa-rametersb1;b22F2m.Notethatrightandleftpartsoftheequality(11)have17dierentarguments,butitdoesnotplayanyroleinourstudyofEA-equivalenceofresultingNihobentfunctions.Equalities(14)-(15)as~c1hbF=hb~c2F;(21)wherehb2f~a;2jg:Andequalities(16)-(17)as(hb1F)0=hb2F0;(22)wherehb1;hb2arethesamegeneratorsfromthesetf~a;2jgwithdierentpa-rametersb1;b22F2m.Tomaketheformulationofthenexttheoremmorevisualinsteadofusingthenotation0wewillusetheinitialone,i.e.'.Wewillalsorefertotheoriginalnotation'insomepartsoftheproofwhenconvenient.Further,by"reduceo-polynomial"wemeanthattheoriginalo-polynomialandthenewone(reduced)deneEA-equivalentNihobentfunctions.Whenwearesaying"deletegenerator"wemeanthatifweskipthisgeneratorthenewo-polynomialwilldeneaNihobentfunctionEA-equivalenttoonegeneratedbytheoriginalo-polynomial.Letibeapositiveintegerandki0.ByHiwedenoteacompositionoflengthkiofg

enerators'and~cfollowingeachotheras
enerators'and~cfollowingeachotherasfollows:Hi='~ci1'~ci2:::|{z}ki(23)Thatis,ifFisano-polynomialandwedenoteTj='~cij,0j(ki+1)=2thenHiF=8&#x]TJ/;ø 9;&#x.962; Tf;&#x 17.;ॗ ;� Td;&#x [00;&#x]TJ/;ø 9;&#x.962; Tf;&#x 17.;ॗ ;� Td;&#x [00;&#x]TJ/;ø 9;&#x.962; Tf;&#x 17.;ॗ ;� Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:Fifki=0;'Fifki=1;T1:::TsiFifki=2si;T1:::Tsi'Fifki=2si+1:Inthetheorembelowweprovethatforagiveno-polynomialwecanderiveallEA-inequivalentNihobentfunctionsonlyusingtransformations',~candtheinversemapinaspecialsequence.Theorem6.LetFbeano-polynomial,gFthecorrespondingNihobentfunctionandGFtheclassofallfunctionso-equivalenttogF.Theno-polynomialsoftheform(H1(H2(H3(:::(HqF)�1:::)�1)�1)�1;(24)whereHiisdenedby(23),foralli2f1:::qg,q1,andki1fori3,ki0fori2,providerepresentativesforallEA-equivalenceclasseswithinGF.Thatis,uptoEA-equivalence,allNihobentfunctionso-equivalenttogFarisefrom(24).Proof.NoterstthatwecangetFitselfintheform(24)ifwetakeq=2,k1=k2=0.ifq=1andk1=0thenwegetF�1.Furtherwehavearestrictionki1fori3toavoidrepetitions.18AccordingtoProposition1anyfunctiono-equivalenttogFcorrespondstoano-polynomialoftheformh1h2:::hkF;(25)whereh1;h2;:::;hk(forsomek0)aregeneratorsofthemodiedmagicactionandtheinversemap.OuraimistosimplifythisexpressiontoexcludeasmanycasesleadingtoEA-equivalentfunctionsaspossible.Thatis,weexcludecertainsequencesofgeneratorswhichsurelyleadtoEA-equivalentNihobentfunctions.Byhijwedenoteageneratorofthesametypeashibutwithadierentparameter.FromTheorem5itfollowsa)Ifh12H,thengh1h2:::hkFEAgh2:::hkFandwecanconsiderreducedo-polynomialh2:::hkF;b)Ifh1istheinversemapandh22Hnf0gthengh1h2:::hkFEAgh1h3:::hkF,sowecanconsiderthereducedo-polynomialh1h3:::hkF.Hence,ifk=1in(25)thenwecangetanEA-inequivalentcaseonlyifh1istheinversemap,anditcorrespondsto(24)wi

thq=1andk1=0.Ifk=2in(25)(anditcannotbere
thq=1andk1=0.Ifk=2in(25)(anditcannotbereducedtothecasek=1)thenwecangetEA-inequivalentcasesonlyifh1istheinversemapandh2=0,anditcorrespondsto(24)withq=1andk1=1.Ifk3wecanreduce(25)untilatsomemomentwewillgetano-polynomialhihi+1:::hkF,wherehiistheinversemapandhi+1=0,thatis,wehave((hi+2:::hkF)0)�1:(26)Notethathereandfurtherweassumethatkislargeenoughtoallowsucharedactionwhileotherwise,itiseasytoseethattheprocesswouldstopandprovideaformula(24)forsomeparameters.Ifhi+22f~a;2jgorhi+2istheinversemapthenwecandeletethegeneratorhi+2andconsiderthereducedo-polynomialhihi+1hi+3:::hkF.Indeed,supposehi+22f~a;2jgthenhihi+1hi+2:::hkF=((hi+2:::hkF)0)�1(22)=(h(i+2)1(hi+3:::hkF)0)�1(20)=h(i+2)2((hi+3:::hkF)0)�1and,accordingto(a),ghihi+1hi+2:::hkFEAghihi+1hi+3:::hkF.Inthecasewhenhi+2istheinversemap,using(18)wegetthesameresultthattheo-polynomials(((hi+3:::hkF)�1)0)�1=((hi+3:::hkF)0)�1)0and((hi+3:::hkF)0)�1=hihi+1hi+3:::hkFdeneEA-equivalentNihobentfunctions.Ifhi+2is0,thenhi+1andhi+2eliminateeachother:hihi+1hi+2:::hkF=hihi+3:::hkF.Ifhi+2=~c,thenwecannoteliminateitfromtheo-polynomialhihi+1hi+2:::hkF.Furtherconsiderano-polynomialhihi+1hi+2:::hkFwherehiistheinversemap,hi+1=0,hi+2=~c,i.e.ano-polynomial((~chi+3:::hkF)0)�1:(27)Whenk=i+2thenweget((~cF)0)�1whichhastheform(24)withq=1andk1=2.Hence,in(27)wecanassumethatki+3.Furtherwecanreducehi+3from(27)unlesshi+3is0.Indeed,considerrsthi+32f~a;2jgthen19((~chi+3:::hkF)0)�1(21)=((hi+3~c1hi+4:::hkF)0)�1(22)=(h(i+3)1(~c1:::hkF)0)�1(20)=h(i+3)2((~c1hi+4:::hkF)0)�1.Thelasto-polynomialdenesaNihobentfunctionEA-equivalenttoonedenedbytheo-polynomial((~c1hi+4:::hkF)0)�1=hihi+1h(i+2)1hi+4:::hkF.Ifhi+3

=~c1,thenusing(8)weimmediatelygethi
=~c1,thenusing(8)weimmediatelygethihi+1hi+2hi+3:::hkF=hihi+1h(i+2)1hi+4:::hkF,whereh(i+2)1=~c+c1.Ifhi+3istheinversemapthenhihi+1hi+2hi+3:::hkF=((~c((hi+4:::hkF)�1))0)�1(20)=(((~c1hi+4:::hkF)�1)0)�1=(((~c1hi+4:::hkF)0)�1)0;denesaNihobentfunctionEA-equivalenttotheonedenedby((~c1hi+4:::hkF)0)�1=hihi+1h(i+2)1hi+4:::hkF.Notethatwecouldeliminatehi+3astheinverseherebecauseitisfollowedbyhi+2=~c,hi+1=0andhiastheinversemap.Hence,if(25)producesaNihobentfunctiongEA-inequivalenttothosecorrespondingtoF,F�1,(F0)�1and((~cF)0)�1thengisEA-equivalenttothefunctioncorrespondingtoano-polynomial('~c0'hl0:::hkF)�1:(28)Nowconsiderano-polynomialoftheform:('~c1'~c2':::hl:::hkF)�1:(29)Case1.Firstwerestricttothecasehl;:::;hk2Hwhenconsidering(29).Notethatiflisanevennumberin(29),thenthegenerator'actsonhl;iflisodd,thenthegenerator~cactsonhl(forsomec2F2m).Weconsiderloddcase,i.e.l=2t+1whileforlevencasetheproofissimilarandweskipit.Ifh2t+12f~a;2jgthen('~c1'~c2':::~cth2t+1:::hkF)�1(21)=('~c1':::'h2t+1~ct1(h2t+2:::hkF))�1(22)=('~c1':::h(2t+1)1'(~ct1(h2t+2:::hkF)))�1(21)=:::(h(2t+1)t('(~c11('(:::(~ct1(h2t+2:::hkF)):::))))�1(20)=h(2t+1)t+1('(~c11('(:::(~ct1(h2t+2:::hkF)):::))))�1,hencewecanreducetheo-polynomial('~c1'~c2':::~cth2t+1:::hkF)�1;andconsider('~c11'~c21':::~ct1h2t+2:::hkF)�1.Ifh2t+1=~ct+1thenobviouslywecanconsidero-polynomial((~c1(~c2(:::(~ct+ct+1(h2t+2:::hkF))0:::)0)0)0)�1.Ifh2t+1=0thenwecannoteliminateit.Continuingthisprocesswegetforthiscasethattheo-polynomial(25)canbereduce

dto('~c1'~c2

dto('~c1'~c2':::F)�1asin(23).Thiscorrespondstothecaseq=1in(24).Case2.Nowweconsider(29)andallowhl;:::;hktobeinversestoo.Westillassumelbeoddand(aswesawearlierintheproof)w.l.o.g.hl;:::;hk2f0;~c;theinversejc2F2mg.Takehltheinverse(theotherpossibilitiesforhlwerediscussedearlierintheproof),i.e.considerthefollowingo-polynomial:('~c1'~c2':::(hl+1:::hkF)�1)�1:(30)20Ifhl+1istheinverse,thenitcancelswithhl.Ifhl+1is~ct+1,then('~c1'~c2':::(~ct+1hl+2:::hkF)�1)�1(20)=('~c1'~c2':::~c(t+1)1(hl+2:::hkF)�1)�1,whichisoftheform(30)withfewertransformationsintheinnerbrackets.Ifhl+1is'thenweget('~c1'~c2':::('hl+2:::hkF)�1)�1.Iffurtherhl+2is~ct+1,then('~c1'~c2':::('~ct+1hl+3:::hkF)�1)�1.Ifhl+2istheinverseorhl+2='thenweget(30).Indeed,ifhl+2='thenitcancelswithhl+1,andifhl+2istheinversethenweget:('~c1'~c2':::('(hl+3:::hkF)�1)�1)�1(18)=('~c1'~c2':::'('hl+3:::hkF)�1)�1.Continuingtheseprocesswewillclearlytransform(30)to(24)inawaythattheseo-polynomialsproduceEA-equivalentNihobentfunctions.Inthispaper,whenwesaythattwoo-polynomialsFandF0denepoten-tiallyEA-inequivalentNihobentfunctionsgFandgF0,itmeansthateitherinsomecasesgFandgF0areEA-inequivalent,oritisnotpossibletodeduceEA-equivalencewiththedevelopedtechnicswhichleavesapossibilitythatgFandgF0maybeEA-inequivalent.Belowweconsidersomeparticularcasesofformula(24).Corollary3.LetFbeano-polynomialdenedonF2m.Theno-polynomialsFc(x)=cFxF1x+c+F(c)�1;c2F2m(31)deneasequenceofNihobentfunctionsgFcpotentiallyEA-inequivalenttoeachotherfordierentc,andEA-inequivalenttoNihobentfunctionsdenedbyF,F�1.Proof.o-polynomial(31)istheexplicitformofo-polynomial(24)forq=1;k1=2.Indeed,((~cF

)0)�1(x)=x~cF1x&
)0)�1(x)=x~cF1x�1=cFxF1x+c+F(c)�1.NotethatFc=(F0)�1forc=0.Hence,theo-polynomial(F0)�1isincludedintheclassofo-polynomialsFc.Forc=1wegetthefunctionF=xF1x+1+1�1studiedin[7]andwhichcandeneaNihobentfunctionEA-inequivalenttothosedenedbyF,F�1and(F0)�1.Forinstance,whenF(x)=x2i,gFisEA-inequivalenttogF,gF�1andg(F0)�1[7].Usingtheequality(8)foreveryc2F2mwecanwrite:Fc=((~cF)0)�1=((~1~c+1F)0)�1=(~c+1F):SinceF,F,F�1and(F0)�1candenefourpotentiallyEA-inequivalentNihobentfunctions,weobtainthatFccandeneNihobentfunctionspotentiallyEA-inequivalenttothosedenedby~c+1F,(~c+1F)�1,((~c+1F)0)�1.Itmeansthat,foranyc2F2maNihobentfunctiongFccanbepotentiallyEA-inequivalenttogF,gF�1andgFc+1.21Corollary4.LetFbeano-polynomialdenedonF2m.Theno-polynomials(Fc)�1=cF0(1+cx)Fx1+cx+cxF1c�1;c2F2m(32)deneNihobentfunctionsg(Fc)�1whichcanpotentiallybeEA-inequivalenttoeachotherfordierentcandEA-inequivalenttoNihobentfunctionsdenedbyF,(F0)�1.Proof.o-polynomial(32)istheexplicitformofo-polynomial(24)forq=1andk1=3.Indeed,((~cF0)0)�1(x)=cF0xF01x+c+F0(c)�1=cF0x1+cxxFx1+cx+cF1c�1=cF0(1+cx)Fx1+cx+cxF1c�1:Notethat(F0)�1=F�1.Sotheo-polynomialF�1isincludedintheclassofo-polynomials(Fc)�1withc=0.Forc=1wegetthefunction(F1)�1=((x+1)F(xx+1)+x)�1alsostudiedin[7],andtheNihobentfunctionassociatedwithitisEA-equivalenttotheonedenedbyF[7].Butinthegeneralcase,forarbitraryc2F2mwecan'tsaythat(Fc)�1denesano-polynomialEA-equivalenttothosedenedbyFandFc.Usingequalities(8)and(31)notethat(Fc)�1=(F0)c=(~c+1F0).Hence,wecansaythat(Fc)�1=(F0)cdenesaNihobentfunctionpotentiallyEA-inequivalenttoNihobentfunctionsdenedbyF0,(F

0)�1and(F0)c+1=(Fc+1)�1.
0)�1and(F0)c+1=(Fc+1)�1.6.3Thecaseofo-monomialsandtheknowno-polynomialsFurtherwestudytheconsequencesoftheobtainedresultsfortheparticularcasesofo-monomialsandtheknowno-polynomials.Lemma7.Forano-monomialF(x)=xd,theNihobentfunctionsdenedbyFcandFareEA-equivalent,foranyc2F2m.Proof.Wehaveforc6=0Fc(x)=('~cF)�1=cFxF1x+c+F(c)�1=cFx1x+cd+cd�1=cFx1+cxxd+cd�1=cFcdx1+cxcxd+1�1=cFcd�1cx1+cxcxd+1�1=1cF1cFcd�1x:FromLemma1itfollowsthatNihobentfunctionsdenedbyFcandFareEA-equivalentforanyc6=0.Fromtheproofofthepreviouslemmaitiseasytoseethatforanyo-monomialF'~cF(x)=c'1F(cx);(33)wherec=cFcd�1;c2F2m.22Lemma8.Forano-monomialF(x)=xd,theNihobentfunctionsdenedby(Fc)�1,(F)�1andFareEA-equivalent,forc2F2m.Proof.F(x)=(x+1)F(xx+1)+x=(x+1)(xx+1)d+x.Forc6=0wehave(Fc)�1(x)=('c'F)�1=cF0(1+cx)Fx1+cx+cxF1c�1=cF0(1+cx)x1+cxd+cx1cd�1=cF01cd(1+cx)cx1+cxd+cx�1=1c(F)�1cdcF0x:UsingLemma1,weconcludethattheNihobentfunctionsdenedby(F)�1and(Fc)�1areEA-equivalentforc6=0.Accordingto[7],theNihobentfunctiondenedby(F)�1andFareEA-equivalent,andtakingintoaccountLemma7,wegetthatNihobentfunctionsdenedby(Fc)�1,(F)�1andFareEA-equivalenttoeachotherforanyc6=0.Fromtheproofofabovelemmaitiseasytoseethatforanyo-monomialF'~c'F(x)= c'1'F(cx):(34)where c=cF0cd�1;c2F2m;F0='F.Furtherwewillneedthefollowingequality,whichholdsforanyo-polynomialF'1'F=1'1F:(35)Indeed,1'1F(x)=(1+x)F11+x+1+1+1=(1+x)Fx1+x+x='1'F(x):Tokeepnotationsassimpleaspossible,

sinceweareinterestedinEA-equivalenceofNi
sinceweareinterestedinEA-equivalenceofNihobentfunctionsandcoecientsofargumentsofo-polynomialdonotaectonEA-equivalenceofNihobentfunctionsaswellascoecientofo-polynomial,theninsteadofaF(bx)=G(x)wewillwriteFGfora;b2F2m.Lemma9.LetFbeano-monomialdenedonF2m.Then'~c1'~c2':::|{z}kF8���������������&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:8&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:1F;ift0mod4;'1F;ift1mod4;1'F;ift2mod4;'1'F;ift3mod4;ifk=2t8&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:1'F;ift0mod4;'1'F;ift1mod4;1F;ift2mod4;'1F;ift3mod4;ifk=2t+1;wheret1.23Proof.Assumethatk=2t,i.e.theorbitinthestatmentofthislemmahastheform'~c1'~c2':::'~ctF.Then1)Fort=1wehave'~c1F(33)'~1F.2)Fort=2,'~c1'~c2F(33)'~c1'1F(??)'1'~c1F(33)'1'1F(35)''&

#14;~1'F1'F.3)F
#14;~1'F1'F.3)Fort=3,'~c1'~c2'~c3F2)'~c11'F'~c1+1'F(??)'1'F4)Fort=4'~c1'~c2'~c3'~c4F3)'~c1'1'F2)1'('F)1F.Thusforevenk,'~c1:::'~ct�3'~ct�2'~ct�1'~ctF4)'~c1:::'~ct�41F'~c1:::'~ct�4+1F4):::8�����&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:1F;ift0mod4;'~c11F1)'1F;ift1mod4;'~c1'~c21F2)1'F;ift2mod4;'~c1'~c2'~c31F3)'1'F;ift3mod4;Notethat'Fisano-monomial,thereforewecanapplythepreviousformulatothecaseofoddk.Indeed,'~c1:::'~ct�3'~ct�2'~ct�1'~ct('F)8&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:1'F;ift0mod4;'1('F);ift1mod4;1'('F)1F;ift2mod4;'1'('F)'1F;ift3mod4;Lemma10.LetFbeano-monomialdenedonF2m.Then'~c1'~c2':::|{z}k('1F)�18����������&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00

;&#x]TJ ;� -1;.93; Td;&#x [00
;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:8&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:('1F)�1;ift0mod3;('1('F)�1)�1;ift1mod3;('1'F�1)�1;ift2mod3;ifk=2t8&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:('1F�1)�1;ift0mod3;('1('F�1)�1)�1;ift1mod3;('1'F)�1;ift2mod3;ifk=2t+1;(36)wheret1.24Proof.Assumethatk=2t,i.e.theorbitinthestatementofthislemmahastheform'~c1'~c2':::'~ct('1F)�1.Then1)Fort=1weget:'~c1('1F)�1(20)'(~c11'1F)�1(18)('('~c11'1F)�1)�1(??)('('1'~c11F)�1)�1(33)('('1'1F)�1)�1(35)('(1'F)�1)�1(20)('1('F)�1)�1:2)Fort=2'~c1'c2('1F)�11)'~c1('1('F)�1)�11)('1('('F)�1)�1)�1(18)('1'F�1)�1:3)Fort=3,'~c1'~c2'c3('1F)�12)'~c1('1'F�1)�11)('1F)�1:Thus,'~c1:::'~ct�2'~ct�1'~ct('1F)�13)'~c1:::'~ct�3('1F)�13):::8��&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:('1F)�1;ift0mod3;'~c1('1F)�11)('1('F)�1)�1;ift1mod3;'~c1'~c2('1F)�12)('1'F�1)�1;ift2mod3:Notethatfrom(18)followsthat'('1F)�1=('(1F)�1)�1=('1F�1)�1.Thereforthecaseofoddkcomesdowntothepreviouscase.Indeed,'~c1:::'~ct�2'~&

#28;ct�1'~ct'('&#
#28;ct�1'~ct'('1F)�13)'~c1:::'~ct�2'~ct�1'~ct('1F�1)�18&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:('1F�1)�1;ift0mod3;('1('F�1)�1)�1;ift1mod3;('1'F)�1;if2mod3:Lemma11.LetFbeano-monomial.Thenforq3(H1(H2(:::(HqF)�1:::)�1)�18�&#x]TJ ;� -1;.93; Td;&#x [00;:1G�1;('1G)�1;'1G;whereG2fF;('F)�1;'F�1;F�1;('F�1)�1;'FgandHiaredenedby(23)forall1iq.Proof.Firstconsiderthefollowingcases:1.q=1.ItiseasytoseethatfromLemma9follows(H1F)�18&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:(1F)�11F�1;('1F)�1;(1'F)�11('F)�1;('1'F)�1;=(1G�1;('1G)�1;(37)25whereG2fF;'Fg2.q=2.ObviouslyfromLemma10wehave(H1('1F)�1)�1='1G;(38)whereG2fF;('F)�1;'F�1;F�1;('F�1)�1;'Fg:Using(37)and(38)weget(H1(H2F)�1)�1378��&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:(H11G�11)�137(1G�12;('1G2)�1;(H1('1G1)�1)�138'1G2;(39)whereG12fF;'Fg;G22fG�11;'G�11g=A1;G22fG1;('G1)�1;'G�11;G�11;('G�11)�1;'G1g=A2:ItiseasytoseethatA1=fF�1;('F)�1;'F�1;('F�1)�1g;A2=fF;('F)�1;'F�1;F�1;('F�1)�1;'Fg:Indeed,ifwetakeG1=FinA2,thenwegetfF;('F)�1;'F�1;F�1;('F�1)�1;'Fg,ifwetakeG1='F,thenwegetthesamesetofo-polynomials,since('('F)�1)�1(18)=(('F�1)�1)�1='F�1:NotethatallfunctionsinthesetsA1andA2areo-monomials.3.q=3,(H1(H2(H3F)�1)�1)�1(39)8�������&#x]TJ ;� -1;.93; Td;&#x [00;

&#x]TJ ;� -1;.93; Td;&#x [00;
&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:(H11G�12)�1(37)(1G�13;('1G3)�1;(H1('1G2)�1)�1(38)'1G3(H1'1G2)�1(37)(1~G�13;('1~G3)�1;whereG32fG�12;'G�12g,G32fG2;'G�12;('G2)�1;G�12;('G�12)�1;'G2g,~G32fG2;'G2g,G22A1,G22A2.Substitutinginthecorrespondingsetso-monomialsfromA1andA2,using(18),wegetthatG3;G3;~G3belongtoA2,therefore(H1(H2(H3F)�1)�1)�18�&#x]TJ ;� -1;.93; Td;&#x [00;:1G�13;'1G3;('1G3)�1;whereG32A2=fF;('F)�1;'F�1;F�1;('F�1)�1;'Fg.Wearegoingtoprovethislemmabyinductiononthelengthoforbitq.Forq=3thestatementofthelemmaistrueaswesawabove.Supposethatitis26trueforanylq�1andl3.Byourassumption:(H1(H2(:::(HqF)�1:::)�1)�18�������&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:(H11G�1)�1(37)(1G�11;('1G1)�1;(H1('1G)�1)�1(38)'1G1;(H1'1G)�1(37)(1~G�11;('1~G1)�1;whereG2A2,G12fG�1;'G�1g,G12fG;('G)�1;'G�1;G�1;('G�1)�1;'Gg,~G12fG;'Gg.BystraightforwardcomputationsitiseasytoseethatallofthesetsareequaltoA2,thus(H1(H2(:::(HqF)�1:::)�1)�18�&#x]TJ ;� -1;.93; Td;&#x [00;:1G�1;('1G)�1;'1G;whereG2fF;('F)�1;'F�1;F�1;('F�1)�1;'Fg,whichprovesourstate-ment.Proposition2.Themodiedmagicactionandtheinversemapappliedtoo-monomialsgiveatmost4EA-inequivalentfunctions.Forano-monomialFthe4potentiallyEA-inequivalentbentfunctionsaredenedb

yF;F�1;(F0)�1andF.Proof.Weuse
yF;F�1;(F0)�1andF.Proof.WeuseLemma11anddiscussthecasesq=1;2andq3separately.1.q=1.Accordingto(37)(H1F)�1hasthefollowingtwoforms1G�1and('1G)�1,whereG2fF;'Fg.TherstfunctionobviouslydenesNihobentfunctionsEA-equivalenttoonedenedbyG�1andthereforetothosede-nedbyF�1and('F)�1.ThesecondfunctiondenesNihobentfunctionsEA-equivalenttoonedenedbyF(byLemma8).2.q=2.From(39)wehave:(H1(H2F)�1)�18�&#x]TJ ;� -1;.93; Td;&#x [00;:1G�12;('1G2)�1;'1G2;whereG22fF�1;('F)�1;'F�1;('F�1)�1g;G22fF;('F)�1;'F�1;F�1;('F�1)�1;'Fg:Obviously,1G�12and'1G2deneNihobentfunctionEA-equivalenttothosedenedbyG�12andG2respectively,whichintheirturndeneNihobentfunctionsEA-equivalenttoF;F�1and(F0)�1.('1G2)�1denesfunctionsEA-equivalenttoonedenedbyF.Indeed,('1G2)�1hasoneofthefol-lowingforms:('1F�1)�1(20)=('(1F)�1)�1(18)='('1F)�1denesNihobentfunctionEA-equivalentto('1F)�1=F('1'F�1)�1,byLemma8denesNihobentfunctionsEA-equivalentto('1F�1)�1=('(1F)�1)�1(18)='('1F)�1,whichdenesfunctionsEA-equivalenttoonedenedby('1F)�1=F;('1('F)�1)�1(20)=('(1'F)�1)�1(18)='('1'F)�1denesNihobentfunctionEA-equivalenttoF(byLemma8);27('1('F�1)�1)�1=('1'('F)�1)�1(35)=(1'1('F)�1)�1(20)=1('1('F)�1)�1denesNihobentfunctionEA-equivalentto('1('F)�1)�1,whichbythepreviouscasedenesNihobentfunctionEA-equivalenttoF.3.Forq3byLemma11,(H1(H2(:::(HqF)�1:::)�1)�18�&#x]TJ ;� -1;.93; Td;&#x [00;:1G�1;('1G)�1;'1G;whereG2fF;('F)�1;'F�1;F�1;('F�1)�1;'Fg.1G�1and'1GdeneNihobentfunctionEA-equivalenttoG�1andGcorrespondingly,which

intheirturndeneNihobentfunctionsEA-
intheirturndeneNihobentfunctionsEA-equivalenttoF;F�1and('F)�1.('1G)�1denesNihobentfunctionsEA-equivalenttoF.Indeed,forGequalstoF�1;('F)�1;'F�1;('F�1)�1,wealreadyproveitinthecaseq=2.IfG='F,then('1G)�1=('1'F)�1whichdenesNihobentfunctionEA-equivalenttoonedenedbyF(byLemma8).IfG=F,then('1F)�1=F:Proposition3.ThemodiedmagicactionandtheinversemapappliedtotheFrobeniusmap,giveexactly3EA-inequivalentfunctionscorrespondingtoF,F�1,(F0)�1.Proof.FortheFrobeniusmapF(x)=x2iwehave:F=(F0)�1=x11�2i.HencebyProposition2,Fcanpotentiallydene3EA-inequivalentNihobentfunctionscorrespondingtoF,F0and(F0)�1.This3o-polynomialsdene3surlyEA-inequivalentNihobentfunctions[7].ThePayneo-polynomialcanberepresentedviaDicksonpolynomials.LetusrecallDicksonPolynomials.Foreverynon-negativeintegerdDicksonpoly-nomialsDd(x)overF2mcanbedenedbyarecursionrelationinthefollowingway:D0(x)=0;D1(x)=x,Dd+2(x)=xDd+1+Dd(x),forallintegersd0.Itsatisesthefollowingroperties:1.DdDd0=Ddd0.2.Ifdisco-primewith2m�1,thenDdisapermutationalpolynomial.UsingDicksonpolynomialswecanprovethefollowingresultsforthePayneo-polynomials.Lemma12.LetF(x)=x16+x12+x56.ThenFc=(Fc)�1foranyc2F2m.Proof.Noterst,thatF(x)=x16+x12+x56=D5(x16).AlsoitiseasytoseethatF0=F.Indeed,F0(x)=xF(x�1)=xD5(x�16)=x(x�16+x�12+x�56)=x16+x12+x56=D5(x16)=F(x):Therefore(F0)�1=F�1,andhence,(Fc)�1=((cF0)0)�1=((cF)0)�1=Fc;foranyc2F2m:28Proposition4.Themodiedmagicactionandtheinversemapappliedtoo-polynomialF(x)=x16+x12+x56canpotentiallygiveEA-inequivalentNihobentfunctionscorrespondingtoo-polynomialsFandFc,c2F2m.Proof.ImmediatelyfollowsfromLemma12.ExampleForm=5wecheckedcomputationallythattheo-polynomialF(x)=D5(x16)overF2mdenes6EA-inequivalentNihobentfunctionscorrespondingtoo-polynomialsF,F�1andFw;Fw3;Fw5,wherewisaprimitiveelementofF2m.RemarkThemodiedmagicactionandtheinversemapappliedtoSubiaco,Adelaideandx2k+x2k+2+x3
2k+4o-polinomialsFcangiveasequence

ofEA-inequivalentfunctionsdenedbyo-
ofEA-inequivalentfunctionsdenedbyo-polynomialsontheorbitsF,F�1,Fc,(~cF)c,(~c(F0))candsoon.7TheKnownHyperovals1Overtwodecades,nitegeometersdeterminedthestabilizersofallknownhy-perovals.Inthissectionweprovideanexplicitlistofallo-polynomialswhichprovideEA-inequivalentNihobentfunctionsforeveryoftheknownhyperoval.WestartbygivinganoverviewoverthenumberofEA-inequivalentNihobentfunctionsforeachknownhyperoval.NameHyperovalConditionNumberRef.Regularx2m=11[23,Th.4.1]m=21[23,Th.4.1]m32[23,Th.4.2]IrregularTranslationx2im33[23,Th.4.3]Segrex6m=52[23,Th.4.4]m�5odd4[23,Th.4.4]GlynnIx3+4m7odd=2(m+1)=24Th.7GlynnIIx+m=7=4=2Th.7m�7odd=2(m+1)=2=2kform=4k�1;=23k+1form=4k+14Th.71SomeoftheresultswillrepeatSection6.2results.Wedecidedtokeepbothofthem,sinceweuseamixofalgebraicandgeometricapproach.29Cherowitzox+x+2+x3+4m=510[23,Th.4.6]m�5prime4m+2m�2mTh.9m�5oddnC(m)[23]Paynex1=6+x3=6+x5=6m5isprime3m+2m�1�1mTh.8m5isoddnP(m)Th.8Lunelli-Sce(Subiaco)m=4prim.root4=+11[23,Th.4.1]Subiacom=6jAutj=603[32,p.98]m=6jAutj=156[32,p.98]moddm=712[34]moddm�7nS(m)Th.11m0(mod4)m�6nS(m)Th.11m2(mod4)m�6jAutj=10eTh.12m2(mod4)m�6jAutj=5e=25-mTh.12Adelaidem=68[34]m�6mevennA(m)Th.10O'Keefe-Penttilam=512[22,Case2]2Below,forgiveno-polynomialsF1andF2,wedenoteF1=F2ifF1andF2deneEA-equivalentNihobentfunctionsgF1andgF2.Notethatamatrixcorrespondingtothetransformation'cis0110
10c1=c110;andthat'~c=cF
('c).Hence,byTheorem3thehyperovaldenedbytheo-polynomialFcisobtainedfromthehyperovaldenedbyFusingthe2Noticethatthereferenceclaims1+110insteadof1+11orbitsduetoatypo.30followingtransformationmatrix(therstmatrixintheproductcorrespondstotheinversetransformation):0@0101000011A
0@0010cFcFF(c)=c1001A=0@0cFcFF(c)=c00110c1A:Thatis,Fc(x)=cFxF1x+c+F(c)�1correspondstothemapAcF:=0@0cFcFF

(c)=c00110c1A:Alsorecallthatthechoiceofa
(c)=c00110c1A:Alsorecallthatthechoiceofano-polynomialforagivenhyperovalHonlydependsonwhichpointofHischosenasnucleus,sotheo-polynomialisdeter-minedbythepreimageof(0;1;0).WehaveAcF(c;F(c);1)T=(cFF(c)+cFF(c)=c;1;c+c)T=(0;1;0):Hence,Fc=Fdifandonlyifh(c;F(c);1)iandh(d;F(d);1)ilieinthesamepointorbitofthestabilizerofH.Tosummarize,wehavethefollowing:(a)Fc=Fdifandonlyifh(c;F(c);1)iandh(d;F(d);1)ilieinthesamepointorbit;(b)F=Fcifandonlyifh(0;1;0)iandh(c;F(c);1)ilieinthesamepointorbit;(c)F�1=Fcifandonlyifh(1;0;0)iandh(c;F(c);1)ilieinthesamepointorbit;(d)F=F�1ifandonlyifh(0;1;0)iandh(1;0;0)ilieinthesamepointorbit.Asguidelinedin[8]weusetheknownresultsonorbitsoftheknownhyper-ovalstogettheexplicitnumbersandrepresentationsforo-polynomialswhichprovideo-equivalentbutEA-inequivalentNihobentfunctionsforeachoftheknownhyperoval.Lemma13.Letm3.Thetwoo-polynomialsobtainedfromtheregularhyperovalH,thatisF(x)=x2,are(uptoEA-equivalenceforthecorrespondingNihobentfunctions)FandF�1.Proof.By[23,Th.4.2],onepointorbitisthenucleusNandtheotherpointorbitisHnfNg.Hence,F�1isarepresentativeofthesecondorbit.Lemma14.Letm3.Thethreeo-polynomialsobtainedfromtheirregulartranslationhyperovalH,thatisF(x)=x2iwithi�1co-primetom,are(uptoEA-equivalenceforthecorrespondingNihobentfunctions)F,F�1andF0.Proof.By[23,Th.4.3],onepointorbitisthenucleusN=(0;1;0),anotherpointorbitisN0:=(1;0;0),andthelastpointorbitisHnfN;N0g.Hence,F,F�1,andF0arerepresentativesofthethreeorbits.31Lemma15.Letm5beodd.ConsidertheSegrehyperovalH,thatisF(x)=x6.(a)Ifm=5,thenthetwoo-polynomialsobtainedfromHare(uptoEA-equivalenceforthecorrespondingNihobentfunctions)FandF1.(b)Ifm�5,thenthetwoo-polynomialsobtainedfromHare(uptoEA-equivalenceforthecorrespondingNihobentfunctions)F,F�1,F0,andF1.Proof.By[23,Th.4.4],form=5thepointorbitsofHaref(1;0;0);(0;1;0);(0;0;1)gandalltheremainingpoints.Hence,(0;1;0)and(1;1;1)arerepresentatives,sowecanchooseFandF1asrepresentatives.Form�5therstorbitsplitsintothreeorbits,sowehavetoaddF�1andF0tothepreviouslist.Theorem7.Thecollineation

stabiliserofaGlynnhyperovalhas4orbitsunl
stabiliserofaGlynnhyperovalhas4orbitsunlessitisoftypeIIandm=7.Proof.FirstconsiderthecaseGlynnI.By[23,Th.4.4]wehave4orbitsunless(3+4)2�(3+4)+10(mod2m�1).Thissimpliesto9
2m+1+21
2(m+1)=2+1331+21
2(m+1)=20(mod2m�1):Onecaneasilycheckthatthisisneversatised.NowconsiderthecaseGlynnII.By[23,Th.4.4]wehave4orbitsunless(+)2�(+)+10(mod2m�1).Form=4k�1,thisis2(3m+7)=4�2(m+1)=4+30(mod2m�1):Equivalityholdsonlyform=7asform�7thelefthandsideissmallerthan2m�1.Thecaluclationform=4k+1issimilar.SimilartoLemma15,weobtainthefollowing.Lemma16.Letm7beodd.ConsiderahyperovalHoftypeGlynnIorGlynnII.(a)Ifm=7,thenthetwoo-polynomialsobtainedfromHare(uptoEA-equivalenceforthecorrespondingNihobentfunctions)FandF1.(b)Otherwise,thefouro-polynomialsobtainedfromHare(uptoEA-equivalenceforthecorrespondingNihobentfunctions)F,F�1,F0,andF1.Theorem8.ThenumberoforbitsofthecollineationstabilizerofthePaynehyperovalHisgivenby3+2m�1mifmisaprime.Moregenerally,thenumberoforbitsaregivenbynP(m):=3+X`jm;�`1F2`n[hj`;hF2h=(2`):ForwaprimitiveelementofFqandc=w2n,wegetFc=Fdifandonlyifd=w2inord=w�2inforsomei2f1;:::;mg.Theo-polynomialsFandF�1deneNihobentfunctionsEA-inequivalenttothosedenedbyallothero-polynomialsfromH.32Proof.By[23,Th.4.5],theorbitsaref(0;1;0)g,f(1;0;0);(0;0;1)g,andsetsHn:=f(wn2i;f(wn2i);1):i=1;:::;mg[f(1;f(wn2i);wn2i):i=1;:::;mg;wherewisaprimitiveelementofFq.NoticethatH0isf(1;1;1)g.FormprimeitiseasytoseethateachorbitHnhaslengthmforn�1,hencethetotalnumberoforbitsis3+2m�1�1m.Ingeneral,ifwn2F`with`jm,thenf(wn)2ig2F`.Thisyieldsthegeneralformula.ThedescriptionoftheequivalenceofFcandFdfollowsdirectlyfromtheexplicitdescriptionoftheorbits.Forexampleform=5,thepreviousresultgivesthefollowingrepresentativesforall6o-polynomialswhichcanbeobtainedfromthePaynehyperoval:F;F�1;F1;Fw;Fw3;Fw5:Theorem9.ThenumberoforbitsofthecollineationstabilizeroftheCherow-itzohyperovalisgivenby4+22m�1�1mifmisaprime.Moregene

rally,thenumberoforbitsaregivenbynC(m):=
rally,thenumberoforbitsaregivenbynC(m):=3+X`jmF(2`)n[hj`;hF2h=`:ForwaprimitiveelementofFqandc=w2n,wegetFc=Fdifandonlyifd=w2inforsomei2f1;:::;mg.TheNihobentfunctionsgFandgF�1arebothEA-inequivalenttoNihobentfunctionsdenedbyallothero-polynomialsfromH.Proof.Corollary4.5in[2]describesthestabilizerasf(x;y;z)7!(x;y;z):2Aut(Fq)g:TherestofthecalculationissimilartothePaynehyperoval,justthatthistimetherstandsecondcoordinatecannotbeinterchanged.Theorem10.Let[1]:=+�1.Forc2Fq,letOc:=fc2h+h�1Xi=1[1]2i:i=0;:::;2m�1g:ThenumberofEA-inequivalentNihobentfunctionsobtainedfromtheAdelaidehyperovalisnA(m):=2+jfOc:c2Fqgj.Inparticular,forxedc2Fq,theNihobentfunctionsdenedbytheo-polynomialsF,F�1,FcarepairwiseEA-inequivalent.Furthermore,gFcandgFdareEA-equivalentifandonlyifd2Oc.Proof.In[31,Eq.(9)](inaslightlydierentrepresentation)thestabilizeroftheAdelaidepolynomialwasdeterminedasthecyclicgroupgeneratedbythemap:x7!0@10[1]01[1]0011A0@xF(x)11A2:33Fromthisitiseasilyveriedthatxes(0;1;0)and(1;0;0),sogFandgF�1arenotEA-equivalenttothosefunctionsdenedbyanyoftheothero-polynomials.Furthermore,itiseasilycheckedthattheorbitof(c;F(c);1)isf(x;F(x);1):x2Ocg:Theorem11.Letm7withm62(mod4),letOc:=fx(�1)i+12i:i=0;:::;2m�1g:ThenumberofEA-inequivalentNihobentfunctionsobtainedfromtheSubiacohyperovalisnS(m):=2+jfOc:c2Fqgj.Inparticular,forxedc6=0;1,theo-polynomialsF,F�1,F0,FcprovidepairwiseEA-inequivalentNihobentfunctions.Furthermore,gFcandgFdareEA-equivalentifandonlyifd2Oc.Proof.By[24,Th.13,Th.16](seealso[15]),thestabilizeroftheSubiacohyperovalHisgeneratedbythemap:x7!0@0010101001A0@xF(x)11A2:Fromthisitiseasilyveriedthatxes(0;1;0),f(1;0;0);(0;0;1)g,(1;1;1),soNihobentfunctionsdenedbyF,F�1=F0,andF1arenotEA-equivalenttothosedenedbyanyothero-polynomialobtainedfromH.Furthermore,itiseasilycheckedthattheorbitof(c;F(c);1)isf(x;F(x);1):x2Ocg:Form2(mod4)therearetwotypesofnon-equivalenthyperovals,see[33].Inpa

rticular,fromTheorem6.6andTheorem6.7in[3
rticular,fromTheorem6.6andTheorem6.7in[33]weobtainthefollowing.Wearenotawareofanynicedescriptionoftheorbitsofthegivengroups,buttheinformationissucienttocalculateallo-polynomialseciently.Theorem12.Letm6withm2(mod4).(a)IfF(x)=2(x4+x)x4+2x2+1+x1=2,thengFisEA-inequivalenttoallgFcandwehaveF�1=F0.Furthermore,Fc=Fdifandonlyif(c;F(c);1)h=(d;F(d);1)foranelementhofthegroup(ofsize10m)generatedby(i)(x;y;z)7!(z;y;x),(ii)(x;y;z)7!(x+z;y+2z;z),(iii)(x;y;z)7!(z2+2x2;z2+y2;z2).(b)IfF(x)=x3+x2+2xx4+2x2+1+x1=2,thengF,gF�1,andgF0arepairwiseEA-inequivalent.Furthermore,Fc=Fdifandonlyif(c;F(c);1)h=(d;F(d);1)hforanelementhofthegroup(ofsize5m=2)generatedby(i)(x;y;z)7!(x;y;z)for2Aut(F)with=,(ii)(x;y;z)7!(z;y+z;x+z).34TheO'Keefe-Penttilahyperovalform=5,whichisnotknowntobelongtoanyinnitefamily,isstabilizedbythegroupgeneratedby0@1011101001A:Hence,mostorbitshavetheformf(c;F(c);1);(1+c�1;1+c�1F(c);1);((1+c)�1;c�1(1+F(c);1)g.Then,representativesforthe14o-polynomialsobtainedfromthehyperovalanddeningEA-inequivalentNihobentfunctionsareF;F�1;Fw;Fw2;Fw4;Fw5;Fw7;Fw8;Fw10;Fw14;Fw16;Fw19:HerewisaprimtiveelementofF25.AcknowledgementTheauthorswouldliketothankAlexanderKholoshaforusefuldiscussions.ThisresearchwassupportedbyTrondMohnStiftelse(TMS)foundation.TheworkofFerdinandIhringerissupportedbyapostdoctoralfellowshipoftheResearchFoundation{Flanders(FWO).References[1]K.Abdukhalikov,"Bentfunctionsandlineoval",FiniteFieldsAppl.,47,pp.97{124,2017.[2]L.Bayens,W.CherowitzoandT.Penttila."GroupsofhyperovalsinDe-sarguesianplanes",Inn.Inc.Geom.,pp.6{7,2007.[3]L.BudaghyanandC.Carlet,\CCZ-equivalenceofsingleandmultioutputBooleanfunctions",AMSContemporaryMath.518,Post-proceedingsoftheconferenceFq9,pp.43{54,2010.[4]L.BudaghyanandC.Carlet,"OnCCZ-equivalenceanditsuseinsecondaryconstructionsofbentfunctions",PreproceedingsofInternationalWorkshoponCodingandCryptographyWCC2009,pp.19{36,2009.[5]L.Budaghyan,A.Kholosha,C.Carlet,andT.Helleseth,\Nihobentfunc-tion

sfromquadratico-monomials",Proceedingsof
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