ASYMPTOTIC BEHAVIOR OF THE EXPONENTIAL SUM OF PERTURBATIONS OF SYMMETRIC POLYNOMIALS

ASYMPTOTIC BEHAVIOR OF THE EXPONENTIAL SUM OF PERTURBATIONS OF SYMMETRIC POLYNOMIALS FRANCIS N. CASTRO AND LUIS A. MEDINA Abstract. In ths paper we consder perturbatons of syetrc boolean functons σ n,k σ n,ks n n-varable and degree k s. We copute the asyptotc behavor of boolean functons of the type σ n,k σ n,ks F X,..., X j for j fxed. In partcular, we characterze all the boolean functons of the type σ n,k σ n,ks F X,..., X j that are asyptotc balanced. We also present an algorth that coputes the asyptotc behavor of a faly of Boolean functons fro one eber of the faly. Fnally, as a byproduct of our results, we provde a relaton between the party of fales of sus of bnoal coeffcents.. Introducton Boolean functons are very portant n the theory of error-correctng codes as well as n cryptography. These functons are beautful cobnatoral objects wth rch cobnatoral propertes. In partcular, syetrc booleans have receved a lot attenton for ther advantage snce they can be dentfed by an n bt vector for exaple, see [3, 7, 9, 0, 9, 6]. One can fnd any papers and books dscussng the propertes of boolean functons see, for exaple, [5, 3,, 6]. The subject can be studed fro the pont of vew of coplexty theory or fro the algebrac pont of vew as we do n ths paper, where we copute the asyptotc behavor of exponental sus of perturbed syetrc boolean functons. The correlaton between two Boolean functons of n nputs s defned as the nuber of tes the functons agree nus the nuber of tes they dsagree all dvded by n,.e.,. CF, F = n Fx,...,xnFx,...,xn. x,...,x n {0,} In ths paper we are nterested n the case when F F can be wrtten as σ n,k σ n,k σ n,ks F X,..., X j, where σ n,k s the eleentary syetrc boolean polynoal of degree k n the n varables X,..., X n and F s a boolean functon n the frst j-varables j fxed. We wrte Cσ n,k σ n,k σ n,ks F nstead of CF, F. In [4], A. Canteaut and M. Vdeau studed n detal syetrc boolean functons. They establshed a lnk between the perodcty of the splfed value vector of a syetrc Boolean functon and ts degree. Recently, a new Date: January 8, 03. 00 Matheatcs Subject Classfcaton. 05E05, T3. Key words and phrases. Syetrc boolean functons, exponental sus, recurrences.

FRANCIS N. CASTRO AND LUIS A. MEDINA cryptographc property have been ntroduced. Ths property s called algebrac unty of boolean functons [0]. All the syetrc boolean functons wth axal algebrac unty have been found [6, 9,, 7]. In [7, ], Carlet and Olejár-Stanek studed the asyptotc nonlnearty of boolean functons. Later, n [3], Roder proved ther results by showng that the nonlnearty of alost all boolean functons s equal to log. In [4, 5], the authors consdered rotaton syetrc Boolean functons of degree 3 n n varables. In [4], Blesch-Cusck-Padgett provded an algorth for fndng a recurson for the truth table of any cubc rotaton syetrc Boolean functon generated by a onoal ther work reduced the coputatonal coplexty. In [5], Cusck proved the coputatonal coplexty found n [4] to soethng lnear n the nuber of varables n assung the ld condton that the roots of the characterstc polynoal are dstnct. In[3], J. Ca et al. coputed a closed forula for the correlaton between any two syetrc Boolean functons. Ths forula ples that Cσ n,k σ n,k σ n,ks satsfes a hoogeneous lnear recurrence wth nteger coeffcents and provdes an upper bound for the degree of the nal recurrence of ths type that Cσ n,k σ n,k σ n,ks satsfes. We obtan a hoogeneous lnear recurrence wth nteger coeffcents satsfed by Cσ n,k σ n,k σ n,ks F. Ths generalze the result of J. Ca et al. In ths paper, we prove that. l Cσ n,k σ n,k σ n,ks F X,..., X j = 0, f and only f CF X,..., X j s balanced or σ n,k σ n,ks s asyptotc balanced. In [], Cusck et al. conjectured that there no nonlnear balanced eleentary syetrc polynoals except for the eleentary syetrc boolean functon of degree k = r n r l varables, where r and l are any postve ntegers. Ths conjecture has been the central topc for several papers. Recent results about the Cusck et al. s conjecture can be found n [7]. In [8], we characterze the asyptotc behavor of the eleentary syetrc boolean functons,.e., l Cσ n,k = wk, where w wk k s the Hang weght of k. In partcular, ths ples that Cusck et al. s conjecture s true for large n. Cobnng the last two results, we have that when k s not a power of two, σ n,k F X,..., X j s asyptotc balanced f and only f CF s balanced. In general, we copute the asyptotc value.3 l Cσ n,k σ n,k σ n,ks F X,..., X j = c 0 k,..., k r SF j, where SF and c 0 k,, k s are defned n secton. We present an algorth to copute the asyptotc behavor of a faly of syetrc boolean functon gven the asyptotc behavor of one of ts ebers. Fnally, we use these asyptotc coeffcents to prove soe results about the party of soe sus of bnoal coeffcents.

ASYMPTOTIC BEHAVIOR OF PERTURBATIONS OF SYMMETRIC FUNCTIONS 3. Prelnares Let F be the bnary feld, F n = {x,..., x n x F, =,..., n}, and F X = F X,..., X n be a polynoal n n varables over F. The exponental su assocated to F over F s:. SF = x F n F x. A boolean functon F X s called balanced f SF = 0. Ths property s portant for soe applcatons n cryptography. In [8], F. Castro and L. Medna studed exponental sus of eleentary syetrc polynoals. Let σ n,k be the eleentary syetrc polynoal n n varables of degree k. Suppose that k < < k s are ntegers. Castro and Medna were able to show that the sequence {Sσ n,k σ n,ks } n N satsfes a hoogeneous lnear recurrence wth nteger coeffcents. They used ths recurrence to study the asyptotc behavor of such sequences. In partcular, they exploted the fact that. l n Sσ n,k σ n,ks = c 0 k,, k s, where.3 c 0 k,, k s = r r k ks, and r = log k s. As part of ther study, they ntroduced the concept of an asyptotcally balanced functon. We say that σ n,k σ n,ks s asyptotcally balanced f =0.4 l n Sσ n,k σ n,ks = 0. Note that f a functon σ n,k σ n,ks s not asyptotcally balanced, then we know that t s not balanced for all suffcently large n. The authors provded fales of syetrc polynoals that were asyptotcally balanced and fales that were not. For exaple, they show that f k < < k s are ntegers wth k s a power of, then the polynoal σ n,k σ n,ks s asyptotcally balanced. In ths artcle, we study soe perturbatons of syetrc functons and extend soe of the results of [8] to the. Recall that σ n,k s the eleentary syetrc polynoal of degree k n the varables X,, X n. Suppose that j < n and let F X be a bnary polynoal n the varables X,, X j the frst j varables n X,, X n. We are nterested n the exponental su of polynoals of the for.5 σ n,k σ n,ks F X, where k < < k s. In partcular, we study the sequence {Sσ n,k σ n,ks F X} n N where k,, k s and F X are fxed and n vares. Of specal nterest s the asyptotc behavor of the exponental su of.5. We want to know the value f exsts of l n Sσ n,k σ n,ks F X.

4 FRANCIS N. CASTRO AND LUIS A. MEDINA Exaple.. Consder the sequence.6 {Sσ n,5 X X X } n N. Usng Matheatca 8.0 wth ts bult-n functon FndLnearRecurrence, we guess that the sequence.6 satsfes the recurrence.7 x n = 6x n 4x n 6x n 3 0x n 4 4x n 5. Moreover, t can be proved, usng soe of the achnery presented n ths paper, that l n Sσ n,5 X X X = 4. In Fgure you can see a graphcal representaton of ths lt. represents Sσ n,5 X X X / n. The lne y = /4 s n red. The blue dots 0.5 0.4 0.3 0. 0. Fgure. Graphcal representaton of Sσ n,5 X X X / n. In the next secton we show that Exaple. s not a concdence, but the rule. In other words, we show that the exponental su of these perturbatons satsfes the sae recurrence as the exponental su of syetrc polynoals. In Secton 4, we study the asyptotc behavor of the. 3. The lnear recurrence Let k < < k s be ntegers and F X be a bnary polynoal n the varables X,, X j j fxed. In ths secton we show that {Sσ n,k σ n,ks F X} n N satsfes the recurrence r 3. x n = r x n, = where r = log k s. We start wth the followng prelnary results. Defnton 3.. For x F n, let w x be the Hang weght of x, n other words, w x s the nuber of entres of x that are one. For exaple, w 0,,, 0, = 3.

ASYMPTOTIC BEHAVIOR OF PERTURBATIONS OF SYMMETRIC FUNCTIONS 5 Theore 3.. Suppose k < < k s are ntegers. Let F X be a bnary polynoal n the varables X,, X j. Defne 3. C F = for = 0,,, j. Then, 3.3 Sσ n,k σ n,ks F X = =0 x F j wth wx= F x j C F S [σ n j,k σ n j,ks ]. Proof. We provde the proof of the case of one eleentary syetrc polynoal,.e. we show that j Sσ n,k F X = C F S σ n j,k =0 s true. The purpose of dong ths s to splfy the notaton and the wrtng of the proof. The general case follows n a slar anner. Recall that Sσ n,k F X = σ n,kxf x. x F n =0 Ths can be re-wrtten as Sσ n,k F X = F x σ n,kx =0 3.4 = x F n F x σ n,kx F x σ n,kx x j 0 Fn x j Fn F x σ n,kx, x j j Fn where x j represents a tuple n Fn that has exactly ones n the frst j entres. Let us assgn values to the frst j entres of the varable X and let the rest of t vary. Fx ths assgnent. Suppose t has ones n the frst j entres,.e. the assgnent has the for δ,, δ j, X j,, X n, where the δ {0, } are fxed, δ δ j =, and X j,, X n are bnary varables. It s not hard to see that n ths case the eleentary syetrc polynoal σ n,k gets transfor to =0 σn j,k, where the varables of σ n j,k are X j,, X n. Thus, for ths partcular assgnent, we have δ,,δ j,x j,,x n F x σ n,kx = F δ,,δj S =0 σ n j,k.

6 FRANCIS N. CASTRO AND LUIS A. MEDINA But ths yelds x j Fn 3.5 F x σ n,kx Note that 3.4 and 3.5 ply the theore. = F x S σ n j,k x F j wth wx= =0 = C F S σ n j,k. Reark. There are three thngs about equaton 3.3. Frst, note that the values of that are nsde the exponental su can be taken odulo two, snce only the party atters. Second, f k l < 0, then the ter σ n j,kl does not exst. Fnally, n the case that k l = 0, then σ n j,0 should be nterpreted as. Corollary 3.3. Let k < < k s be ntegers and F X a bnary polynoal n the varables X,, X j j fxed. Consder the sequence =0 3.6 {Sσ n,k σ n,ks F X} n N and let r = log k s. Then, 3.6 satsfes recurrence 3.. In partcular, 3.7 l n Sσ n,k σ n,ks F X exsts. Proof. Theore 3. ples Sσ n,k σ n,ks F X = j C F S [σ n j,k σ n j,ks ]. =0 However, we know that each S =0 [σn j,k σ n j,ks ] satsfes recurrence 3., see [8] for detals. Thus, Sσ n,k σ n,ks F X satsfes the sae recurrence. Fnally, snce the bggest odulo of the roots of the characterstc polynoals assocated to 3. s see [8], then 3.8 l n Sσ n,k σ n,ks F X exsts. Exaple 3.4. Consder the perturbaton =0 σ n,5 σ n,3 X X X 3 X 4 X 5 X 6. Apply Theore 3. to get the coeffcents C 0 F = C F = 6 C F = 5 C 3 F = 0 C 4 F = 5 C 5 F = 6 C 6 F =.

ASYMPTOTIC BEHAVIOR OF PERTURBATIONS OF SYMMETRIC FUNCTIONS 7 Take each bnoal coeffcent nsde the exponental sus odulo to obtan Sσ n,5 σ n,3 X X X 3 X 4 X 5 X 6 = Sσ n 6,5 σ n 6,5 6Sσ n 6,5 σ n 6,4 σ n 6,3 σ n 6, 5Sσ n 6,5 σ n 6, 0Sσ n 6,5 σ n 6,4 σ n 6, 5Sσ n 6,5 σ n 6,3 σ n 6, 6Sσ n 6,5 σ n 6,4 σ n 6,3 σ n 6, σ n 6, Sσ n 6,5. Note that the thrd and ffth coeffcents appear as negatve nstead of postve. The reason for ths s that the ter σ n 6,0 = appears nsde the correspondng exponental su and SGX = SGX for any bnary polynoal G. Now, Corollary 3.3 tells us that l n Sσ n,5 σ n,3 X X X 3 X 4 X 5 X 6 exsts. If fact, n the next secton we show that ths lt s 3/64. Below you can see a graphcal representaton of ths fact. The blue dots correspond to Sσ n,5 σ n,3 X X X 3 X 4 X 5 X 6 / n. The red lne correspond to y = 3/64. 0.7 0.6 0.5 0.4 0.3 Fgure. Graphcal representaton of Sσ n,5 σ n,3 X X X 3 X 4 X 5 X 6 / n. Exaple 3.5. Let us go back to the sequence n Exaple.,.e. {Sσ n,5 X X X } n N. Followng Theore 3. wth F X = X X X, we get C 0 F = C F = 0 C F =. Ths ples that Sσ n,5 X X X = Sσ n,5 Sσ n,5 σ n,3.

8 FRANCIS N. CASTRO AND LUIS A. MEDINA Now, usng the theory presented n [8], t can be showed that both, Sσ n,5 and Sσ n,5 σ n,3, satsfy recurrence.7. Snce Sσ n,5 X X X s a lnear cobnaton of the, then Sσ n,5 X X X satsfes the sae recurrence. Also, l n Sσ n,5 X X X = l = l n Sσ n,5 Sσ n,5 σ n,3 Sσn,5 4 n Sσ n,5 σ n,3 n = 4 c 05 c 0 5, 3 = 4 = 4, where the values of c 0 5 and c 0 5, 3 can be obtaned fro.3. Reark. Note that Corollary 3.3 tells us that Sσ n,5 X X X satsfes the recurrence 3.9 x n = 8x n 8x n 56x n 3 70x n 4 56x n 5 8x n 6 8x n 7, but we proved that ths sequence satsfes recurrence.7, whch has less order. Therefore, even though Corollary 3.3 provdes us wth a lnear recurrence for these perturbatons, ths recurrence s not necessary the nal one. It s nterestng to know what s the nal hoogeneous lnear recurrence wth nteger coeffcents that these sequences satsfy. However, our an focus n ths artcle s to study the asyptotc behavor of these perturbatons and knowng that they satsfy a lnear recurrence s enough. Thus, n ths paper we do not consder the proble of fndng the nal hoogeneous lnear recurrence wth nteger coeffcents that they satsfy. 4. Asyptotc behavor In ths secton we study the asyptotc behavor of the exponental su of perturbatons of type.5. We are nterested n the queston of when are these perturbatons asyptotcally balanced. We start wth the followng exaple. Exaple 4.. Consder the polynoal σ n,6 X X X 3 X 4 X 5. We already know that σ n,6 s asyptotcally balanced. In ths case, t turns out that the perturbaton s also asyptotcally balanced. In Fgure 3 you can see a graphcal representaton of ths fact. The blue dots represent Sσ n,6 / n and the red dots represent Sσ n,6 X X X 3 X 4 X 5 / n. When we started the study of these perturbatons, we observed that n the partcular case when a syetrc polynoal was dsturbed by a lnear polynoal,.e. 4. σ n,k σ n,ks X X j, the result was an asyptotcally balanced functon. Exaple 4.. Consder the perturbaton 4. σ n,7 X X X 3.

ASYMPTOTIC BEHAVIOR OF PERTURBATIONS OF SYMMETRIC FUNCTIONS 9.0 0.5 00 00 300 400 500 600 0.5.0 Fgure 3. Graphcal representaton of Sσ n,6 / n Sσ n,6 X X X 3 X 4 X 5 / n red. blue vs It turns out that ths perturbaton s asyptotcally balanced, even though σ n,7 s not. Fgure 4 shows a graphcal representaton of the exponental sus of σ n,7 and σ n,7 X X X 3. The blue dots represent Sσ n,7 / n and the red dots represent Sσ n,7 X X X 3 / n..0 0.8 0.6 0.4 0. 0 0 30 40 50 60 Fgure 4. Graphcal representaton of Sσ n,7 / n blue vs Sσ n,7 X X X 3 / n red. Perturbatons of the for σ n,k σ n,ks X X j are not the only exaples n whch you start wth a functon that s not asyptotcally balanced, but after you perturb t the result s an asyptotcally balanced functon. Exaple 4.3. Consder the perturbaton σ n,5 X X X X 3 X X 3. The polynoal σ n,5 s not asyptotcally balanced, but ths perturbaton s. Ths s a strkng exaple snce the asyptotc behavor of σ n,5 s close to, but addng X X X X 3 X X 3 to σ n,5 akes the dstrbuton of 0 s and s close to each other. Fgure 5 s graphcal representaton of the exponental sus of σ n,5 and σ n,5 X X X X 3 X X 3. The blue dots correspond to σ n,5 and the red dots to the perturbaton. The two exaples above have soethng n coon: the functon F X s balanced. As we entoned before, n Exaple 4. we start wth an asyptotcally balanced functon and after perturbaton, we stll get an asyptotcally balanced functon. In Exaples 4. and 4.3, we start wth a polynoal that s not asyptotcally

0 FRANCIS N. CASTRO AND LUIS A. MEDINA.0 0.8 0.6 0.4 0. Fgure 5. Graphcal representaton of Sσ n,5 / n Sσ n,5 X X X X 3 X X 3 / n red. blue vs balanced, however, after we perturb t wth a balanced functon F X, the result s asyptotcally balanced. In the theore below we show that these are the only ways to obtan an asyptotcally balanced functon. In other words, a perturbaton of the for σ n,k σ n,ks F X s asyptotcally balanced f and only f σ n,k σ n,ks s asyptotcally balanced or F X s balanced. Theore 4.4. Let k < < k s be ntegers and F X a bnary polynoal n the varables X,, X j j fxed. Let r = log k s. Then, 4.3 Sσ n,k σ n,ks F X = c 0 k,, k s SF π n j n O cos r. In partcular, 4.4 l n Sσ n,k σ n,ks F X = c 0 k,, k s SF j. Proof. We already know that { } 4.5 n Sσ n,k σ n,ks F X s a convergent sequence. The dea n ths proof s to construct a subsequence of 4.5 for whch we can calculate the lt. Defne B n F n to be the set of all x such that σ n,k x σ n,ks x =. Note that Sσ n,k σ n,ks F X = F x F x x F x B n 4.6 = n j SF F x. x B n Suppose that r k s < r and consder the subsequence n N {Sσ r r,k σ r r,k s F X} N.

ASYMPTOTIC BEHAVIOR OF PERTURBATIONS OF SYMMETRIC FUNCTIONS It s not hard to see that Sσ r r,k σ r r,k s n = k r r ks. =0 Let,, p be all ntegers between and r such that l k l k s od. We know that k k s od s perodc and the perod s a dvsor of r. Therefore, k k s od f and only f l od r for soe l {,, p }. Ths ples that x F r r s n B r r precsely when w x l od r. Therefore, Sσ r r,k σ r r,k s F X p = r r j SF Consder the su w x l od r F x = q=0 l= w x l od r F x. w x= r q l F x. Note that there are j r r j s r q l s tuples wth w x = r q l and exactly s s n the frst j entres. The values of F X on these tuples su to r r j C s F r. q l s Hence, w x l od r F x = = q=0 s=0 s=0 j r r j C s F r q l s j r r j C s F r. q l s Recall the seres ultsecton for the su of bnoal coeffcents [30] n n n = s n πj πn tj cos cos. t t s t s s s s Ths ples that q=0 4.7 r r j r q l s = r r a=0 j=0 πa cos r q=0 r r j cos π r r j l sa r.

FRANCIS N. CASTRO AND LUIS A. MEDINA Note that the bggest ter of the su above s r r r j, and t occurs when a = 0. Moreover, all other ters n the su are exponentally saller than r j r. However, j F x r r j 4.8 = C s F r q w x l od r s=0 q=0 l s and 4.7 ples j r r j C s F r q l s s=0 4.9 q=0 Equatons 4.8 and 4.9 ply l r r = j C s F r r j r o r r s=0 = r r j r j C s F o r r s=0 = r r j r SF o r r. w x l od r F x = j r SF. We conclude that 4.0 l r r Sσ r r,k σ r r,k s F X p = l r r j SF F x r r l= w x l od r p = j SF j r SF l= = j p j r SF = p r j SF. Fnally, t s not hard to see, c 0 k,, k s = r r k =0 ks = r r p = p r. Snce { } r r Sσ r r,k σ r r,k s F X N s a subsequence of { n Sσ n,k σ n,ks F X} n N,

ASYMPTOTIC BEHAVIOR OF PERTURBATIONS OF SYMMETRIC FUNCTIONS 3 then we conclude that l n Sσ n,k σ n,ks F X = c 0 k,, k s SF j. Reark. Theore 4.4 generalzes the asyptotc verson of Cusck s conjecture,.e. for k not a power of two, σ n,k s asyptotcally not balanced. Note that when k s not a power of two, then Theore 4.4 ples that σ n,k F X s asyptotcally balanced f and only f F X s a balanced functon. Corollary 4.5. Suppose that k < < k s are ntegers and F X a bnary polynoals n the varables X,, X j, wth j fxed. Then, the polynoal σ n,k σ n,ks F X s asyptotcally balanced f and only f σ n,k σ n,ks s asyptotcally balanced or F X s a balanced functon. Proof. Ths s a drect consequence of Theore 4.4. Exaple 4.6. Consder the followng perturbaton a generalzaton to the perturbaton n Exaple 3.4, The reader can check that Also, and so Theore 4.4 tells us that l σ n,5 σ n,3 X X X 3 X j. SX X X 3 X j = j. c 0 5, 3 =, c 0 5, 3 SF j = j j. n Sσ n,5 σ n,3 X X X 3 X j = j j. 5. A result about bnoal coeffcents In ths secton we present a relaton between the asyptotc coeffcents of syetrc polynoals. Ths relaton, n turns, wll provde us wth a relaton about the party of soe sus of bnoal coeffcents. Recall that f k < < k s and r = log k s, then where l n Sσ n,k σ n,ks = c 0 k,, k s, 5. c 0 k,, k s = r r k ks. =0

4 FRANCIS N. CASTRO AND LUIS A. MEDINA We say that the degree of the asyptotc coeffcent s k s,.e. the degree of an asyptotc coeffcent s the bggest arguent. We ntroduce the followng notaton. Consder the expresson 5. c 0 k, We nterpret However, f then the ter c 0 k, 4 k, k, as k,, k, k. od. 0 od, k s not an arguent of c 0. For exaple, 3 3 c 0 k, k, k, k 3 = c 0 k, k, k, k 3 4 4 k, k 3, k 4 = c 0 k, k 4. 3 We now present one of the an results of ths secton. Theore 5.. Suppose that k be an nteger and let < k. Then, c 0 k = c 0 k, k, k,, k, k. Proof. Apply Theore 3. wth F X = X X j to get j l j l Sσ n,k X X j = l S σ n j,k. l l=0 Dvde each sde of the above equaton by n, let n, use the fact that F X = X X j s balanced, and apply Theore 4.4 to get 5.3 0 = j j j l l l=0 c 0 k, =0 l l k,, k l, k l. l We use 5.3 to prove the theore by nducton. Suppose that =. Note that f j =, then 5.3 ples 0 = c 0 k c 0 k, k. Therefore, the theore s true f =. Suppose the theore holds for j, that s c 0 k = c 0 k, k = c 0 k, k = = c 0 k, k, Let j =. Then, 0 = l l l=0 c 0 k, k,, k, k. l l k,, k l, k l. l

ASYMPTOTIC BEHAVIOR OF PERTURBATIONS OF SYMMETRIC FUNCTIONS 5 Ths equvalent to 0 = l l l=0 c 0 k, c 0 k, l l k,, l l k,, Apply the nducton hypothess to get 0 = l c 0 k l l=0 c 0 k, k,, l = c 0 k c 0 k, We conclude that c 0 k = c 0 k, and the theore holds. k l, k l k, k k, k k,,. k, k. l k,, k, k l There 5. can be generalzed. We frst extend the defnton of c 0 k,, k s, whch s orgnally defned for k < < k s. Note that f we allow k to be zero, then, by defnton 5., we obtan 5.4 c 0 0, k,, k s = c 0 k,, k s. Ths s consstent wth the nterpretaton σ n,0 = and SGX = SGX. Moreover, f we allow soe not all of the k to be negatve, let say k < k < < k j < 0 < k j < k s, then 5. ples 5.5 c 0 k,, k j, k j,, k s = c 0 k j,, k s. Ths s consstent wth the fact that f k < k < < k j < 0 < k j < k s, then σ n,k σ n,kj σ n,kj σ n,ks = σ n,kj σ n,ks, snce the ters σ n,k,, σ n,kj do not exst. Also, repettons can be allowed. Let say that k = k, then 5. ples 5.6 c 0 k, k, k 3,, k s = c 0 k 3,, k s. Ths s consstent wth the fact that Sσ n,k σ n,k3 σ n,ks = Sσ n,k3 σ n,ks. The sae happens f one of the k s s repeated an even aount of tes as an arguent of c 0. On the other hand, f one of the k s s repeated an odd aount of tes, say k = k = k 3, then 5.7 c 0 k, k, k, k 4, k s = c 0 k, k 4,, k s. Ths s consstent wth the fact that S3σ n,k σ n,k4 σ n,ks = Sσ n,k σ n,k4 σ n,ks. In suary, f k s repeated an even aount of tes, then we drop t fro the arguents. On the other hand, f k s repeated an odd aount of tes, then we leave t as an arguent, but we only wrte t once.

6 FRANCIS N. CASTRO AND LUIS A. MEDINA We are now n poston of provdng a generalzaton of Theore 5.. But, before dong ths, we ntroduce yet another notaton. We wrte [a ] n = to represent the lst a, a,, a n. In other words, For exaple, Next, s a generalzaton of Theore 5.. [a ] n = = a, a,, a n. [ ] 4 = =, 4, 9, 6 [ ] 5 = = 3, 5, 7, 9, [b ] 3 = = b, b, b 3. Theore 5.. Suppose that k < < k s and be any postve nteger. Then, c 0 k,, k s = [ c 0 [k ] s =, k ] s,, = [ ] s k, [k ] s =. = Proof. The proof s bascally the sae nducton as n the proof of Theore 5.. The only dfference s that now we use the nterpretatons 5.4, 5.5, 5.6, and 5.7. Reark. Theores 5. and 5. provde a ethod to copute the asyptotc behavor for all the syetrc boolean functons of degree k s fro only knowng few of the. See next exaples. Exaple 5.3. Consder the case when the degree s 3. In ths case, Theores 5. and 5. ply c 0 3 = c 0 3, c 0 3 = c 0 3, 3 c 0 3 = c 0 3,, = c 0 3, 3,, 0 = c 0 3,,, 0 = c 0 3,,. Ths covers all cases. Snce c 0 3 = /, then we conclude that c 0 3, = c 0 3, = / and c 0 3,, = /. Exaple 5.4. Consder now the case when the degree s 5. In ths case, there are 6 asyptotc coeffcents. We only need to know two of the n order to get the rest. Start wth c 0 5 = / and apply Theores 5. and 5. to get when = : c 0 5 = c 0 5, 4 when = : c 0 5 = c 0 5, 3 when = 3 : c 0 5 = c 0 5, 4, 3, when = 4 : c 0 5 = c 0 5, when = 5 : c 0 5 = c 0 5, 4,, 0 = c 0 5, 4, when = 6 : c 0 5 = c 0 5, 3,, = c 0 5, 3, when = 7 : c 0 5 = c 0 5, 4, 3,,, 0,, = c 0 5, 4, 3,,.

ASYMPTOTIC BEHAVIOR OF PERTURBATIONS OF SYMMETRIC FUNCTIONS 7 The reader can check that these are all the asyptotc coeffcents that can be obtaned fro c 0 5,.e. 8 produces a coeffcent that s already lsted above. To proceed, now choose a coeffcent that s not lsted above, let say c 0 5, 4, 3 = 0. Apply Theore 5. to get when = : c 0 5, 4, 3 = c 0 5, 4, 4, 3, 3, = c 0 5, when = : c 0 5, 4, 3 = c 0 5, 3, 4,, 3, = c 0 5, 4,, when = 3 : c 0 5, 4, 3 = c 0 5, 4, 3,, 4, 3,,, 3,,, 0 = c 0 5, 3, when = 4 : c 0 5, 4, 3 = c 0 5,, 4, 0, 3, = c 0 5, 4, 3, when = 5 : c 0 5, 4, 3 = c 0 5, 4,, 0, 4, 3, 0,, 3,,, = c 0 5,, when = 6 : c 0 5, 4, 3 = c 0 5, 3,,, 4,, 0,, 3,,, 3 = c 0 5, 4, when = 7 : c 0 5, 4, 3 = c 0 5, 4, 3,,, 0,,, 4, 3,,, 0,,, 3, 3,,, 0,,, 3, 4 = c 0 5, 3,,. Ths copletes the lst of all asyptotc coeffcents. Reark. It turns out that, usng our ethod, all the 3 asyptotc coeffcents of degree 6 can be obtaned fro c 0 6, c 0 6, 5, 4, c 0 6, 5, 3, and c 0 6, 5,. It appears that the aount of asyptotc coeffcents needed for our ethod to copute all asyptotc coeffcents of degree k s grows exponentally n k s. Theores 5. and 5. also provde us wth a relatonshp between soe sus of bnoal coeffcents. Suppose that k < < k s and r = log k s. Recall that c 0 k,, k s = r r =0 k Note that c 0 k,, k s counts the nuber of tes k k s s odd when runs fro 0 to r. Ths has the followng plcaton. Theore 5.5. Let k < < k s and r = log k s. If two asyptotc coeffcents of degree k s are equal, say c 0 k l, k l,, k s = c 0 k j, k j,, k s, then both lsts [ ] r k l k l k s =0 and [ ] r k j k j k s =0 have the sae aount of odd nubers. On the other hand, f c 0 k l, k l,, k s = c 0 k j, k j,, k s, then the aount of odd nubers n [ ] r s the sae as the aount of even nubers n [ and vceversa. k l k j k l k j k s k s =0 ] r Proof: Ths s a drect consequence of Theores 5. and 5.. =0 ks.

8 FRANCIS N. CASTRO AND LUIS A. MEDINA Reark. Lucas Theore can be used to obtan the party of a bnoal coeffcent. However, our ethod avods the dffculty of the calculaton of the party of each ndvdual bnoal coeffcent to obtan the result of Theore 5.5. Exaple 5.6. Consder the case of degree 5. The coplete lst of asyptotc coeffcents of degree fve appears n Exaple 5.4. Note that c 0 5, 4, 3, = c 0 5,. Therefore, the lsts [ 5 4 3 ] 7 =0 and [ 5 7 ] contan the sae =0 aount of odd nubers whch n ths case s : [ ] 7 = 0, 0,, 4,, 6, 56, 5 4 3 =0 [ ] 7 = 0,,, 3, 4, 6,, 8. 5 =0 We also know that c 0 5, 4, 3, = c 0 5, 4, 3,,. Therefore, the aount of odd nubers n [ 5 4 3 7 ] s the sae as the aount of even nubers n [ =0 5 4 3 7 ], whch n ths case s 6: =0 [ ] 7 = 0,, 3, 7, 5, 3, 6, 9. 5 4 3 Acknowledgents. We would lke to thank Professor Thoas W. Cusck for hs helpful coents and suggestons n a prevous verson of ths paper. =0 References [] C. Bey and G. M. Kyureghyan, On Boolean functons wth the su of every two of the beng bent, Des. Codes Cryptogr., 49, 34 346, 008. [] R. E. Canfeld, Z. Gao, C. Greenhll, B. McKay and R. W. Robnson, Asyptotc enueraton of correlaton-une Boolean functons, Cryptogr. Coun.,, -6, 00 [3] J. Y. Ca, F. Greeen, and T. Therauf. On the Correlaton of Syetrc Functons. Theory of Coputng Systes, 9, 45 58, 996. [4] A. Canteaut and M. Vdeau, Syetrc Boolean Functons, IEEE Transactons on Inforaton Theory, 5, 79 807, 005. [5] C. Carlet, Boolean Functons for Cryptography and Error Correctng Codes, Boolean Models and Methods n Matheatcs, Coputer Scence, and Engneerng, Eds. Yves Craa and Peter Haer, Cabrdge Unversty Press, 00. [6] C. Carlet, X. Zeng, C. L, and L. Hu, Further propertes of several classes of Boolean functons wth optu algebrac unty, Des. Codes Cryptogr., 5, 303-338, 009. [7] C. Carlet, On the Degree, Nonlnearty, algebrac thnckness and non-noralty of Boolean Functons, wth Developents on Syetrc Functons, IEEE Trans. Infor. Theory 50, 78-85, 004. [8] F. Castro and L. A. Medna. Lnear recurrences and asyptotc behavor of exponental sus of syetrc boolean functons. Elec. J. Cobnatorcs, 8:#P8, 0. [9] T. W. Cusck, and Y. L, kth Order Syetrc SAC Boolean Functons and Bsectng Bnoal Coeffcents, Dscrete. Appl. Math. 49, 73-86, 005. [0] Y. L and T. W. Cusck, Lnear Structures of Syetrc Functons over Fnte Felds, Inf. Processng Letters 97, 4-7, 006. [] T. Cusck, Y. L, and P. Stăncă, Balanced Syetrc Functons over GF p. IEEE Trans. on Inforaton Theory, 54, 304-307, 008. [] T. Cusck, Y. L, and P. Stăncă, On a conjecture for balanced syetrc Boolean functons, J. Math. Crypt., 3, -8, 009.

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