On maximally nonlinear and extremal balanced Boolean functions

Transcription

1 O imally oliear ad extremal balaced Boolea uctios Michel Mitto DCSSI/SDS/Crypto.Lab. 18, rue du docteur Zameho 9131 Issy-les-Moulieaux cedex, Frace Abstract. We prove a ew suiciet coditio or a Boolea uctio to be extremal balaced or imally oliear, i odd or eve dimesio. Uder this coditio, we deduce the balaced coverig radius ρ B () ad the coverig radius ρ(). We prove some geeral properties about imally oliear or extremal balaced uctios. Fially, a applicatio to eve weights Boolea uctiosisgive. Keywords: Boolea uctios, extremal balaced uctios, imally oliear uctios, bet uctios, coverig radius, balaced coverig radius, Walsh ad Fourier trasorms. 1 Itroductio This paper ivestigates the coverig radius ad the balaced coverig radius or Boolea uctios. From Rothaus [8], the coverig radius is kow i eve dimesio. The exact value o balaced coverig radius is ukow i both eve or odd dimesio except a iite umber o small dimesios, but it has bee previously studied by Dobberti [3] ad Seberry, Zhag, Zheg [9] where a lower boud, which is the best achieved so ar, was derived. This problem is kow to be diicult, ad ay ew approach to it potetially brigs the problem closer to its solutio. I this respect, we preset a ew coditio or the study o imally oliear Boolea uctios. The coditio takes two orms () ad (Q) depedig o whether the uctios are balaced or ot, respectively. I particular, i eve dimesio this coditio gives a ew characterizatio o bet uctios. However, or the balaced uctios i eve dimesio ad or the imally oliear uctios i odd dimesio, the coditio is geerally oly suiciet ad we prove that it is oly veriied i low dimesios. Uder this coditio, we compute the values o the coverig ad balaced coverig radii. We iish with 1

2 a applicatio to eve weights Boolea uctios i which we lik the distace to the aie uctios set with the degree o the algebraic ormal orm o these uctios. Our approach is based o the study o the kerel o the Walsh spectrum. I Carlet [1], the size o this kerel was previously show to be relevat i the cotext o partially-bet uctios. It is iterestig to see that it comes up agai i the cotext o our paper. Basic Deiitios ad Notatio I this documet, the iite ield (Z/Z, +,.) with his addititive ad multiplicative laws will be deoted by F ad the F algebra o Boolea uctios i variables will be deoted by F(F, F ). For F(F, F ) ad a F, recall that the set 1 (a) is deied by 1 (a) ={u F (u) =a}. We will use #E to deote the umber o elemets o the set E. A uctio F(F, F ) is said balaced i # 1 (0) = # 1 (1) = 1. The Hammig distace betwee ad g F(F, F ) deied by #( +g) 1 (1) will be deoted by d(,g). W (a) is the Walsh spectrum o F(F, F ) to a poit a =(a 0,...,a 1 ) F deied by W (a) = X (x)( 1) <a,x>. (1) I this ormula, the sum is calculated i Z, ad x F <a,x>= a 0 x a 1 x 1 is the scalar product o F. The kowledge o the spectrum (W (a)) a F is equivalet to the kowledge o by the ollowig iversio theorem valid or each x F (x) = X W (a)( 1) <a,x>. () a F Walsh ad Fourier spectrums are equivalet sice we have the ollowig passage ormula 1 δ a 0 W (a) = (a) (3) valid or each a F, i which δ b a is the Kroecker s symbol. I the sequel, we will use the otatio W (a) = 1 δ a 0 W (a). Each F(F, F ) veriies the arseval s relatio (W (a)) = ( 1). a F x deotes the absolute value o the real umber x, ad dxe the iteger mi { N x} or x positive real umber. For each iteger i [0, 1 ], we will have to cosider the sets W 1 (i) deied by W 1 (i) ={a F W (a) = i}.

3 The aie uctio F(F, F ) deied by (x) =< α,x > +λ,with α,x F ad λ F, will be deoted by l α + λ. Thedistacedeied by mi d(,l α + λ) betwee F(F α F,λ F, F ) ad the aie uctios set, will be deoted by δ(). It is easy to prove that δ() = 1 W a F (a). Thus whe is balaced δ() = 1 W (a). a F {0} The iteger δ() will be deoted by ρ(). I theory o Error F(F,F ) Correctig codes [6], ρ() is called the coverig radius o the irst Reed-Muller code o legth. Auctio will be called imally oliear i δ() =ρ(). The iteger F(F,F), balacedδ() will be deoted by ρ B() ad will be called the balaced coverig radius i dimesio. O course, we have ρ B () ρ(). A balaced uctio F(F, F ) will be called extremal i δ() =ρ B (). The subset o F(F, F ) cotaiig all the the imally oliear (resp. extremal balaced) uctios will be deoted by C() (resp. E()). 3 A Suiciet Coditio or Maximally Noliearity ad Extremality 3.1 The case o extremal balaced uctios Theorem 1 Let g F(F, F ) be a balaced uctio such that The g E(). # Wg 1 ( 1 δ(g)) #W. () roo. Cosider a balaced uctio g F(F, F ) which veriies the coditio (). I a Wg 1 ( 1 δ(g)),we have Wg (a) = 1 δ(g) 6= 0, ad thus Wg 1 ( 1 δ(g)) F Wg (0). This implies the iequality #W g (0) # W g 1 ( 1 δ(g)) (4) O the other had, the arseval s idetity implies or each, g F(F, F ) X X ( 1) = (Wg (a)) = (W (b)). g (0) Now, we choose E() such that b/ W #W (0) = #Wh (0) (5) h E() 3

4 I we deote p = #Wg (0) ad q = #W (0), (), (4) ad (5) imply q p. Thereore or each idexatio o p elemets o F Wg (0) ad q elemets o F W (0), we have rom arseval s relatio px (Wg (a k )) = k=1 which implies successively qx (W (b k )) k=1 qx [(Wg (a k )) (W (b k )) ]+ k=1 qx [(W (b k )) (Wg (a k )) ]= k=1 (whe p = q, px k=q+1 px k=q+1 (W g (a k )) =0 (W g (a k )) 0 (6) q [(W (b k)) (Wg (a k )) ]=0). We also remark that the k=1 choice o veriyig (5), together with the iequality (), implies the ollowig property: # Wg 1 ( 1 δ(g)) q (7) Now, we choose the idexatio o p elemets o F W g (0) by decreasig values o Wg (a k ). So rom (6) we have at least oe iteger k [1,q] such that (W (b k )) (Wg (a k )) 0. The it ollows rom (7) that or ay k [1,q], Wg (a k ) = 1 δ(g). For k = k, W (b k ) W g (a k ) = 1 δ(g) ad we obtai the ollowig three properties: 1 W b F (b) 1 W (b k ) 1 W (b k ) δ(g) 1 W b F (b) = δ() =ρ B() sice is extremal balaced These properties imply ρ B () δ(g), ad thus sice g is balaced, ρ B () =δ(g). Fially g is balaced with δ(g) =ρ B (), thereore g E(). At sectio 3, we will see that there exists uctios veriyig (). From Theorem 1, we ca deduce the ollowig Corollary: Corollary I there exists a balaced uctio g veriyig (), the roo. Obvious. # Wg 1 ( 1 ρ B ()) #W. 4

5 3. The case o imally oliear uctios Now, cosider a uctio g F(F, F ) which veriies the ew coditio # Wg 1 ( 1 δ(g)) #W C() The we observe that the proo o Theorem 1, suitably adjusted, is valid whe replacig E() by C(). Weget: Theorem 3 Let g F(F, F ) be a uctio such that # Wg 1 ( 1 δ(g)) #W. (Q) C() The g C(). roo. The same as Theorem 1. As see previously at Corollary, we have the ollowig result: Corollary 4 I there exists a uctio g veriyig (Q), the # Wg 1 ( 1 ρ()) #W. C() roo. Obvious. Remark that these results are idepedat o the hypothesis o to be odd or eve. We also have the ollowig result: Corollary 5 I there exists a uctio g F(F, F ) such that 1 # Wg 1 ( 1 δ(g)) #W, C() the ρ() =ρ B (). roo. We kow that g veriies (Q) ad thereore we have (0) # Wg 1 ( 1 δ(g)). C() #W But g also veriies the coditio # Wg 1 ( 1 δ(g)) 1, the (0) 1. Thus there exists oe uctio C() at least such C() #W that W (0) 6=. Thereore, let a be a elemet i F such that W (a) =0. The it is easy to prove that the uctio +l a is balaced ad such that δ( +l a )=δ() =ρ(). Thus this uctio is balaced ad imally oliear ad we have proved the result. Our aim is ow to use these results to compute ρ B () ad ρ() uder () ad (Q) hypothesis, respectively. But beore, we give some examples o uctios which veriy these () ad (Q) coditios. 5

6 3.3 Some examples o uctios For eve : Whe is eve, bet uctios [6, 8] are deied as boolea uctios havig uiorm Walsh spectrum W (a) = 1 or each a F. The it is easy to see that is bet i ad oly i is imally oliear. We have see, rom Theorem 3, that i a uctio veriies (Q), this uctio is imally oliear. We have the ollowig coverse result: Corollary 6 I is eve, g F(F, F ) is bet i ad oly i g veriies (Q). roo. From Theorem 3, whe g veriies (Q) g is imally oliear (or eve or odd ) ad i particular bet or eve. Now, let g be a bet uctio. I this case, we kow that his Walsh spectrum is such that Wg (a) = 1 or each a F. So we have δ(g) = 1 1 ad # Wg 1 ( 1 δ()) = # Wg 1 ( 1 )=. O the other had, we also have #W = 0 sice or eve, C() W (a) = 1 6=0or each a F ad each C(). The we obtai # Wg 1 ( 1 δ(g)) = = C() veriies (Q) For odd : #W (0) which proves that g Usig classical costuctios o bet uctios i eve dimesio, or istace Maioraa-MacFarlad uctios [, 5], we are able to costruct two bet uctios g 1,g F(F 1, F ) such that Wg 1 (0) = Wg (0) = 1 1. Let us cosider the ew uctio g F(F, F ) deied by g(x 1,..., x )=x g 1 (x 1,..., x 1 )+(x +1)g (x 1,..., x 1 ). The we have the ollowig properties or g: g is balaced, Wg (a) =0or 1 or each a F δ(g) = 1 1, #W g (0) = 1,, # W g 1 ( 1 δ(g)) = 1. Moreover, it is kow rom [5] that ρ(3) = = , ρ(5) = 1 = , ρ(7) = 56 = , ad thus the uctios g or =3, 5, 7 are extremal balaced. For these three values o, siceg E() we have (0) #W (0) = 1 ad ially #W g #W # Wg 1 ( 1 δ(g)) = 1. Thereore, whe =3, 5, 7 these uctios g veriy (). 4 ρ() ad ρ B () Computatios 6

7 Theorem 7 I there exists a balaced (resp. ay) uctio g F(F, F ) veriyig () (resp. (Q)), we have ρ B () = 1 (resp. ρ() = 1 1 ( #W) 1 1 ( #W) 1 C() roo. Let be a uctio i F(F, F ). The arseval s relatio (W (a)) = ( 1) implies the existece o a/ W (0) such that (W (a)) We have δ() = 1 ( 1) #W ( #W ) 1, so W (a) 1 W (a) 1 1 I we choose E() (resp. C()), weseethat ρ B () 1 or each E() 1 ( #W ) 1 1 ( #W ) 1 (resp. ρ() 1 or each C()). From this, oe ca deduce the irst iequality Ã! ρ B () mi 1 1 ( #W (0)) 1 (resp. ρ() = 1 mi C() = 1 1. (8).) (9) ( #W ) 1. (10) ( #W) 1! Ã 1 1 ( #W (0)) 1 1 ( #W) 1 C(). ). (11) We ow suppose the existece o g F(F, F ), balaced (resp. ay) uctio veriyig (): # Wg 1 ( 1 δ(g)) #W. (resp. (Q): # Wg 1 ( 1 δ(g)) #W ) C() This uctio g satisies ( 1) = X (Wg (a)) = a F 1 δ(g) X i=1 # W g 1 (i) i ( 1 δ(g)) # W g 1 ( 1 δ(g)) (1). 7

8 ad rom theorem 1(resp. theorem 3), g is such that δ(g) =ρ B () (resp. δ(g) =ρ()). Soweobtai ( 1) ( 1 ρ B ()) # W g 1 ( 1 ρ B ()) (resp. ( 1) ( 1 ρ()) # W g 1 ( 1 ρ()) ). As()(resp.(Q)) is veriied, the Corollary (resp. Corollary 4) applied to the above iequality implies ( 1) ( 1 ρ B ()) ( #W) (resp. ( 1) ( 1 ρ()) ( #W ). C() Thus we get ad ially 1 ( 1 ρ B ()) ( #W) 1 (resp. 1 ( 1 ρ()) ( #W) 1 ) C() ρ B () 1 (resp. ρ() 1 1 ( #W) 1 1 ( #W) 1 C() (13) ). (14) Combiig the iequalities (10) ad (13) (resp. proved the theorem. (11) ad (14)), we have Remark 8 Whe is eve, rom corollary 6, we recover the well-kow value ρ() = 1 1 because #W (0) = 0 or ay C(). 5 Cosequeces or Balaced ad Maximally Noliear Fuctios ropositio 9 For ay odd iteger 1, we have #Wg (0) 1 ad #Wg (0) 1. (15) g E() g C() roo. Suppose there exists a balaced uctio F(F, F ) such that the absolute value o his Walsh spectrum is costat o his support: c N {0}, a / W (0), W (a) = c. The ay uctio g E() is such that δ(g) δ() = 1 c, ad thus Wg (a) c. g (0) By the arseval s relatio we also have 8

9 ( 1) = g (0) ( 1) #W g (0). W g (a) #W g (0) c, the c O the other had, the same arseval s relatio o implies the existece ³ o a / W (0) such that W (a ) ( 1), ad the hypothesis o #W ³ implies W ) (a = c. These properties give us the iequalities that imply #W g I we deote A() ={ F(F, F ) ( 1) #Wg (0) c ( 1) (0) #W (0) or ay g E(). balaced ad c N {0}, a / W the above result implies #Wg (0) mi A() ad thus #W #W (0), W (a) = c}, (0) or each g E(), #Wg (0) mi #W (16) g E() A() Thereore, ³ i A() 6=, each A() veriies #W (0) c = ( 1). I that case, there exists a iteger i which veriies the ollowig two coditios c = i (17) #W (0) = ( 1) i. (18) It ollows rom (17) that i is eve, ad rom (18 ) that #W (0) = ( 1) i > 0 sice 0 W (0) ( is balaced). Thus we have i 1 ad, i there exists a uctio A() such that c = 1, we see that #W (0) equals #W (0) = ( 1) i calculated or mi A() i = 1. So is ecessarily odd. But as see at.3., we are able to costruct such uctios A() whe is odd: the uctios (x 1,..., x )=x 1 (x 1,..., x 1 )+(x +1) (x 1,...,x 1 ), with 1, F(F 1, F ) bet uctios ad W 1 (0) = W (0) = 1 1, are elemets o A() with c = 1, so A() 6= whe is odd. We deduce rom this that mi A() #W or odd. It implies rom(16)the irst iequality proo (0) =[ ( 1) i ] i= 1 = 1 #Wg (0) 1. (19) g E() Now, cosiderer the set B() ={ F(F, F ) c N {0}, a / W (0), W (a) = c}. B() is ever empty sice, i eve C() B(), ad i odd A() B(). So let s B(). 9

10 For ay uctio g C() we have δ(g) δ() = 1 c ad thus Wg (a) c. Thereore, we have g (0) ( 1) = such that g (0) ( 1) #W g (0). W g (a) #W g (0) c, so c The same arseval s relatio o implies the existece o a ³ W (a ) ( 1), ad rom the hypothesis o, ³ W (a ) ( 1) #W #W g #W = c. The c ( 1) #W g (0) c ( 1) (0) #W #W which implies ad thereore (0) or ay g C(). so we have / W (0) #Wg (0) mi #W (0) g C() B() ³ But, or each B(), we have #W (0) implies the existece o a iteger i such that c = ( 1) which c = i (1) #W (0) = ( 1) i. () It ollows rom (1) that i is eve, ad rom () that #W (0) = ( 1) i 0. This implies i. The three coditios is odd, i is eve ad i, imply i 1, ad A() B() with A() 6= implies also i = 1. Thereore, rom (0) we obtai #Wg (0) [ ( 1) i ] i= 1 = 1 ad the secod iequality g C() is proved. Corollary 10 For odd 1, i () true or g E(), the # Wg 1 ( 1 ρ B ()) 1. I () alse, we have #W < 1. I (Q) true or g C(), the # Wg 1 ( 1 ρ()) 1. I (Q) alse, we have #W < 1. C() roo. From Corollary (resp. Corollary 4), i () (resp. (Q)) true or g balaced, we have # Wg 1 ( 1 ρ B ()) #W (resp. # Wg 1 ( 1 ρ()) #W (0)). O the other had, C() sice is odd, rom ropositio 9 we have #W (0) 1 10

11 (resp. (0) 1 ) ad the results are proved whe () (respectively (Q)) is true. Now, i () (resp. (Q)) is alse, there exists o balaced C() uctio g (resp. o uctio g) suchthat Wg 1 ( 1 δ(g)) #W (resp. # Wg 1 ( 1 δ(g)) #W ). C() So or ay balaced uctio g (resp. ay uctio g) wehave # Wg 1 ( 1 δ(g)) < #W (resp. Wg 1 ( 1 δ(g)) < #W (0)). I particular whe C() is odd, we ca use g balaced such that δ(g) = 1 1 ad # Wg 1 ( 1 )= 1 (geeralized uctios o A()). Thereore Wg 1 ( 1 δ(g)) = # Wg 1 ( 1 )= 1 < #W (resp. # Wg 1 ( 1 δ(g)) = # Wg 1 ( 1 )= 1 < #W) C() ad the corollary is proved. #W Corollary 11 I there exists a balaced uctio g F(F, F ) veriyig (), we have or eve, ρ B () = 1 ad #W = 1 +, or odd, ρ B () = 1 1 ad #W = 1. roo. Cosider g balaced veriyig (). From Theorems 1 ad 7 we have δ(g) =ρ B () = 1 1, ad thus ( 1) = a iteger j such that ( µ #W The j is eve ad #W) 1 ( 1 ρ B ()). Thereore, there exists ( 1 ρ B ()) = j (0) = ( 1) j #W #W (0) = ( 1) j ]0, [ sice the balacedess o each E() implies 0 W (0) ad ot everywhere equal to zero. So we have j ], ( 1)], ad ially j [ 1, ( 1)]. Thereore, ρ B () = 1 j oraeveitegerj [ 1, ( 1)].We have the ollowig two cases: I is odd, the ropositio 9 implies #W (0) 1 ad, sice #W (0) = ( 1) j or j eve, j 1, we also obtai j 1 ad ially j = 1. Thus #W (0) = 1 ad ρ B () =

12 I is eve, sice the iteger j 1 such that ρ B () = 1 j will be also eve, we have ecessarily j 6= 1, ad the j [, ( 1)]. Thus we obtai ρ B () 1. But, as see at.3., or eve we are able to costruct balaced uctios such that W (a) = or a/ W (0). Thus A() 6=. ³ The arseval s relatio applied to uctios A() give us the equality #W (0) = ( 1), which implies #W (0) = 1 +. From this we deduce #W (0) 1 + ad rom (16) we also #W g E() mi A() obtai g (0) 1 +. The ( 1) j 1 +, ad thereore j. Fially j = ad we see that ρ B () = 1 ad #W = ( 1) j = = 1 +. Corollary 1 I 1 is odd, ad i there exists a uctio g F(F, F ) veriyig (Q), we have ρ() =ρ B () = 1 1 ad #W = 1. E() roo. For this uctio g, Theorems 3 ad 7 imply δ(g) =ρ() = 1 1, so ( 1) = iteger j such that µ C() ( #W C() ) 1 #W (0) 1 ρ() adthuswehavea ( 1 ρ()) = j #W = ( 1) j C() Thereore j is eve, (0) = ( 1) j 0 implies j, ad C() ρ() = 1 j. As is odd, the two coditios j eve ad j imply j 1. So ρ() = 1 j 1 1. But or odd, we have costruct balaced uctios such that δ() = 1 1 ad thus ρ B () 1 1. Combiig these properties, we obtai 1 1 ρ B () ρ() 1 1 so ρ B () =ρ() = 1 1, j = 1 ad ially #W = 1. C() We coclude this sectio by the two ollowig results: Corollary 13 For ay eve iteger, 6, ad or ay balaced uctio g F(F, F ) we have # Wg 1 ( 1 δ(g)) < #W. #W I particular, or ay g E() # Wg 1 ( 1 ρ B ()) < E() #W E() (0). 1

13 roo. From Corollary 11 or eve, wehaveρ B () = 1 i there exists a balaced uctio g veriyig (). From [3, 9], i we write = s t or s 1 ad t 1 odd, we kow that ρ B () 1 ( sx i 1 ) 1 ( 1) s i=1 Thereore i g veriies () we will have sx ( i 1 )+ 1 ( 1) s As this iequality is ot veriied or 6, we obtai the result. i=1 Thereore, or eve 6, there exists o balaced uctios veriyig (). Whe is odd, we have the ollowig similar result: Corollary 14 For ay odd iteger, 15, ad or ay uctio g F(F, F ) we have # Wg 1 ( 1 δ(g)) < #W. I particular, or ay g C() # W g 1 ( 1 ρ()) < C() #W C() (0). roo. From Corollary 1 or odd, ithereexistsg uctio veriyig (Q), we have ρ() = 1 1. O the other had, atterso-wiedema [7] have proved that ρ() or ay odd 15. Soorodd 15, i there exists a uctio g veriyig (Q), we must have ,ad we see that this is alse. So (Q) is alse or odd 15, ad the corollary is proved. Because (Q) alse or odd 15, ote that the result o the Corollary 10 is veriied. 6 Applicatio to Boolea Fuctios o Degree at most Structure o δ() Let us cosider F(F, F ) as a elemet o F [x 1,..., x ] ad suppose that d () 1. Let l α + λ be oe o its earest aie uctios. We have (see sectio ) δ() =d(, l α + λ) = 1 W (α) with W (α) = W a F (a). Let us cosider the uctios amily ( Ω ) Ω F deied by Ω = + l Ω+α. We will use the ollowig techical lemma: 13

14 Lemma 15 W (Ω + α) = 1 W Ω (0) or ay Ω F {0}. roo. For α F ad λ F, let us cosider g = + l α + λ. I we deote p =#g 1 (1) the weight o g, we have g 1 (1) = {x 1,...,x p } or p elemets x i F. The i we itroduce the uctios δ x deied by δ x (y) =δ y x, we ca write g = δ x δ xp.so = l α + λ + δ x δ xp. Moreover, or all a F, a direct calculatio proves the ormula W (a) = δ a 0 +( 1) λ+1 δ a px α 1 +( 1) λ ( 1) <α+a,xk>. (3) k=1 Sice δ() =d(, l α + λ) =d(g + l α + λ,l α + λ), we also have δ() =p ad i a =0, W = 1+( 1) λ+1 δ 0 px α 1 +( 1) λ ( 1) <α,xk>. (4) The Ω = + l Ω+α = + l Ω + l α = l Ω + λ + δ x δ xp. For each Ω 6= 0, we observe that (4) implies W Ω (0) = [W ] α=ω = 1 +( 1) λ ( 1) <Ω,x1> ( 1) <Ω,xp>. Thus usig (3) with Ω = a + α 6= 0, we ially obtai W Ω (0) 1 =( 1) λ ( 1) <Ω,x1> ( 1) <Ω,x p> = W (Ω + α) 1 δ Ω+α 0 = W (Ω + α). Theorem 16 Let F(F, F ) be a oaie uctio such that d () 1 or 3. There exists a iteger m veriyig 0 <m ³ µ d e d () +#W (0) + 1 k=1 1 such that δ() = 1 d d () e 1 m. (5) roo. Cosider agai the amily ( Ω ) Ω F with Ω = + l Ω+α. Sice is oaie we have d ( Ω )=d (), ad uctios Ω veriy d ( Ω ) 1. As d ( Ω ) 1, the theorem o Ax-Katz ([10] pp. 51-5, or [4] pp. 319) applied to Ω implies» ¼ b # 1 Ω (0) with b = d 1, or ay Ω F (6) ( Ω ) For 3, we have b 1 < 1. As # 1 Ω (0) = W Ω (0), (6) implies b W Ω (0). So whe W Ω (0) 6= 1,W Ω (0) 1 will be divisible by b. Thus i we deote Ω = a + α 6= 0, the Lemma 15 implies b W (a) or ay a 6= α such that W (a) 6= 0. (7) 14

15 a F ³ W (a) = ( 1) together with the equality The arseval s idetity a W (a) = W (α) implies W (α) 6= 0, ad we also have ( 1) ³W (α) = ( 1) 1 δ() = { a6=α} (W (a)). Usig property (7) or ay a 6= α such that W (a) 6= 0, there exists q a Z {0} such that W (a) =b q a.thusiwedeote q = X { a6=α} we obtai the ollowig equatio or δ() : q a (8) ( 1) 1 δ() = b q Sice is oaie, we have δ() > 0, ad thereore 1 δ() = ³ ( 1) b 1 q with q < implies e d d () which ially δ() = 1 ³ ( 1) b q 1 l = 1 d () m ³ l 1 µ d () m q 1 (9) ³ e with d d () >q > 0. Moreover, déiitio (8) also implies Now, sice 1 ( 1) (x) = q #W (0) 1 (30) we may write at the poit x =0, 1 ( 1) (0) = W (a)( 1)<a,x> ad α / W (0), b ( q a,a6=α W (α) = 1 δ() = 1 ( 1) (0) b ( q a = b b 1 ( 1) (0),a6=α qa.,a6=α O the other had, it ollows rom (9) that µ l 1 1 δ() = b ( m) d () q, ad ially ) )+W (α) so, µ ( l d () m) q 1 = ( 1) (0) l m d () X qa,a6=α N. 15

16 Thereore, i we deote m = µ ( l d () m) q 1 = ( 1) (0) d d ()e qa,a6=α ad i we use the iequality o q together with (30), we obtai l ³ Theorem. d () m +#W (0) + 1 m > 0 adwehaveprovedthe Corollary 17 For 3, let F(F, F ) be a oaie uctio o eve weight # 1 (1). The δ() veriies (5). roo. I we cosider (x 1,..., x ) = a i1...i x i x i as i 1 {0,1},...,i {0,1} elemet o F Ã[x 1,..., x ],wehave! a i1...i = (x 1,..., x ) mod. Thus 0 x à 1 i 1,...,0 x i! a = (x 1,..., x ) mod = # 1 (1) mod. x 1 F,...,x F The # 1 (1) is eve i ad oly i a =0, idem d () 1, ad the Corollary results o Theorem Applicatio to balaced uctios Sice # 1 (1) = 1, all the balaced uctios are o eve weights, ad we iish with the ollowig result: Corollary 18 For odd, 3, i there exists a extremal balaced uctio o degree, the ρ B () = 1 1. roo. Sice is odd, the ropositio 9 implies For ay E(), it ollows rom Theorem 16 that 0 <m ³ µ d ³ µ d e d () +#W (0) #W 1. e d () 1 +1 I there exists E() with d () =, we obtai or this uctio ³ e d d () e = +1 ad the d d () 1 +1=1. Thus m =1ad the ρ B () value results o (5). 7 Coclusio 1, We have studied a ew suiciet coditio or imal oliear ad extremal balaced Boolea uctios. For eve, this coditio characterizes the bet 16

17 uctios. For eve or odd, uder orms () ad (Q) o the coditio we have computed ρ B () ad ρ(), respectively. Later, these values are proved oly valid i low dimesios, so i a subsequet study oe may ask how to geeralize () ad (Q). I high odd or eve dimesios, we have deduced some ew iequalities o the size o the Walsh spectrum s kerel o uctios i E() ad C(). I a secod part, or eve weights uctios, a geeral orm or δ() icludig d () is give. Ackowledgemets. We would like to thak. Camio, A. ad F. Le Leslé, or their costructive commets which greatly improved the presetatio o the paper. 8 Reereces 1. Carlet, C.: artially-bet uctios. Desigs, Codes ad Cryptography, 3, , (1993). Dillo, J.F.: Elemetary Hadamard dierece sets. hd thesis, Uiversity o Marylad, Dobberti, H.: Costructio o imally oliear uctios ad balaced boolea uctios with high oliearity. I roc. Fast Sotware Ecryptio, 61-74, Berli Heidelberg New-York: Spriger Lidl, R., Niederreiter, H.: Fiite ields. Cambridge Uiversity ress Mc Farlad, R.L.: A amily o ocyclic dierece sets. J. Comb. Th. (Series A) 15, 1-10 (1973) 6. Mc Williams, F.J., Sloae, N.J.A.: The theory o Error-Correctig codes. Amsterdam: North-Hollad atterso, N.J., Wiedema, D.H.: The coverig radius o the ( 15, 16) Reed-Muller code is at least IEEE Tras. Iorm. Theory, IT-9, (1983). 8. Rothaus, O.S.: O bet uctios. J. Comb. Th. (Series A) 0, (1976) 9. Seberry, J., Zhag, M., Zheg, Y.: Noliearly balaced boolea uctios ad their propagatio characteristics. I roc. CRYTO 93, 49-60, Berli Heidelberg New-York: Spriger Small, C.: Arithmetic o iite ields. Marcel Dekker