%42) *OURNAL VOLUME NUMBER $ECEMBER
#ORRELATION )MMUNE &UNCTIONS WITH #ONTROLLABLE .ONLINEARITY
3EONGTAEK #HEE 3ANGJIN ,EE +WANGJO +IM AND $AEHO +IM
#/.4%.43 !"342!#4
) ).42/$5#4)/. )N THIS PAPER WE CONSIDER THE RELATIONSHIP BETWEEN NONLINEARITY AND CORRELATION IMMU )) 02%,)-).!2)%3 NITY OF "OOLEAN FUNCTIONS )N PARTICULAR WE ))) #/22%,!4)/. )--5.)49 !.$ DISCUSS THE NONLINEARITY OF CORRELATION IMMUNE ./.,).%!2)49 FUNCTIONS SUGGESTED BY 0 #AMION ET AL &OR )6 $%3)'. /& #/22%,!4)/. )--5.% THE ANALYSIS OF SUCH FUNCTIONS WE PRESENT A &5.#4)/.3 SIMPLE METHOD OF GENERATING THE SAME SET OF FUNCTIONS WHICH MAKES IT POSSIBLE TO CON 6 #/.3425#4)/. /& #/22%,!4)/. STRUCT CORRELATION IMMUNE FUNCTIONS WITH CON )--5.% &5.#4)/.3 7)4( TROLLABLE CORRELATION IMMUNITY AND NONLINEAR #/.42/,,!",% ./.,).%!2)49 ITY !LSO WE `ND A BOUND FOR THE CORRELATION 6) ./.,).%!2)49 !.$ #/22%,!4)/. IMMUNITY OF FUNCTIONS HAVING MAXIMAL NONLIN M )--5.)49 /& #N K EARITY 6)) #/.#,53)/.3 !00%.$)8 !#+./7,%$'-%.4 2%&%2%.#%3 3EONGTAEK #HEE ET AL %42) *OURNAL VOLUME NUMBER $ECEMBER
) ).42/$5#4)/. OUR METHOD FOR CONSTRUCTING CORRELATION IM MUNE FUNCTIONS AND DISCUSS SOME PROPERTIES #RYPTOGRAPHIC "OOLEAN FUNCTIONS PLAY OF THE GENERATED FUNCTIONS THAT WERE ALREADY AN IMPORTANT ROLE IN THE DESIGN OF NONLIN ANALYZED IN ;= WHICH IS RATHER COMPLICATED EAR `LTER FUNCTIONS OR NONLINEAR COMBINERS COMPARED TO OURS 7E ALSO SUGGEST A CON IN STREAM CIPHER AS WELL AS PRIMITIVE LOGICS DITION FOR OBTAINING MAXIMAL NONLINEARITY IN BLOCK CIPHERS OF THE FUNCTIONS )N 3ECTION 6 WE PRESENT )N PARTICULAR THE FUNCTION WHOSE OUT A SYSTEMATIC METHOD TO OBTAIN CORRELATION PUT LEAKS NO INFORMATION ABOUT ITS INPUT IMMUNE FUNCTIONS WITH CONTROLLABLE NONLIN VALUES IS OF GREAT IMPORTANCE 3UCH FUNC EARITY AND GIVE AN EXAMPLE 3ECTION 6) DE TIONS CALLED CORRELATION IMMUNE FUNCTIONS SCRIBES THE RELATIONSHIP BETWEEN THE CORRE WERE `RSTLY INTRODUCED BY 4 3IEGENTHALER LATION IMMUNITY AND NONLINEARITY OF FUNC ;= 3INCE THEN THE TOPIC HAS BEEN AN AC TIONS DISCUSSED IN 3ECTION 6 )N PARTICULAR TIVE RESEARCH AREA ;= ;= AND MANY STREAM WE DISCUSS THE RANGE OF THE CORRELATION IM CIPHERS HAVE EMPLOYED THE CORRELATION IM MUNITY OF FUNCTIONS THAT HAVE MAXIMAL NON MUNE FUNCTIONS 0 #AMION ET AL ;= PRE LINEARITY 4HE CONCLUSIONS ARE ADDRESSED IN SENTED A METHOD FOR CONSTRUCTING BALANCED 3ECTION 6)) CORRELATION IMMUNE FUNCTIONS * 3EBERRY ET AL ;= DISCUSSED THE NONLINEARITY AND PROP AGATION CHARACTERISTICS OF SUCH FUNCTIONS )) 02%,)-).!2)%3 4HE OBJECTIVE OF THIS PAPER IS TO DISCUSS N THE RELATIONSHIP BETWEEN CORRELATION IMMU ,ET : BE THE N DIMENSIONAL NITY AND NONLINEARITY OF "OOLEAN FUNCTIONS VECTOR SPACE WITH THE BINARY )N PARTICULAR WE FOCUS OUR ATTENTION ON THE N TUPLES OF ELEMENTS X X FUNCTIONS GENERATED BY 0 #AMION ET ALS qqqXN &OR A AqqqAN B BqqqBN N METHOD )N ORDER TO ACHIEVE SUCH A GOAL IN : A q B AB wqqqwANBN IS THE INNER WE PRESENT A SIMPLE METHOD OF GENERATING PRODUCT OF TWO VECTORS THE SAME SET OF FUNCTIONS WHICH MAKES IT !FUNCTIONF IS SAID TO BE BALANCED POSSIBLE TO CONSTRUCT CORRELATION IMMUNE IF FX J FX G FX J FX G! N FUNCTIONS WITH CONTROLLABLE CORRELATION IM FUNCTION F ON : IS K TH ORDER CORRELATION IMMUNE f K f N IFFX IS STATISTICALLY MUNITY AND NONLINEARITY INDEPENDENT OF ANY SUBSET OF K INPUT VARI 4HE REST OF THIS PAPER IS ORGANIZED AS ABLES XI qqqXIK f I qqq IK f N ANDK FOLLOWS 3ECTION )) INTRODUCES NOTATIONS IS CALLED THE CORRELATION IMMUNITY OF F4HE AND DE`NITIONS THAT ARE NEEDED IN THIS PA ALGEBRAIC NORMAL FORM OF F IS AS FOLLOWS PER )N 3ECTION ))) WE DERIVE AN UPPER BOUND FOR THE NONLINEARITY OF CORRELATION IM FXqqqXN A wAX wqqqwANXN
MUNE FUNCTION )N 3ECTION )6 WE DESCRIBE wAXX wqqqwANpNXNpXN %42) *OURNAL VOLUME NUMBER $ECEMBER 3EONGTAEK #HEE ET AL
wAXXX wqqq &>W HOLDSFORANYW WITH f WTW f
wANpNpNXNpXNpXN K WHEREWTW IS THE (AMMING WEIGHT OF W
wAqqqNXX qqqXN ,EMMA ,ET F BE A "OOLEAN FUNCTION OF N VARIABLES 4HE NONLINEARITY OF F IS 4HE ALGEBRAIC DEGREE OF A "OOLEAN FUNC Np > TION DENOTED BY DEGF IS DE`NED AS THE .F p MAX J &W J W MAXIMUM OF THE ORDER OF ITS PRODUCT TERMS THAT HAVE A NONZERO COEbCIENT IN THE AL 4HEOREM 0ARSEVALS 4HEOREM ,ET F BE GEBRAIC NORMAL FORM ! "OOLEAN FUNCTION A "OOLEAN FUNCTION OF N VARIABLES THEN WITH DEGF f IE FX A wA X wqqqw 8 N ANXN IS SAID TO BE AbNE )N PARTICULAR IF &> W W: N A IT IS SAID TO BE LINEAR &OR TWO "OOLEAN FUNCTIONS F AND G WE DE`NE THE DISTANCE BETWEEN F AND G BY DFG FX J FX GX G4HEMINIMUM ))) #/22%,!4)/. )--5.)49 DISTANCE BETWEEN F AND THE SET OF ALL AbNE !.$ ./.,).%!2)49 FUNCTIONS t IE MINgt DFg IS CALLED THE NONLINEARITY OF F AND DENOTED BY .F )N )T IS WELL KNOWN THAT THE CORRELATION IM N MOST CASES IT WILL BE MORE CONVENIENT TO MUNITY OF A FUNCTION ON : AND ITS ALGE DEAL WITH F>X p FX WHICH TAKES VAL BRAIC DEGREE D ARE CONSTRAINED BY THE RELA UES IN FpG TION K D f N ;= .ATURALLY WE CAN IMAG 4HE DE`NITIONS OF BALANCEDNESS CORRELA INE THAT THERE MAY BE A SIMILAR RELATIONSHIP TION IMMUNITY AND NONLINEARITY CAN BE DE BETWEEN THE CORRELATION IMMUNITY AND NON RIVED FROM THE NOTIONS OF 7ALSH (ADAMARD LINEARITY TRANSFORMS )N THIS SECTION WE DERIVE AN UPPER $E`NITION ,ET F BE A "OOLEAN FUNCTION IN BOUND FOR THE NONLINEARITY OF THE CORRELA N THE VECTOR SPACE : 4HE 7ALSH (ADAMARD TION IMMUNE FUNCTIONS > > TRANSFORM OF F IS THE REAL VALUED FUNCTION & N N ,EMMA )F F : : THEN OVER THE VECTOR SPACE : DE`NED AS 8 Np &>W F>X p WqX . f Np p P F c X
N > ,EMMA !"OOLEANFUNCTIONF IS BALANCED WHERE c FW : J &W G IF AND ONLY IF &> 0ROOF "Y 4HEOREM WE HAVE ,EMMA ;= &OR A "OOLEAN FUNCTION F F IS 8 c q MAX J &>W Jg &>W N W K TH ORDER CORRELATION IMMUNE IF AND ONLY IF W 3EONGTAEK #HEE ET AL %42) *OURNAL VOLUME NUMBER $ECEMBER
N (ENCE MAX J &>W JgP 4HEREFORE BY )6 $%3)'. /& #/22%,!4)/. W c ,EMMA WE HAVE )--5.% &5.#4)/.3
Np > .F p MAX J &W J 4HROUGHOUT THIS SECTION WE SET THE FOLLOW W ING NOTATION Np fNp p P c t N AN INTEGER WITH N g K AN INTEGER WITH f K f Np N M AN INTEGER WITH f M NpK 4HEOREM )F F : : IS A K TH ORDER M NpM CORRELATION IMMUNE FUNCTION THEN l A FUNCTION ON : INTO : WITH M Np WTlY g K FORALL Y : Np .F f p P p NpM TX l X X: |K d e d e d e T MAXTX N N N N O X WHERE | N p qqq ! lY K K Y 0ROOF3INCEF IS K TH ORDER CORRELATION IM .OW WE PRESENT A METHOD OF CONSTRUCTING COR MUNE BY ,EMMA &>W HOLDSFORANY RELATION IMMUNE FUNCTIONS W WITH f WTW f K (ENCE N 4HEOREM $E`NE A "OOLEAN FUNCTION F : N N > c pFW : J &W G : BY N N f pFW : J f WTW f KG FYX !Y qX d e d e d e N N N N p qqq WHERE Y Y qqqY :M X X qqq K M NpM XNpM : 4HEN THE FOLLOWINGS HOLD |K I F IS BALANCED 4HEREFORE THE ASSERTION HOLDS BY ,EMMA t II F IS K TH ORDER CORRELATION IMMUNE
Np NpMp III .F pT )F F IS BALANCED AND K TH ORDER CORRELA > IV ,ET !YI BETHEI TH COMPONENT OF !Y TION IMMUNE THEN & IE c f |K p )F w ! I FORSOME I f I f N p (ENCEWEHAVETHEFOLLOWING Y Y M THEN DEGF M #OROLLARY &OR A BALANCED AND K TH ORDER N CORRELATION IMMUNE FUNCTION F : : 0ROOF I 3INCE WT! g K WEHAVE ! Np 8 Y Y Np ! qX .F f p P 4HUS p Y 4HEREFORE WE | p K X HAVE %42) *OURNAL VOLUME NUMBER $ECEMBER 3EONGTAEK #HEE ET AL
8 8 &> p FYX p !YqX )F wY!YI THEN IN THE ABOVE EXPRES YX YX SION THE TERM Y Y qqqY X IS NOT CAN 88 M p !YqX CELLED (ENCE DEGF M t Y X M #OROLLARY &OR ANY R : : IFWE "Y ,EMMA F IS BALANCED DE`NE FYX !Y qXwRY THEN PROPERTIES N I II AND IV OF 4HEOREM HOLD !ND THE II &OR ANY BA : WITH f WTB A f K NOTETHAT NONLINEARITY OF F IS BOUNDED BY 8 FYX BA qYX Np NpMp &>BA p p .F f p YX 8 p !YqXp BqYwAqX WHERE EQUALITY HOLDS IF AND ONLY IF l IS IN YX 8 8 JECTIVE IE T p BqY p !YwA qX Y X 0ROOF 7E ONLY PROVE SINCE THE PROOFS OF RESTS ARE TRIVIAL "Y WE HAVE 3INCE f WTA f K AND WT! g K Y 8 > NpM BqYwRY 8WE HAVE A w !Y 4HUS &BA p ! wA qX p Y 4HEREFORE BY FYJ!Y AG X > > NpM WE HAVE &BA "Y ,EMMA 4HEN MAX J &BA Jg AND .F f BA F IS K TH ORDER CORRELATION IMMUNE Np pNpMp III "Y WE KNOW THAT 3UPPOSE l IS INJECTIVE THEN CLEARLY WE . Np p NpMp 8 8 HAVE F BY #ON &>BA p BqY p !YwA qX VERSELY ASSUME THAT l IS NOT INJECTIVE AND Y X MAXJ&>BA J NpM 4HEN BY 0ARSEVALS 8 BA NpM p BqY 4HEOREM WE HAVE FYJ!Y AG 8 (ENCEWEHAVE N &>BA BA 8 8 MAX J &>BA J MAXJ &>BA J TqNpM > BA B A & BA B AIMl Np NpMp M M NpM N "Y ,EMMA .F pTq
IV 7E NOTE THAT 4HIS IS A CONTRADICTION 4HEREFORE EQUALITY
FYX Y w Y w qqqYM w ! qX IN HOLDS IF AND ONLY IF l IS INJECTIVE t wY w Y w qqqY ! qX M 3IMILAR RESULTS ARE STUDIED IN ;= 4HE MAIN ADVANTAGE OF OUR METHOD IS THAT IT IS
wYY qqqYM!Mp q X SIMPLE ENOUGH TO ANALYZE THE RELATIONSHIP 3EONGTAEK #HEE ET AL %42) *OURNAL VOLUME NUMBER $ECEMBER
BETWEEN CORRELATION IMMUNITY AND NONLIN 0ROOF "Y ,EMMA WE HAVE d e d e d e EARITY !ND WE DERIVE THE EXACT NONLIN N N N qqq EARITY WHILE ONLY A LOWER BOUND WAS GIVEN K K N d e d e d e d e IN ;= 4HIS FACT DRIVES US TO STUDY UNDER N N N N qqq K K K K WHAT CONDITIONS WE CAN MAKE THE NONLIN d e d e N N EARITY MAXIMAL N Np M d e d e d e &OR CONVENIENCE WE DENOTE BY #N K THE N N N qqq SET OF "OOLEAN FUNCTIONS GENERATED BY 4HE K K N d e d e d e OREM L N N N qqq K K Np 4HE FOLLOWING LEMMA IS USEFUL TO `ND d e d e d e d e N N N N CONDITIONS FOR MAXIMAL NONLINEARITY OF A qqq t K K N K M FUNCTION IN #N K ,EMMA )F L g LT THEN ,EMMA &OR GIVEN POSITIVE INTEGERS N AND d e d e d e L L L KN g f K f Np AND ANY POSITIVE IN T qqq g NpL K K L TEGER T LETLT BE THE SMALLEST L SUCH THAT d e d e d e WHERE LT IS THE VALUE DE`NED IN ,EMMA L L L T qqq g NpL K K L 0ROOF "Y THE DE`NITION OF LT d e d e d e LT LT LT NpL 4HEN WE HAVE L f T q LT IE MINFTq LT JT T qqq g T K K LT qqqG L 3INCE L g L WEHAVE 4O PROVE ,EMMA WE NEED SOME LEM T d e d e d e d e L L L L MAS ,EMMA IS WELL KNOWN AND WE OMIT T qqq qqq K K L L ITS PROOF d e d e dT e L L L gT T T qqq T ,EMMA &OR POSITIVE INTEGERS N AND K K K LT THE FOLLOWING HOLDS gNpLT g NpL t d e d e d e N Np Np .OW WE ARE READY TO PROVE ,EMMA K K K p 0ROOFOF,EMMA)F T THENTq LT L 3O LETS SHOW THAT T q LT g L WHEN T g ,EMMA &OR POSITIVE INTEGERS N AND K )F T g THEN THERE IS P P g SUCH THAT P P THE FOLLOWING HOLDS f T )FL pP f LT THEN d e d e d e N N N Tq LT g Tq LpP g PLpP L qqq K K N d e d e d e N N N (ENCE IF L pP f LT THE PROOF IS COMPLETED f qqq K K N )T REMAINS TO SHOW THAT THE CASE LT L pP %42) *OURNAL VOLUME NUMBER $ECEMBER 3EONGTAEK #HEE ET AL
CAN NOT HAPPEN 3UPPOSE LT L p P IE CANBEOBTAINEDIFM N pL T WHERE
LT f L pP p 4HEN BY ,EMMA L IS THE VALUE DE`NED IN ,EMMA d e d e L pPp L pPp 0ROOF "Y THE DE`NITION OF !Y T AND M T qqq K K SATISFY THE FOLLOWING INEQUALITY d e L pPp d e d e d e NpLpPp NpM NpM NpM g T qqq g M L pPp K K NpM !LSO BY ,EMMA d e d e d e L pPp L pPp L pPp T qqq )N IF WE SUBSTITUTE NpM WITH L THEN K K LpPp d e d e d e d e d e L pPp L pPp L L L NpL fP qqq T qqq g K K K K L d e L pPp LpPp Np Lp d e d e d e AND .F p T q BY 4HEOREM L pP L pP L pP fP qqq III (ENCE FOR EACH T T qqq THE K K L pP MAXIMUM NONLINEARITY IS OBTAINED IF L IS "Y APPLYING ,EMMA TO WE OBTAIN THE SMALLEST VALUE SATISFYING 4HAT d e d e d e Np LTp L pPp L pPp L pPp IS MAX.F MAX.F p T q T qqq M L K K LpPp 4HEREFORE BY ,EMMA d e d e d e L p L p L p f qqq MAX . K K L p MT F
Np LTp (ENCE BY MAXLT .F pMINT Tq d e d e d e p p Lp Lp Lp N pL qqq gNpLpPp t K K Lp 3INCE P g WE HAVE d e d e d e Lp Lp Lp qqq g NpL P 6 #/.3425#4)/. /& K K Lp #/22%,!4)/. )--5.% g NpLp &5.#4)/.3 7)4( "Y THE DE`NITION OF L L f L p "UT THIS #/.42/,,!",% IS A CONTRADICTION t ./.,).%!2)49 4HE FOLLOWING THEOREM IS ONE OF THE MA )N THIS SECTION BY USING 4HEOREM WE JOR RESULTS IN THIS PAPER SUGGEST A METHOD FOR CONSTRUCTING CORRE M 4HEOREM &OR F #N K THEMAXIMAL LATION IMMUNE FUNCTIONS WITH CONTROLLABLE Np Lp NONLINEARITY OF F IS .F p AND IT NONLINEARITY 3EONGTAEK #HEE ET AL %42) *OURNAL VOLUME NUMBER $ECEMBER
4HAT IS f K f NppD f Np WHERE THE -ETHOD FOR CONSTRUCTING K TH ORDER CORRELATION SMALLEST N IS IMMUNE FUNCTIONS 4HE SECOND IN 3TEP BY THE ABOVE K
Np Lp f Np (ENCE WITH NONLINEARITY .F p d e d e d e Np Np Np qqq )NPUT N N g THE NUMBER OF INPUT VARIABLES OF K K Np d e d e Np Np "OOLEAN FUNCTION g g K K K f K f Np CORRELATION IMMUNITY (ENCEWECAN`NDLL f N p SATISFYING 3TEP &OR K f L f N `ND THE SMALLEST L IN 3TEP SATISFYING 4HETHIRD IN3TEP THENUMBEROFVEC L d e d e d e TORS IN : WITH WEIGHT GREATER THAN OR EQUAL L L L qqq g NpL TO K IS K K L d e d e d e L L L qqq AND CALL THIS VALUE L K K L
NpL 3TEP #HOOSE VECTORS ! ! qqq! NpL p AND IT IS GREATER THAN OR EQUAL TO NpL BY
L NpL IN : WITH WEIGHT GREATER THAN OR 3TEP (ENCE WE CAN CHOOSE VECTORS WITH WEIGHT GREATER THAN OR EQUAL TO K EQUAL TO K 4HEFUNCTIONDE`NEDIN3TEPFUL`LLS N 3TEP $E`NE F : : BY THE REQUIREMENTS OF 4HEOREM 3O THE BAL ANCED K TH ORDER CORRELATION IMMUNE FUNC FYqqqYNpL XqqqXL FYX !Y q X Np TION IN 3TEP HAS NONLINEARITY .F p Lp 7E NOW DISCUSS THE ABOVE METHOD STEP BY STEP &IRST FOR THE INPUT SINCE A FUNC %XAMPLE 7E CONSTRUCT A BALANCED TION WITH ALGEBRAIC DEGREE IS CONSTANT IT AND NONLINEAR ST ORDER CORRELATION IMMUNE IS NOT BALANCED !ND A FUNCTION WITH ALGE FUNCTION F : : BRAIC DEGREE IE AN AbNE FUNCTION IS OF NONLINEARITY -OREOVER THE SUM OF COR )NPUT N K d e d e RELATION IMMUNITY K AND ALGEBRAIC DEGREE 3TEP 3INCE g p AND d e d e d e D FOR A FUNCTION F IS LESS THAN OR EQUAL TO g p WE HAVE L THE NUMBER OF INPUT VARIABLES N AND IN PAR TICULAR IF F IS BALANCED K D f N p ;= (ENCE FOR A BALANCED NONLINEAR AND CORRE 3TEP #HOOSE VECTORS !I : WITH LATION IMMUNE FUNCTION WE HAVE WT!I g SAY
K D f NpKg Dg ! ! %42) *OURNAL VOLUME NUMBER $ECEMBER 3EONGTAEK #HEE ET AL
! ! 4HUS d e d e d e ! ! L L L L pNpL g qqq K ! !
3INCE THE RIGHT HAND SIDEJ OFK IS POSITIVE 3TEP $E`NEF : : AS FOLLOWS N L NpL )N ALL L g t FYX !Y q X Y w Y w Y w X wX ,EMMA 4HE NECESSARY AND SUbCIENT wY w Y w YX wX CONDITION FOR TWO INTEGERS N AND} X TOh SATISFY wY w Y Y w X wX N THE FOLLOWING EQUATION IS X f wY w Y Y X wX d e d e d e N N N wY Y w Y w X wX qqq g Np X X N
wYY w YX wX 0ROOF )F N IS EVEN THEN wY Y Y w X wX wX d e d e d e N N N wY Y Y X wX wX qqq Np N N N d e d e d e 4HEN F IS BALANCED ST ORDER CORRELA N N N Np N N qqq N TION IMMUNE WITH .F p !ND } h SINCE wY!Y wY!Y wY!Y N 3O HOLDS IF X f !ND IFN IS AND wY!Y BY 4HEOREM IV ODD THEN DEGF NpL d e d e d e N N N Np N N qqq N 6) ./.,).%!2)49 !.$ } h N #/22%,!4)/. )--5.)49 3O HOLDS IF X f t M /& #N K 4HEOREM &OR A "OOLEAN FUNCTION F IN M Np B N C 7E NOW DISCUSS THE RELATIONSHIP BETWEEN #N K WE HAVE .F f p ANDTHE CORRELATION IMMUNITY AND NONLINEARITY OF EQUALITY HOLDS IF AND ONLY IF K SATIS`ES THE M FOLLOWING FUNCTIONS IN #N K N N N ,EMMA ,ET N K L BE GIVEN AS ,EMMA " # " # " # N p J K qqq g N ! ! ! 4HEN L g N K K 0ROOF "Y THE DE`NITION OF L IF N IS EVEN d e d e d e L L L J K NpL N qqq g K f IF N IS ODD K K L 3EONGTAEK #HEE ET AL %42) *OURNAL VOLUME NUMBER $ECEMBER
J K N M 0ROOF "Y ,EMMA SINCE L g WE IS THE MAXIMUM AMONG FUNCTIONS IN #N K HAVE THE RANGE OF CORRELATION IMMUNITY GIVEN IN #OROLLARY IS ACTUALLY CORRECT EXCEPT ONLY N Np Lp Np B C .F p f p FOR THE CASE N )N THAT CASE GIVES J K N K f WHERE INFACT K f AND THE EQUALITY HOLDS IF L (ENCE )T IS NATURAL TO QUESTION WHAT IS THE BY THE DE`NITION OF L EQUALITY HOLDS IF THE FOLLOWINGS ARE SATIS`ED RANGE OF K IF THE NONLINEARITY OF FUNCTION M du e du e du e IN #N K IS `XED 4HE FOLLOWING COROLLARY N N N qqq u WHICH CAN BE VERI`ED SIMILARLY TO #OROLLARY K K N SOLVES THE PROBLEM Np N g B C M u #OROLLARY )F THE FUNCTION IN #N K HAS d u N e d u N e d N e Np N Np Lp qqq u B C NONLINEARITY .F p THEN THE AP K K N PROXIMATE RANGE OF THE CORRELATION IMMU 3INCE HOLDS FOR ANY K EQUALITY HOLDS IF NITY K IS AS FOLLOWS AND ONLY IF HOLDS )F N IS ODD SINCE THE } P h B N C L p L p RIGHT HAND SIDE OF IS BY,EMMA Z NpLp HOLDS IF AND ONLY IF } P h L p f f NpL u } h K LZ N Np JNK K f WHERE 0: g Z_ _ AND : i . JNK IE K f t &OR THE CASE WHERE A FUNCTION IN #MK N JNK HAS MAXIMUM CORRELATION IMMUNITY WE "Y 4HEOREM WE CAN CONSTRUCT HAVE THE FOLLOWING TH ORDER CORRELATION IMMUNE FUNCTIONS WITH Np B N C NONLINEARITY p IF N IS ODD )F N IS #OROLLARY ,ET K N p 4HEN THE EVEN WE HAVE THE FOLLOWING MAXIMUM NONLINEARITY OF "OOLEAN FUNCTION F #MK IS. Np pNp Np Np N F #OROLLARY )F N IS EVEN AND .F p B N C THE APPROXIMATIVE UPPER BOUND OF THE &IGURE REPRESENTS THE RELATIONSHIP BET CORRELATION IMMUNITY K IN 4HEOREM IS WEEN CORRELATION IMMUNITY AND NONLINEARITY } R h M N N OF FUNCTIONS IN #N K FORN 3INCE THE K f NONLINEARITY IS TOO BIG TO INVESTIGATE ITS BE HAVIOR PRECISELY AS K INCREASES WE PRESENT 4HE PROOF OF #OROLLARY IS LEFT TO THE THE RELATIONSHIP BETWEEN THE CORRELATION IM APPENDIX MUNITY AND L p WHICH DETERMINES THE )N FACT FOR N qqq IF A FUNC NONLINEARITY IN &IG AS A SOLID LINE )N Np N TION HAS NONLINEARITY .F p WHICH &IG THE DOTTED CURVE WHICH IS DERIVED %42) *OURNAL VOLUME NUMBER $ECEMBER 3EONGTAEK #HEE ET AL
6)) #/.#,53)/.3 /
4HE MAIN RESULTS OF THIS PAPER ARE AS /! SOCIATED WITH STUDYING THE RELATIONSHIP BE / TWEEN CORRELATION IMMUNITY AND NONLINEAR /
ITY OF FUNCTIONS IN ;= ;= 3UCH RESULTS WERE
/ POSSIBLE BY MAKING THE GENERATING METHOD " 6 " 6 " 0 !!2 SIMPLE AND CLEAR ENOUGH TO DISCUSS SEVERAL PROPERTIES 4HIS PAPER MAY PROVIDE US WITH A NEW AVENUE TOWARDS STUDYING THE RELA &IG 2ELATIONSHIP BETWEEN NONLINEARITY AND COR RELATION IMMUNITY TIONSHIP BETWEEN CORRELATION IMMUNITY AND NONLINEARITY
" !00%.$)8 !
0ROOFOF#OROLLARY )F A RANDOM VARIABLE
8 HAS THE BINOMIAL DISTRIBUTION WITH PA )*)+$,- d e $ )*)+$,- RAMETERS N AND IE 8 i B N THEN " . ))-$ " 6 " 6 " N d ed e 0 !!2 8 N N 08 g X K K X &IG 2ELATIONSHIP BETWEEN L p LOG Np p (ENCE THE CONDITION FOR 4HEOREM HOLDS IF .F AND CORRELATION IMMUNITY 08 g K g FROM 4HEOREM IS CONCERNED WITH @GENERAL d e CORRELATION IMMUNE FUNCTIONS 7E NOTICE N WHERE 8 i B 4HEN THE FOLLOW THAT THE NONLINEARITY DOES NOT DECREASE UN d e N N TIL THE CORRELATION IMMUNITY K REACHES ABOUT ING HOLDS IF 8 i B AND IS BN C LARGE ENOUGH &ROM THE ABOVE TWO COROLLARIES WE CAN N CONSTRUCT "OOLEAN FUNCTIONS WITH CONTROL " K p # LABLE NONLINEARITY AND CORRELATION IMMU 08 f K 0 ": f R # N ! NITY %VEN THOUGH THE RANGE IN THE #OROL LARIES AND ARE APPROXIMATIVE THE ERROR | N c )N GENERAL WE ASSUME THAT IE RANGE WAS VERI`ED AS SMALL ENOUGH BY OUR N SIMULATIONS 3EONGTAEK #HEE ET AL %42) *OURNAL VOLUME NUMBER $ECEMBER
WHERE : i . 3INCE 0: f ;= 9 8IAN <#ORRELATION IMMUNITY OF "OOLEAN WEHAVE FUNCTIONS %LECTRONICS ,ETTERS VOL PP [ N ;= ' 8IAO AND * -ASSEY CRYPTOGRAPHY AND CRYPTANALYSIS PP [ ;= 0 #AMION # #ARLET 0 #HARPIN AND . +WANGJO +IM RECEIVED THE 3ENDRIER
INFORMATION SECURITY CRYPTOLOGY AND MICROWAVE COM MUNICATIONS (E IS NOW A MEMBER OF )%%% )!#2 AND)%)#%ANDADIRECTOROF+))3#
$AEHO +IM RECEIVED -3 DE GREE IN ELECTRICAL ENGINEERING FROM (ANYANG 5NIVERSITY IN (E JOINED %42) IN WHERE HE IS CURRENTLY WORKING AS $IRECTOR OF #OD ING 4ECHNOLOGY $EPARTMENT (IS RESEARCH INTERESTS INCLUDE CODING THEORY INFORMA TION SECURITY AND CRYPTOLOGY (E IS A REGULAR MEMBER OF THE +OREAN #OMMUNICATION 0ROFESSIONAL %NGINEER !SSOCIATION