COMBINATORIAL METHOD OF POLYNOMIAL EXPANSION OF SYMMETRIC BOOLEAN FUNCTIONS

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1 COMBINATORIAL MTHOD O POLYNOMIAL XPANSION O SYMMTRIC BOOLAN UNCTIONS Dala A. Gorodecky The Uted Isttute of Iformatcs Prolems of Natoal Academy of Sceces of Belarus, Msk,, Belarus, dala.gorodecky@gmal.com. Astract A ovel polyomal expaso method of symmetrc Boolea fuctos s descred. The method s effcet for symmetrc Boolea fucto wth small set of valued umers ad has the lear complexty for elemetary symmetrc Boolea fuctos, whle the complexty of the kow methods for ths class of fuctos s quadratc. The proposed method s ased o the cosequece of the comatoral Lucas theorem. Keywords: polyomal expaso, symmetrc Boolea fucto, carrer vector, reduced Zhegalk spectrum, complexty. Itroducto The polyomal expaso s amog the most complex tasks of the dscrete mathematcs. The polyomal expaso ca e used to defe the ffty-ffty dstruto of ad the Stehaus tragle, to sythesze modular summators, to fd a algerac mmuty cryptography ad to solve varous theoretcal prolems ad practcal applcatos. Because of hgh computatoal complexty of geerato of the polyomal for a artrary Boolea fucto the uversal methods of the polyomal expaso are ot effectve. Therefore the methods of geerato of expasos for varous classes of Boolea fuctos are more effectve. Oe of these classes s symmetrc Boolea fuctos (SB). It s kow may methods of the polyomal expaso of SB. Oe of the most effectve methods s the traseut tragle method []. It has the complexty O. The kow methods have the redudat computatos,.e. the termedate computatos should e produced to geerate the polyomal expaso. The artcle represets the method of the polyomal expaso wth the complexty O partcular cases. The method could e appled to solve the task of polyomal expaso, as well as the reverse task,.e. represetato of the fucto descred y the polyomal. The method s ased o the cosequece of the comatoral Lucas theorem, sce t s referred as the comatoral method.. Ma deftos A artrary Boolea fucto X of the varales, where X x x..., wth uchaged value after swappg ay couple of varales a, a a,..., x ad,, x, x, where j ad, j,,...,, s called SB. SB of the varales s charactersed y the set of valued umers A a, a,..., a r. The fucto s equal f ad oly f the set of varales x, x,..., x has exactly a umers of s, where a, r ad r. These SBs are referred r a as. If r, the a fucto X s called elemetary SB (SB). There s oe-to-oe correspodece etwee SB a, a a,..., r ad j ts ary code,,..., the carrer vector [] (or the reduced truth vector [3]), where the th etry

2 s a value of the fucto wth the umers of s, where f ad oly f the s the valued umer of the SB. The followg formula s true for a artrary SB : X X X. I other words,. () Postve polarty Reed-Muller polyomal (all varales are ucomplemeted) s called as Zhegalk polyomal ad s referred as P. SB of the varales s called the polyomal-uate SB (PUSB or homogeeous SB [4]), f the Zhegalk polyomal form P cotas rak products wth the postve lterals, where. Ths fucto referred as, x x... x, xx... xx... x x,... xx... x.. Hece t follows I geeral case, the polyomal form P of SB X P x x... x xx... xx... xx... xx... x, where,,..., follows ca e represeted as:, s the reduced Zhegalk (Reed-Muller) spectrum of SB. It X X. () rom the other had PUSB of the varales s charactersed y the set of polyomal B,...,. The j th etry of the reduced Zhegalk spectrum umers..., q,,, s equal f ad oly f, where j q ad q. If q, the a fucto j s called the elemetary PUSB (PUSB). The artcle provdes the method of the trasformato of the reduced truth vector the reduced spectrum a a,,..., a r.,.e.,,..., q to,,..., q 3. Comatoral method of geerato of the carrer vector to a a, ad ackwards,.e., a,..., r to The comatoral method of the geeratg of the reduced truth vector,,..., q a a a reduced spectrum,,..., r s proposed elow. 3.. Geerato of the carrer vector ad the

3 The process of the geeratg of the carrer vector demostrated o the example. of the PUSB could e xample. Let s assume that t s ecessary to get the carrer vector,,..., the PUSB X X. rom the codto t follows that,,,,,, ad x x x 3 x x4 x x 5 x x x x 3 x x4 x x 5 x x x 3 x 4 x 3 x 5 x 3 x x 4 x 5 x 4 x x 5 x for Note, that umer of the colum s equal to the umer of factors the colum whch are. The polyomal P cotas 5 cluded the polyomal of the fucto X rak products. To geerate the carrer vector,,,,, defed wth the followg argumets, where, : assumg the cotas 3 4 5, the th etry should e. But t s mpossle, ecause the polyomal (3) of X does t cota term. I ths case ad therefore ; assumg the cotas. Accordg to the defto of the SB polyomal (3) s equal for x ad x x3... x. But t s mpossle. I ths case ad therefore ; assumg the cotas polyomal (3) s equal for x. Accordg to the defto of the SB x ad x x x x. Thus the oly factor from the x x3 x 4 x5 x frst colum of the polyomal (3) s equal. I ths case ad therefore ; assumg 3 the cotas. Accordg to the defto of the SB polyomal (3) s equal for x ad. Thus the factors from the frst ad secod colums of the polyomal (3) are equal. Sce the umer of the uty compoets s the odd umer, the ths case ad therefore ; assumg 4 the 4 cotas (3). Accordg to the defto of the SB polyomal (3) s equal for x x x3 x4 ad x 5 x. Sce the factors from the frst, secod ad thrd colums of the polyomal (3) are equal, the the umer of the uty compoets s the eve umer. I ths case ad therefore ; assumg 5 the cotas Accordg to the defto of the SB polyomal (3) s equal for x x x3 x4 x5 ad x. Sce the factors from the frst, secod, thrd ad fourth colums of the polyomal (3) are equal, the the umer of the uty compoets s the eve umer. I ths case ad therefore ; 5 3

4 assumg the cotas 3. Accordg to the defto of the SB polyomal (3) s equal for x x x3 x4 x5 x. Sce the factors from all colums of the polyomal (3) are equal, thr the umer of the uty compoets s the odd umer. I ths case ad therefore. As the result the carrer vector of the PUSB 3 s,,,,,, ad P. It s worth to pay atteto to the fact that the value of the polyomal depeds oly o the party umer of uty factors. The reasog used the example may e summarzed wth the theorem. Theorem. The th etry x, x,..., x s calculated y usg the formula: of the carrer vector,..., of the PUSB,, f mod ; (4) otherwse, where,. Proof. Let s cosder three cases of relatos ad. The frst case where. The the umer of the uty terms s less the the rak products ad (see the frst ad secod cases of the example ). Therefore. just oe The secod case where rak term of the PUSB the example ). I ths case. fucto The thrd case where,.e. x x... x ad x x... x. Thus s equal ad x x x P... (see the thrd case of,.e. x x... x... x ad x... x. Thus the s represeted y the polyomal x x... x xx... xx... x x... x x... xx... P P s determed y eve-odd of. Thus, f mod ; otherwse. The statemet of the theorem s proved. As a result of Theorem the carrer vector of the PUSB followg form x. Sce the value of, f mod ; I ths case otherwse. correspods to the,,...,,,,...,. (5)

5 Cosequece of the Lucas theorem s helpful for calculato usg the formula (4). It determes the eve-odd of the umer ad as follows. Theorem. (Cosequece of the Lucas theorem) [5]. The umer mod each t of s o more tha the same t of, where decmal represetato. Note, that the ary legth,..., ad,,..., s defed as log ad log respectvely. xample. Let s defe eve-odd of the umer usg Theorem ad assumg ad for two cases a) ad ) 5. log,,, ad The legth of s 4, the 4, 3 a) log, the, ; ) 5 3, the,. log 3, or the case a) the ary represetatos ad are comparale ad satsfy the codto of Theorem, as pctured gure a). or the case ) the ary represetatos of ad are ot comparale ad do ot satsfy the codto of Theorem, as pctured gure ). 4 3 a) ) gure. Defto of eve-odd of the umers a) ; ) 5 As a result, the case a) the umer s the odd ad thus mod ; the case ) the umer s the eve ad thus mod. 5 5 Let's geerate the carrer vector for the fucto aove example usg Theorem ad Theorem.. xample 3. Let s assume that t s ecessary to geerate rom the codto t follows,,,,,,, ad,,,,, Accordg to the formula (5) 3 4 5, 3 4, 5, ot volved the comparso satsfed the codto of Theorem does ot satsfy the codto of Theorem Therefore, order to fd, the eve-odd order of the Bomal coeffcets 3 4 5,,, respectvely should e defed. rom the Theorem they could e defed as show o fgure.

6 The fgure s aalogous to represetato as follows: 3 3 ; ; 5 ;. As the result the carrer vector s,,,,,, ad 3 P. The procedure of calculatg of the etres of the carrer vector usg cosequece of the Lucas theorem s called comatoral methods mod mod mod mod ot volved the comparso does ot satsfy the codto of Theorem satsfy the codto of Theorem gure. Defto of eve-odd of the umers 3.. Geerato of the carrer vector q,,..., 3 4 5,,, The comatoral method of geeratg of the carrer vector for the PUSB x x,..., x ca e geeralzed for a artrary PUSB,...,,,,...,, q PUSB q x, x,..., x wth the followg theorem. Theorem 3. The q,,..., th etry q of the carrer vector,,...,..., s calculated wth the followg formula:,, of the, f... mod ; q () otherwse, where,. Note, that for j s meagless, where j, q, therefore let s assume j for j. j The proof of Theorem 3 follows from Theorem. xample 4. Let s assume that t s ecessary to geerate rom the codto t follows,,,,,,,,,,.. Accordg to the formula (5) t follows ad. Thus,,,,,,,,, , 3 4 5

7 Accordg to the formula () ad Theorem t s easy to defe, 7, 8, 9 ad as show o the fgure mod mod mod mod mod mod mod mod mod mod mod mod mod mod 9 mod ot volved the comparso does ot satsfy the codto of Theorem mod satsfy the codto of Theorem gure 3. Calculatg of the, 7, 8, 9, The fgure 3 s aalogous to represetato as follows: mod 5 ; 7 7 mod mod mod 5 7 for 7 ;

8 mod mod mod mod mod mod mod mod ; mod mod mod mod Thus ad. As the result the carrer vector of the PUSB s,,,,,,,,,, ad P. 4. Geerato of the reduced spectrum q,,..., a a a The comatoral method of the geerato of the carrer vector,,..., r a a a,,..., r a, a,..., ar, where a a To solve the task of the geeratg of the reduced spectrum, a,..., r appled to the geerato of the reduced spectrum Theorem 3 ca e adapted to the two followg forms. Theorem 4. The th etry a a ca e s the SB. of the reduced spectrum,..., x, x,..., x s calculated wth the followg formula: where SB a,. Theorem 5. The th etry a a a,,..., r s calculated wth the followg formula: where a,. Note, that ad a j. a r a, a,..., Theorem ad of the PUSB,, f mod ; a (7) otherwse, a, a ar of the reduced spectrum,...,,..., of the,, f... mod ; a a ar (8) otherwse, a j for a j s meagless, where j, r, therefore let s assume of the SB Accordg to Theorem 4 ad Theorem 5 the reduced spectrum,..., a r correspods to the followg form,

9 ,,...,,, a,...,. (9) a a The example of the applcato of Theorem 5 wll e cosdered.,3 xample 5. Let s geerate the reduced spectrum. 7,3 rom the codto t follows the carrer vector s 7,,,,,,, the formula (9) t follows,,3 ad,,,,,, , 7. Accordg to Accordg to the formula (8) ad Theorem 5 t s easy to defe 3, 4, 5, ad 7 as show elow: 3 3 mod 3 mod mod mod 3 mod mod 4 ; 5 5 mod 3 mod mod 5 mod 3 mod mod 7 7 mod 3 mod mod 7,3 As the result the reduced spectrum of the fucto 7,,,,,,, ad P, The complexty of the comatoral method,3 7 s The complexty of the proposed method ca e defed as the umer of the ary a operatos XOR (or OR) ad s referred S for the PUSB (or SB ) ad S for the PUSB,,..., a, a,..., a q r (or SB ). The postve relatoshp of two ary vectors s x, x,..., x y, y y t t,...,, f x y, where, t. I ths way to defe the relatoshp x y the followg codto should e satsfed t t x y. () Therefore accordg to the codto of Theorem the complexty (the umer of operatos mod s log. rom Theorem t follows ()) of the computato of the umer the complexty of the computato of the carrer vector,..., s, S log. () rom Theorem 3 t follows the complexty of the computato of the carrer vector q,,...,,..., s,

10 S log log... log log q q 3... q q q log q log log. log a a The complexty of the calculato of ad, a,..., ar ad () respectvely.. Dscusso () ca e calculated wth () There are some effectve methods to solve the task of polyomal expaso,.e. geeratg a, a,..., ar,..., ad the reverse task of geeratg of the of the reduced spectrum,,,..., r carrer vector,...,,. Oe of these methods s the traseut tragle method. It was orgally proposed y V.P. Supru for SB []. The the method was geeralzed for artrary Boolea fuctos []. The traseut tragle method s the most effectve method for polyomal expaso of the SB x x,..., ad has the complexty, x O. The traseut tragle form s geerated from the upper ase to the ottom usg the XORoperato (see example ). Thus the umer of XOR-operatos defes the complexty S T of the traseut tragle method ad s S T. (3) There s the example of the mplemetato of the traseut tragle method for geeratg of the carrer vector,,...,. xample. Let s assume the reduced spectrum,,,,,,. I order to geerate the traseut tragle method wll e used. rom the codto the upper ase of the tragle wll,,,,,, the form as follow:,.e. ad ad t takes Accordg to the traseut tragle method the left sde of the tragle correspods to the,,,,,,. Therefore 3. reduced carrer vector ad t s equal to

11 Usg the formula (3) the complexty of the computato of the wth the traseut tragle method s S T 5. O the other sde, to complexty of performg the same task usg the comatoral method, accordg to the formula () ad as show example 3, s S 8. rstly let s compare the complexty S (formula ()) of the comatoral method proposed the artcle ad the complexty S T (formula (3)) of the traseut tragle method for a a SB (or ). The llustrato of the comparso of S ad S T s show o the gure 4. As t ca e see at fgure 4 the comatoral method for PUSB has the lear complexty. The proposed method ca provde te tmes more effcecy for 8 varales comparso wth the traseut tragle method. The complexty of the comatoral method (formula ()) s calculated for the worst case,.e. for the PUSB, where. The complexty of the comatoral method comparso wth the complexty of the traseut tragle method for,,..., r PUSB, where r, strogly depeds o umers cluded the set of the polyomal umers. As a result, the tale demostrates the threshold of the effcecy of the comatoral method comparso wth the traseut tragle method. gure 4. The comparso of the complexty S The secod colum cotas of the power set of the polyomal the comatoral method ad the complexty S T of umers B for whch the the traseut tragle method complexty S ad S T s approxmately equal. The thrd colum cotas the set of the polyomal umers B for whch the complextes of oth methods are the same. Ay other set of the polyomal umers B provdes a lower complexty of the comatoral method for the umer of the varales specfed the frst colum. The fourth colum shows the rato of the set of the polyomal umers to all varales specfed the frst colum. Two rght colums show the comparale complextes of the comatoral ad the traseut tragle methods.

12 Tale The effcecy of the comatoral method ad the traseut tragle method Numer of the varales that the fucto,,..., r Numer r of the polyomal umers,..., for, r Set of the polyomal umers B,...,, r Percetage of the umer r of the varales, % Complexty of the comatoral method, depeds o S S T 3 {,3,4 } { 4,...,8 } { 4,...,9} { 8,..., 4} {,..., 8} 75 9 {,...,4} {,...,5} {,...,} {,...,7} {,...,8} { 3,...,57} { 4,..., 7} { 8,...,3} { 5,...,39} { 5,...,753} Coclusos S Complexty of the traseut tragle method, S The comatoral method s a ew method of geeratg of the carrer vector q,,..., a a a ad the reduced spectrum q,,..., of SB,.e. polyomal expaso of SB. The proposed method s the lear complexty ad the complexty of the kow methods (for example, traseut tragle method) s quadratc for PUSB (or SB). The comatoral method provdes hgh effcecy for small umer of varales for PUSB (or SB). Refereces [] V.P. Supru Polyomal xpresso of Symmetrc Boolea uctos, Sovet Joural of Computer ad Systems Scece, Vol. 3,, Nov.-Dec. 985, p (traslato from Izv. Adad. SSSR, Tekhcheskaa Keretka, 985, 4, p. 3-7). [] J.T. Butler, G.W. Dueck, S.N. Yaushkevch, V.P. Shmerko O the use of traseut tragle to sythess fxed-polarty Reed-Muller expasos, Proceedgs of the Reed-Muller Workshop 9, May 3-4, 9, Naha. Okawa, Japa, p. 9-. [3] T. Sasao, J.T. Butler The egefucto of the Reed-Muller trasformato, Proceedgs Reed-Muller 7 Workshop, May, 7, Oslo, Norway. [4] A. Braeke, B. Preeel O the Algerac Immuty of Symmetrc Boolea uctos, Progress Cryptology INDOCRYPT 5, Lecture Notes Computer Scece, Vol. 3797, 5, p [5] A. Gravlle Arthmetc Propertes of Bomal Coeffcets I: Bomal coeffcets modulo prme powers, Caada Mathematcal Socety Coferece Proceedgs, 997, p [] V.P. Supru Tale Method for Polyomal Decomposto of Boolea uctos, Keretka,, 987, p. -7 ( Russa). T