A New Class of Hyper-bent Boolean Functions with Multiple Trace Terms

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1 1 A New Class of Hyper-bet Boolea Fuctios with Multiple Trace Terms Baocheg Wag, Chumig Tag, Yafeg Qi, Yixia Yag, Maozhi Xu Abstract Itroduced by Rothaus i 1976 as iterestig combiatorial objects, bet fuctios are maximally oliear Boolea fuctios with eve umbers of variables whose Hammig distace to the set of all affie fuctios equals 1 ± 1. Not oly bet fuctios are applied i cryptography, such as applicatios i compoets of S-box, block cipher ad stream cipher, but also they have relatios to codig theory. Hece a lot of research have bee paid o them. Youssef ad Gog itroduced a ew class of bet fuctios the so-called hyper-bet fuctios which have stroger properties ad rarer elemets. It seems that hyper-bet fuctios are more difficult to geerate. Moreover, (hyper)-bet fuctios are ot classified. Charpi ad Gog studied a class of hyper-bet fuctios f defied o F by f = Tr 1 (a r x r(m 1) ), = m ad a r F, where R is a subset of a set of represetatives of the cyclotomic cosets modulo m +1 for which each coset has the full size. Further, Mesager cotributed to the kowledge of a class of hyper-bet fuctios f b defied over F by f b = Tr 1 (a r x r(m 1) ) + Tr 1(bx 1 3 ), b F 4, = m ad a r F m. I this paper, we study a ew class of the hyper-bet fuctios f b defied over F by f b = Tr 1 (a r x r(m 1) ) + Tr 4 1(bx 1 ), b F 16, = m ad a r F m. Idex Terms Boolea fuctios, bet fuctios, hyper-bet fuctios, Walsh-Hadamard traformatio, Dickso polyomials. I. INTRODUCTION Bet fuctios with eve umbers of variables are maximally oliear Boolea fuctios, that is, their hammig distace to the set of all affie fuctios equals 1 ± 1. Bet fuctios were defied ad amed by Rothaus [8] i the study of combiatorial objects. They have bee extesively studied for their applicatios i cryptography, but have also bee applied to spread spectrum, codig theory [3], [4] ad combiatorial desig. The recet study of bet fuctios alog with properties ad costructios of bet fuctios ca be foud i [], [11], [4]. A bet fuctio ca be seemed as a fuctio defied o F, F m F m, or F ( = m). Thaks to the differet structures of the vectorspace F ad the Galois field F, bet fuctios ca be well studied. However, it is ot yet clear o the geeral structure of bet fuctios over F. Further, it seems impossible to classify bet fuctios. As B. Wag ad Y, Yag are with Iformatio Security Ceter, Beijig Uiversity of Posts ad Telecommuicatios ad Research Ceter o fictitious Ecoomy ad Data Sciece, Chiese Academy of Scieces, Beijig, , Chia C. Tag, Y. Qi ad M. Xu are with Laboratory of Mathematics ad Applied Mathematics, School of Mathematical Scieces, Pekig Uiversity, , Chia C. Tag s tagchumigmath@163.com a result, may research works are devoted to the descriptio of ew class of bet fuctios [1], [6], [7], [9], [10], [1], [13], [18], [19], [], [3], [], [4], [6], [30]. Youssef ad Gog [9] itroduced a class of bet fuctios called hyperbet fuctios, which achieve the maximal miimum distace to all the coordiate fuctios of all bijective moomials (i.e., fuctios of the form Tr 1 (ax i ) + ϵ, gcd(c, 1) = 1). Actually, it is Gog ad Golomb [14] who, based o a property of the exteded Hadamard trasform of Boolea fuctios, preseted the defiitio of hyper-bet fuctios. The classificatio of hyper-bet fuctio has ot bee achieved yet. May related problems are still ope. May research focus o the characterizatio of betess of Boolea fuctios. The moomial bet fuctios i the form Tr 1 (ax s ) are cosidered i [1], [18]. Leader [18] described the ecessary coditios for s such that Tr 1 (ax s ) is a bet fuctio. I particular, whe s = r( m 1) ad (r, m + 1) = 1, the moomial fuctios Tr 1 (ax s ) (i.e., the Dillo fuctios) were extesively studied i [6], [9], [18]. A class of quadratic fuctios over F i polyomial form 1 i=1 a i Tr 1 (x 1+i ) + a Tr 1 (x +1 ) (a i F ) was described ad studied i [8], [1], [16], [17], [0], [30]. Dobberti et al. [1] costructed a class of biomial bet fuctios of the form Tr 1 (a 1 x s1 + a x s ), (a 1, a ) (F ) with Niho power fuctios. Garlet ad Mesaager [] studied the duals of the Niho bet fuctios i [1]. I [], [3], [6], Mesager cosidered the biomial fuctios of the form Tr 1 (ax r(m 1) ) + Tr 1(bx 1 3 ), where a F ad b F 4. The he gave the lik betwee the betess property of such fuctios ad Kloosterma sums. Leader ad Kholosha [19] geeralized oe of the costructios prove by Dobberti et al. [1] ad preseted a ew primary costructio of bet fuctios cosistig of a liear combiatio of r Niho expoets. Carlet et al. [4] computed the dual of the Niho bet fuctio with r expoets foud by Leader ad Kholosha [19] ad showed that this ew bet fuctio is ot of the Niho type. Charpi ad Gog [6] preseted a characterizatio of betess of Boolea fuctios over F of the form Tr 1 (a r x r(m 1) ), where R is a subset of the set of represetatives of the cyclotomic cosets modulo m + 1 of maximal size. These fuctios iclude the well-kow moomial fuctios with the Dillo expoet as a special case. The they described the betess of these fuctios with the Dickso polyomials. Mesager et al. [4], [] geeralized the results of Charpi ad Gog [6] ad cosidered the betess of Boolea fuctios over F of the

2 form Tr 1 (a r x r(m 1) ) + Tr 1(bx 1 3 ), where = m, a r F m ad b F 4. Further, they preseted the lik betwee the betess of such fuctios ad some expoetial sums (ivolvig Dickso polyomials). I this paper, we cosider a class of Boolea fuctios over F i D. These Boolea fuctios are give by the form Tr 1 (a r x r(m 1) ) + Tr 4 1(bx 1 ), where = m, m (mod 4), a r F m ad b F 16. Whe b = 0, Charpi ad Gog [6] described the betess ad hyper-betess of these fuctios with some character sums ivolvig Dickso polyomial. Geerally, it is elusive to give a characterizatio of betess ad hyper-betess of Boolea fuctios i D. This paper presets the betess ad hyper-betess of fuctios i D i two cases: (1) b = 1 ad b 4 + b + 1 = 0; () a r F m. The rest of the paper is orgaized as follows. I Sectio II, we give some otatios ad review some kowledge o bet fuctios. I Sectio III, we cosider the betess ad hyper-betess of Boolea fuctios i D i two cases: (1) b = 1 ad b 4 + b + 1 = 0; () a r F m. The betess ad hyper-betess of these fuctios for the two cases are related to some character sums ivolvig Dickso polyomials ad some equatios o the weights of some Boolea fuctios. I Sectio IV, we list some examples. Fially, Sectio V makes a coclusio for the paper. II. PRELIMINARIES Let be a positive iteger. F is a -dimesioal vector space defied over fiite field F. Take two vectors i F x = (x 1,, x ) ad y = (y 1,, x ). Their dot product is defied by x, y := x i y i. i=1 F is a fiite field with elemets ad F is the multiplicative group of F. Let F k be a subfield of F. The trace fuctio from F to F k, deoted by Tr k, is a map defied as Tr k(x) := x + x k + x k + + x k. Whe k = 1, Tr 1 is called the absolute trace. The trace fuctio Tr k satisfies the followig properties. Tr k(ax + by) = atr k(x) + btr k(y), a, b F k, x, y F. Tr k(x k ) = Tr k(x), x F. Whe F k F r F, the trace fuctio Tr k satisfies the followig trasitivity property. Tr k(x) = Tr r k(tr r (x)), x F. A Boolea fuctio over F or F is a F -valued fuctio. The absolute trace fuctio is a useful tool i costructig Boolea fuctios over F. From the absolute trace fuctio, a dot product over F is defied by x, y := Tr 1 (xy), x, y F. A Boolea fuctio over F is ofte represeted by the algebraic ormal form (ANF): f(x 1,, x ) = x i ), a I F. i I a I ( I {1,,} i I Whe I =, let = 1. The terms x i are called moomials. The algebraic degree of a Boolea fuctio f is the globe degree of its ANF, that is, deg(f) := max{#(i) a I 0}, where #(I) is the order of I ad #( ) = 0. Aother represetatio of a Boolea fuctio is of the form f(x) = 1 j=0 i I a j x j. I order to make f a Boolea fuctio, we should require a 0, a 1 F ad a j = a j, where j is take modulo 1. This makes that f ca be represeted by a trace expasio of the form f(x) = 1 (a j x j ) + ϵ(1 + x 1 ) j Γ Tr o(j) called its polyomial form, where Γ is the set of itegers obtaied by choosig oe elemet i each cyclotomic class of module 1 (j is ofte chose as the smallest elemet i its cyclotomic class, called the coset leader of the class); o(j) is the size of the cyclotomic coset of modulo 1 cotaiig j; a j F o(j); ϵ = wt(f) (mod ), where wt(f) := #{x F f(x) = 1}. Let wt (j) be the umber of 1 s i its biary expasio. The {, ϵ = 1 deg(f) = max{wt (j) a j 0}, ϵ = 0. The sig fuctio of f is defied by χ(f) := ( 1) f. Whe f is a Boolea fuctio over F, the Walsh Hadamard trasform of f is the discrete Fourier trasform of χ(f), whose value at w F is defied by χ f (w) := ( 1) f(x)+ w,x. x F Whe f is a Boolea fuctio over F, the Walsh Hadamard trasform of f is defied by χ f (w) := ( 1) f(x)+tr 1 (wx), x F where w F. The we ca defie the bet fuctios. Defiitio A Boolea fuctio f : F F is called a bet fuctio, if χ f (w) = ± ( w F ). If f is a bet fuctio, must be eve. Further, deg(f) []. Hyper-bet fuctios are a importat subclass of bet fuctios. The defiitio of hyper-bet fuctios is give below.

3 3 Defiitio A bet fuctio f : F F is called a hyperbet fuctio, if, for ay i satisfyig (i, 1) = 1, f(x i ) is also a bet fuctio. [3] ad [9] proved that if f is a hyper-bet fuctio, the deg(f) =. For a bet fuctio f, wt(f) is eve. The ϵ = 0, that is, f(x) = 1 (a j x j ). j Γ Tr o(j) If a Boolea fuctio f is defied o F F, the we have a class of bet fuctios. Defiitio The Maioraa-McFarlad class M is the set of all the Boolea fuctios f defied o F F of the form f(x, y) = x, π(y) + g(y), where x, y F, π is a permutatio of F ad g(x) is a Boolea fuctio over F. For Boolea fuctios over F F, we have a class of hyper-bet fuctios PS ap [3]. Defiitio Let = m, the PS ap class is the set of all the Boolea fuctios of the form f(x, y) = g( x y ), where x, y F m ad g is a balaced Boolea fuctios (i.e., wt(f) = m 1 ) ad g(0) = 0. Whe y = 0, let x y = = 0. xy Each Boolea fuctio f i PS ap satisfies f(βz) = f(z) ad f(0) = 0, where β F m ad z F m F m. Youssef ad Gog [9] studied these fuctios over F ad gave the followig property. Propositio.1: Let = m, α be a primitive elemet i F ad f be a Boolea fuctio over F such that f(α m +1 x) = f(x)( x F ) ad f(0) = 0, the f is a hyper-bet fuctio if ad oly if the weight of (f(1),f(α), f(α ),, f(α m )) is m 1. Further, [3] proved the followig result. Propositio.: Let f be a Boolea fuctio defied i Propositio.1. If f(1) = 0, the f is i PS ap. If f(1) = 1, the there exists a Boolea fuctio g i PS ap ad δ F satisfyig f(x) = g(δx). Let PS # ap be the set of hyper-bet fuctios i the form of g(δx), where g(x) PS ap, δ F ad g(δ) = 1. Charpi ad Gog expressed Propositio. i a differet versio below. Propositio.3: Let = m, α be a primitive elemet of F ad f be a Boolea fuctio over F satisfyig f(α m+1 x) = f(x) ( x F ) ad f(0) = 0. Let ξ be a primitive m + 1-th root i F. The f is a hyper-bet fuctio if ad oly if the cardiality of the set {i f(ξ i ) = 1, 0 i m } is m 1. I fact, Dillo [9] itroduced a bigger class of bet fuctios the Partial Spreads class PS tha PS ap ad PS # ap. Theorem.4: Let E i (i = 1,,, N) be N subspaces i F of dimesio m such that E i E j = {0} for all i, j {1,, N} with i j. Let f be a Boolea fuctio over F. If the support of f is give by supp(f) = N Ei, where i=1 E i = E i \{0}, the f is a bet fuctio if ad oly if N = m 1. The set of all the fuctios i Theorem.4 is defied by PS. Now we recall the kowledge of Dickso polyomials over F. For r > 0, Dickso polyomials are give by D r (x) = r i=0 r r i ( r i i ) x r i, r =, 3,. Further, Dickso polyomials ca be also defied by the followig recurrece relatio. with iitial values D i+ (x) = xd i+1 + D i (x) D 0 (x) = 0, D 1 (x) = x. Some properties of Dickso polyomials are give below. deg(d r (x)) = r. D rp (x) = D r (D p (x)). D r (x + x 1 ) = x r + x r. The first few Dickso polyomials with odd r are D 1 (x) = x, D 3 (x) = x + x 3, D 7 (x) = x + x + x 7, D 9 (x) = x + x + x 7 + x 9, D 11 (x) = x + x 3 + x + x 9 + x 11. III. THE BENTNESS OF A NEW CLASS OF BOOLEAN FUNCTIONS WITH MULTIPLE TRACE TERMS A. Boolea fuctios i D Let = m ad m (mod 4). Let E be the set of represetig elemets i each cyclotomic class of module 1. Let D be the set of Boolea fuctios f b over F of the form f b (x) := Tr 1 (a r x r(m 1) ) + Tr 4 1(bx 1 ) (1) where R E, a r is i F m ad b F 16. Note that the cyclotomic coset of module 1 cotaiig 1 is { 1, 1, 1, 3 1 }. The its size is 4, that is, o( 1 ) = 4. Hece, the Boolea fuctio f b is ot i the class cosidered by Charpi ad Gog [6]. From m (mod 4), m (mod ). The every Boolea fuctio f b satisfies f b (α m +1 x) = f b (x), x F, where α is a primitive elemet of F. Note that f b (0) = 0. The the hyper-betess of f b ca be characterized by the followig propositio. Propositio 3.1: Let f b D. Set the character sum of the form Λ(f b ) := u U χ(f b (u)) () where U is the group of m +1-th roots of uity i F, that is, U = {x F x m +1 = 1}. The f b is a hyper-bet fuctio if ad oly if Λ(f b ) = 1. Further, a hyper-bet fuctio f b lies i PS ap if ad oly if Tr 4 1(b) = 0.

4 4 Proof: From Propositio.3, we have that f b is a hyperbet fuctio if ad oly if its restrictio to U has Hammig weight m 1. From the defiitio of Λ(f b ), Λ(f b ) = x U χ(f b (u)) = #{u U f b (u) = 0} #{u f b (u) = 1} = #U #{u f b (u) = 1} = m + 1 #{u f b (u) = 1}. Hece, the restrictio of f b to U has Hammig weight m 1 if ad oly if Λ(f b ) = 1. As a result, f b is a hyper-bet fuctio if ad oly if Λ(f b ) = 1. As for the secod part of the propositio, we compute f b (1) = Tr 1 (a r ) + Tr 4 1(b) = Tr m 1 (a r + a m r ) + Tr 4 1(b) =Tr 4 1(b). Hece, f b (1) = 0 if ad oly if Tr 4 1(b) = 0. From Propositio., we have a hyper-bet fuctio f b lies i PS ap if ad oly if Tr 4 1(b) = 0. B. The characterizatio of Boolea fuctios i D Our goal is to preset a characterizatio of hyper-betess of Boolea fuctios f b (b 0) i D. I this sectio we aalyze properties of Λ(f b ) for the characterizatio. We ow give some otatios first. Let α be a primitive elemet i F. The β = α 1 is a primitive -th root of uity i U ad U is a cyclic group geerated by ξ = α m 1. Let V be a cyclic group geerated by α (m 1). The U = 4 i=0ξ i V, F = F m U. Next, we itroduce the character sums Note that S i = χ(f 0 (ξ i v)). S 0 + S 1 + S + S 3 + S 4 = u U χ(f 0 (u)) = Λ(f 0 ). (3) For ay iteger i, S i = S i (mod ). The followig lemma gives the property of S i. Lemma 3.: S 1 = S 4, S = S 3. Proof: Noth that Tr 1 (x m ) = Tr 1 (x), the S i = = χ( Tr 1 (a r (ξ i v) r(m 1) )) χ( Tr 1 (a m r (ξ im v m ) r(m 1) )). From a r F m, a m r = a r. Sice m (mod 4) ad m 1 (mod ), hece i m i (mod ) ad ξ im v m = ξ i (ξ i(m +1) v m ), where ξ i(m +1) V. The map v ξ i(m +1) v m is a permutatio of V. Cosequetly, S i = χ( Tr 1 (a r (ξ i v) r(m 1) )) = S i. We just take i = 1,. The this lemma follows. From Lemma 3. ad (3), the followig corollary follows. Corollary 3.3: S 0 + (S 1 + S ) = Λ(f 0 ). Geerally, Λ(f b ) is a liear combiatio of S 0, S 1 ad S. Propositio 3.4: Λ(f b ) ca be expressed by a liear combiatio of S 0, S 1 ad S, that is, Λ(f b ) =χ(tr 4 1(b))S 0 + (χ(tr 4 1(bβ )) + χ(tr 4 1(bβ 3 )))S 1 + (χ(tr 4 1(bβ)) + χ(tr 4 1(bβ 4 )))S. Proof: From (), we have Λ(f b ) = u U = u U 4 = = = i=0 χ(f 0 (u) + Tr 4 1(bu 1 )) χ(tr 4 1(bu 1 ))χ(f 0 (u)) 4 i=0 χ(tr 4 1(b(ξ i v) 1 ))χ(f 0 (ξ i v)) (From (3)) χ(tr 4 1(b(α i(m 1) ) 1 ))χ(f 0 (ξ i v)) (ξ = α m 1 ) 4 χ(tr 4 1(bβ i(m 1) ))χ(f 0 (ξ i v)) i=0 (β = α 1 ) Sice m (mod ), hece m 1 3 (mod ). We have 4 Λ(f b ) = χ(tr 4 1(bβ 3i ))χ(f 0 (ξ i v)) = i=0 4 i=0 χ(tr 4 1(bβ 3i )) From the defiitio of S i, we obtai 4 Λ(f b ) = χ(tr 4 1(bβ 3i ))S i. i=0 χ(f 0 (ξ i v)) From Lemma 3., this propositio follows. Assume that a r F m 1 for every r R, where m 1 = m/. Further, we have the followig propositio. Propositio 3.: Assume a r F m 1, where r R, m 1 = m/, the S 1 = S, S 0 + 4S 1 = Λ(f 0 ). Proof: From a r F m 1, Tr 1 (a r x r(m 1) ) = Tr 1 (a r x m 1 r( m 1) ). The we have S i = = I particular, take i = 1, the S 1 = χ( Tr 1 (a r (ξ i v) r(m 1) )) χ( Tr 1 (a r (ξ m 1 i v m 1 ) r(m 1) )). χ( Tr 1 (a r (ξ m 1 v m 1 ) r(m 1) )).

5 Sice m (mod ), ( m1 ) 1 (mod ) ad m 1 ± (mod ). Whe m 1 (mod ), the The map S 1 = χ( Tr 1 (a r (ξ ξ m 1 v m 1 ) r(m 1) )). v ξ m 1 v m 1 is a permutatio of V. Cosequetly, S 1 = χ( Tr 1 (a r (ξ v) r(m 1) )) = S. Whe m 1 (mod ), we ca similarly obtai S 1 = S 3. As a result, S 1 = S. From Corollary 3.3, S 0 + 4S 1 = Λ(f 0 ). For Λ(f b ), the propositio below gives some properties. Propositio 3.6: Λ(f b ) satisfies the followig properties. (1) Λ(f b 4) = Λ(f b ). () If b a primitive elemet i F 16 Λ(f b ) = Λ(f b ) = S 0. ad Tr 4 1(b) = 0, the Proof: From b F 16, Tr 4 1(b 4 ) = Tr 4 1(b). Further, Tr 4 1(b(β + β 3 )) = Tr 4 1(b 4 (β 8 + β 1 )) = Tr 4 1(b 4 (β + β 3 )) ad Tr 4 1(b(β + β 4 )) = Tr 4 1(b 4 (β 4 + β 16 )) = Tr 4 1(b 4 (β + β 4 )). From the expressios of Λ(f b 4) ad Λ(f b ) i Propositio 3.4, Λ(f b 4) = Λ(f b ). () For a elemet b i F 16 such that Tr 4 1(b) = 0, it is easy to verify that b satisfies the followig equatio. Hece, we have b 4 + b + 1 = 0. Tr 4 1(b(β + β 3 )) =Tr 1(b 4 (β + β 3 ) + b(β + β 3 )) =Tr 1((b + b 4 )(β + β 3 )) =Tr 1(β + β 3 ). The miimal polyomial of β over F is β 4 +β 3 +β +β+1 = 0. Hece, Tr 1(β + β 3 ) = β + β 3 + β 4 + β 6 = 1. The we have Tr 4 1(b(β + β 3 )) = 1. Similarly, Tr 4 1(b(β + β 4 )) = 1. Therefore, we obtai χ(tr 4 1(bβ )) + χ(tr 4 1(bβ 3 )) = 0 ad χ(tr 4 1(bβ)) + χ(tr 4 1(bβ 4 )) = 0. From Propositio 3.4, Λ(f b ) = S 0. If b is a primitive elemet i F 16 such that Tr 4 1(b) = 0, b is also a primitive elemet i F 16 such that Tr 4 1(b) = 0. Naturally, we obtai Λ(f b ) = Λ(f b ) = S 0. I fact, we have more explicit results o Λ(f b ). Propositio 3.7: Let b F 16, the (1) If b = 1, the Λ(f b ) = S 0 (S 1 + S ) = S 0 Λ(f 0 ). () If b {β + β, β + β 3, β + β 4, β 3 + β 4 }, that is, b is a primitive elemet such that Tr 4 1(b) = 0, the Λ(f b ) = S 0. (3) If b = β or β 4, the Λ(f b ) = S 0 S 1. (4) If b = β or β 3, the Λ(f b ) = S 0 S. () If b = 1 + β or 1 + β 4, the Λ(f b ) = S 0 + S 1. (6) If b = 1 + β or 1 + β 3, the Λ(f b ) = S 0 + S. (7) If b = β + β 4, the Λ(f b ) = S 0 + S 1 S. (8) If b = β + β 3, the Λ(f b ) = S 0 S 1 + S. Proof: From the expressio of Λ(f b ) i Propositio 3.4, these results follows. Assume that a r F m 1 for every r R. We have more simplified results tha Propositio 3.7. Propositio 3.8: Assume that a r F m 1, where r R, the (1) If b = 1, the Λ(f b ) = S 0 Λ(f 0 ). () If b {β, β, β 3, β 4 }, the Λ(f b ) = S 0 S 1 = S 0+Λ(f 0 ). (3) If b {1 + β, 1 + β, 1 + β 3, 1 + β 4 }, the Λ(f b ) = S 0 + S 1 = 3S0 Λ(f0). (4) If b {β+β, β+β 3, β +β 4, β 3 +β 4, β+β 4, β +β 3 }, the Λ(f b ) = S 0. Proof: Propositio 3. gives that S 1 = S. From Propositio 3.7, these results i Propositio 3.8 follow. Corollary 3.9: Assume a r F m 1, where r R, the Λ(f b ) = Λ(f b ). Proof: From Propositio 3.8, this corollary follows. To characterize the hyper-betess of f b with character sums over F m, we ow itroduce some results o the character sums by Mesager []. Lemma 3.10: Let = m. f 0 is the fuctio over F defied by (1) with b = 0. Let g 0 be a Boolea fuctio defied o F m by g 0 (x) = Tr m 1 (a r D r (x)), where D r (x) is the Dickso polyomial of degree r. U is the group of m +1-th roots of uity i F. The for ay positive iteger p, we have χ(f 0 (u p )) = 1 + χ(g 0 (D p (x))). u U From Lemma 3.10, we have the followig propositio. Propositio 3.11: f b is the fuctio defied by (1). Let g 0 be a Boolea fuctio defied o F m by g 0 (x) = Tr m 1 (a r D r (x)), (4) where D r (x) is the Dickso polyomial of degree r. The (1) If b is a primitive elemet i F 16 such that Tr 4 1(b) = 0, the Λ(f b ) = 1 [1 + χ(g 0 (D (x)))]. () If b = 1, the Λ(f b ) = 1 [4 10 χ(g 0 (D (x))) χ(g 0 (x)) 3].

6 6 Proof: (1) From Propositio 3.7, whe b is the primitive elemet such that Tr 4 1(b) = 0, we have Λ(f b ) = S 0 = From Lemma 3.10, we obtai Λ(f b ) = 1 [1 + χ(f 0 (v)) = 1 χ(f 0 (u )). u U χ(g 0 (D (x)))]. () From Propositio 3.7, whe b = 1, we have Λ(f b ) =S 0 Λ(f 0 ) From Lemma3.10, we obtai = χ(f 0 (v)) χ(f 0 (u)) u U = χ(f 0 (u )) χ(f 0 (u)). u U u U Λ(f b ) = [1 + [1 + = 1 [4 10 χ(g 0 (D (x)))] χ(g 0 (x))] χ(g 0 (D (x))) χ(g 0 (x)) 3]. To have aother versio of Propositio 3.11, we first itroduce the followig lemma. Lemma 3.1: For ay Boolea fuctio g(x) over F m, χ(g(x)) = 1 [ χ(g(x)) χ(tr m 1 (x 1 ) + g(x))]. Proof: For ay x, y F, χ(x + y) = χ(x) + χ(y). The we have χ(g(x)) χ(tr m 1 (x 1 ) + g(x)) = = + = χ(g(x)) χ(g(x)) ( χ(tr m 1 (x 1 ))χ(g(x)),tr m 1 (x 1 )=0 ( 1)χ(g(x))) χ(g(x)). χ(g(x)) Hece, this lemma follows. Propositio 3.13: f b ad g 0 are fuctios defied by (1) ad (4) respectively. The, (1) If b is a primitive elemet i F 16 such that Tr 4 1(b) = 0, the Λ(f b ) = 1 [1 + χ(g 0 (D (x))) () If b = 1, the Λ(f b ) = 1 [ χ(tr m 1 (x 1 ) + g 0 (D (x)))]. χ(g 0 (D (x))) +g 0 (D (x))) + χ(g 0 (x)) χ(tr m 1 (x 1 ) + g 0 (x)) 3]. χ(tr m 1 (x 1 ) Proof: From Propositio 3.11 ad 3.1, this propositio follows. Note that for ay Boolea fuctio g(x) over F m, χ(g(x)) = m wt(g(x)). Hece, we have the followig corollary. Corollary 3.14: f b ad g 0 are fuctios defied by (1) ad (4) respectively. The, (1) If b is a primitive elemet i F 16 such that Tr 4 1(b) = 0, the Λ(f 0 ) = 1 [1+wt(Trm 1 (x 1 )+g 0 (D (x))) wt(g 0 (D (x)))]. () If b = 1, the Λ(f 0 ) = 1 [4wt(Trm 1 (x 1 ) + g 0 (D (x))) 4wt(g 0 (D (x))) + 10wt(g 0 (x)) 10wt(Tr m 1 (x 1 ) + g 0 (x)) 3]. Proof: From Propositio 3.13, this corollary follows. C. The hyper-betess of Boolea fuctios i D I this subsectio, we give a characterizatio of hyperbetess of Boolea fuctios i D. Theorem 3.1: Let = m ad m (mod 4). Let b is a primitive elemet i F 16 such that Tr 4 1(b) = 0, that is, b 4 + b + 1 = 0. f b is the fuctio defied o F by (1). Let g 0 be a Boolea fuctio defied o F m by g 0 (x) = x R Trm 1 (a r D r (x)), where D r (x) is the Dickso polyomial of degree r. The, the followig assertios are equivalet. (1) f b is hyper-bet. () χ(g 0 (D (x))) =. (3) wt(tr m 1 (x 1 ) + g 0 (D (x))) wt(g 0 (D (x))) =, where Tr m 1 (x 1 ) + g 0 (D (x)) ad g 0 (D (x)) are fuctios over F m. Proof: From Propositio 3.1, Propositio 3.11 ad Corollary 3.14, this theorem follows. Whe b = 1, we have the followig theorem. Theorem 3.16: Let = m ad m (mod 4). f 1 is the fuctio defied o F by (1) with b = 1. Let g 0 be a Boolea fuctio defied o F m by g 0 (x) = x R Trm 1 (a r D r (x)),

7 7 where D r (x) is the Dickso polyomial of degree r. The, the followig assertios are equivalet. (1) f 1 is hyper-bet. () χ(g 0 (D (x))) χ(g 0 (x)) = 4. (3) wt(tr m 1 (x 1 ) + g 0 (D (x))) wt(g 0 (D (x))) + wt(g 0 (x)) wt(tr m 1 (x 1 ) + g 0 (x)) = 4. Proof: From Propositio 3.1, Propositio 3.11 ad Corollary 3.14, this theorem follows. The followig propositio gives relatios of the hyperbetess of differet fuctios i D. Propositio 3.17: Let = m ad m (mod 4). Let d be a positive iteger coprime to m +1. Let b be a primitive elemet i F 16 such that Tr 4 1(b) = 0, that is, b 4 +b+1 = 0. f b is the fuctio defied by (1). Let h b be a Boolea fuctio defied by Tr 1 (a r x dr(m 1) ) + Tr 4 1(bx 1 ), where a r F m. The, h b is hyper-bet if ad oly if f b is hyper-bet. Proof: From Propositio 3.1 ad () i Propositio 3.7, h b is hyper-bet if ad oly if χ(h 0 (v)) = 1 for the fuctio h 0 = Tr 1 (a r x dr(m 1) ). Sice d ad the cardiality of V m +1 are coprime, the map v v d is a permutatio of V. Therefore, χ(h 0 (v)) = χ( Tr 1 (a r v dr(m 1) )) = χ( Tr 1 (a r v r( 1) )) = χ(f 0 (v)). Cosequetly, χ(h 0(v)) = 1 if ad oly if χ(f 0 (v)) = 1. From Propositio 3.1 ad () i Propositio 3.7, h b is hyper-bet if ad oly if f b is hyper-bet. Whe b = 1, we have the followig propositio. Propositio 3.18: Let = m ad m (mod 4). Let d be a positive iteger coprime to m + 1. f 1 is the fuctio defied by (1) with b = 1. Let h 1 be a Boolea fuctio defied by Tr 1 (a r x dr(m 1) ) + Tr 4 1(x 1 ), where a r F m. The, h 1 is hyper-bet if ad oly if f 1 is hyper-bet. Proof: Set h 0 = Tr 1 (a r x dr( 1) ). From Propositio 3.1 ad (1) i Propositio 3.7, h 1 is hyper-bet if ad oly if χ(h 0 (v)) χ(h 0 (u)) = 1. From the process u U of proof i Propositio 3.17, χ(h 0 (v)) = χ(f 0 (v)). Sice (d, m + 1) = 1, we ca have u U χ(h 0 (u)) = u U χ(f 0 (u)). Therefore, h 1 is hyper-bet if ad oly if f 1 is hyper-bet. Further, we assume d i Propositio The we ca get the followig propositio. Propositio 3.19: Let = m ad m (mod 4). Let d be a positive iteger coprime to m +1 ad d. Let b be a primitive elemet i F 16 such that Tr 4 1(b) = 0, that is, b 4 +b+ 1 = 0. f b is the fuctio defied by (1). Let h b be a Boolea fuctio defied by Tr 1 (a r x dr(m 1) ) + Tr 4 1(b x 1 ), () where b F 16. The (1) h 0 ad h β i(i = 0, 1,, 3, 4) are ot bet fuctios, where β is the primitive -th root of uity i F 16. () h b (b F 16 \{0, 1, β, β, β 3, β 4 }) have the same hyper-betess. Further, they are hyper-bet if ad oly if f b is hyper-bet. Proof: Set h 0 (x) = Tr 1 (a r x d(m 1) ). Let S i := χ(h 0 (ξ i v)). The S i = χ( Tr 1 (a r (ξ id v d ) r(m 1) )). Sice d ad (d, m +1 ) = 1, the map v ξ id v d is a permutatio of V. Therefore, S i = χ( Tr 1 (a r v r(m 1) )) = χ(f 0 (v)) = S 0. From (3), Λ(h 0 ) = S i = S 0. From (1), (3) ad (4) i Propositio 3.7, Λ(h β i) = 3S 0. Obviously, S 0 ad 3S 0 are ot equal to 1. Sice S 0 is odd. From Propositio 3.1, h 0 ad h β i(i = 0, 1,, 3, 4) are ot bet fuctios. From (), (), (6), (7) ad (8) i Propositio 3.7, whe b F 16 \{0, 1, β, β, β 3, β 4 }, Λ(h b ) = S 0 = S 0. The from Propositio 3.1 ad () i 3.7, () i this propositio follows. Assume a r F m 1 for ay r R. We have the hyperbetess of f b (b {β, β, β 3, β 4 }) i the followig theorem. Theorem 3.0: Let = m, m (mod 4) ad m = m 1. Let b {β, β, β 3, β 4 }. f b is the fuctio defied o F by (1), where a r F m 1 ad r R. Let g 0 be a Boolea fuctio over F m defied by g 0 (x) = Tr m 1 (a r D r (x)), where D r (x) is the Dickso polyomial of degree r. The the followig assertios are equivalet. (1) f b is hyper-bet. () χ(g 0 (D (x))) + χ(g 0 (x)) = 8. (3) wt(tr m 1 (x 1 ) + g 0 (D (x))) wt(g 0 (D (x))) + wt(tr m 1 (x 1 ) + g 0 (x)) wt(g 0 (x)) = 8. Proof: From () i Propositio 3.8, Λ(f b ) = 1 (S 0 + Λ(f 0 )) = 1 (S 0 Λ(f 0 )) 3 S 0. From (1) i Propositio

8 8 3.7 ad () Propositio 3.11, S 0 Λ(f 0 ) = 1 [4 χ(g 0 (D (x))) 10 χ(g 0 (x)) 3]. (6) The from (6) ad (7), Λ(f b ) = 1 [3 χ(g 0 (D (x))) χ(g 0 (x)) 1]. From () i Propositio 3.7 ad (1) i 3.11, Hece, S 0 = 1 [1 + Λ(f b ) = 1 [ + χ(g 0 (D (x)))]. (7) χ(g 0 (D (x))) χ(g 0 (x)) + 3]. The from Propositio 3.1, f b is hyper-bet if ad oly if () i this theorem holds. Further, from Propositio 3.11 ad Corollary 3.14, χ(g 0 (D (x))) =wt(tr m 1 (x 1 ) + g 0 (D (x))) ad wt(g 0 (D (x))) (8) χ(g 0 (x)) =wt(tr m 1 (x 1 ) + g 0 (x)) wt(g 0 (x)) (9) Cosequetly, assertios () ad (3) i this theorem are equivalet. Hece, this theorem follows. If b {1 + β, 1 + β, 1 + β 3, 1 + β 4 }, we have the followig theorem correspodig to Theorem 3.0. Theorem 3.1: Let = m, m (mod 4) ad m = m 1. Let b {1 + β, 1 + β, 1 + β 3, 1 + β 4 }, that is, b is the primitive elemet i F 16 such that Tr 4 1(b) = 1. f b is the fuctio over F by (1), where a r F m 1 ad r R. Let g 0 be a Boolea fuctio defied o F m by g 0 (x) = Tr m 1 (a r D r (x)), where D r (x) is the Dickso polyomial of degree r. The the followig assertios are equivalet. (1) f b is hyper-bet. () 3 χ(g 0 (D (x))) χ(g 0 (x)) = 4. (3) 3wt(Tr m 1 (x 1 ) + g 0 (D (x))) 3wt(g 0 (D (x))) wt(tr m 1 (x 1 ) + g 0 (x)) + wt(g 0 (x)) = 4. Proof: From (3) i Propositio 3.8, Λ(f b ) = 1 (3S 0 Λ(f 0 )) = 1 (S 0 Λ(f 0 )) 1 S 0. From Propositio 3.1, f b is hyper-bet if ad oly if () i this theorem holds. Further, from (8) ad (9), f b is hyper-bet if ad oly if (3) i this theorem holds. Hece, this theorem follows. If b {β+β, β+β 3, β +β 4, β 3 +β 4, β+β 4, β +β 3 }, we have the followig theorem correspodig to Theorem 3.0. Theorem 3.: Let = m, m (mod 4) ad m = m 1. Let b {β+β, β+β 3, β +β 4, β 3 +β 4, β+β 4, β +β 3 }, that is, b is the primitive elemet such that Tr m 1 (b) = 1 or a primitive 3-th root of uity. f b is the fuctio over F by (1), where a r F m 1 ad r R. Let g 0 be a Boolea fuctio defied o F m by g 0 (x) = Tr m 1 (a r D r (x)), where D r (x) is the Dickso polyomial of degree r. The f b is hyper-bet if ad oly if () ad (3) i Theorem 3.1 hold. Proof: From (4) i Propositio 3.8, Λ(f b ) = S 0. Hece, from Theorem 3.1, this theorem follows. IV. EXAMPLES OF HYPER-BENT FUNCTIONS IN D I this sectio, we list some istaces of hyper-bet fuctios i D. Let m 1 = 3, the m = 6 ad = 1. F 6 = F [x]/(x 6 + x 4 +x 3 +x+1), F 4 = F [x]/(x 4 +x+1), Let α 6 be a root of x 6 + x 4 + x 3 + x + 1 = 0. Let α 4 be a root of x 4 + x + 1 = 0. Take R = {1}, a 1 = α6 3 ad b = α 4 i (1). From Theorem 3.1, we have a hyper-bet fuctio of the form Tr 1 1 (α6 3 x m 1 ) + Tr 4 1(α 4 x 1 ). Take R = {1, 3}, a 1 = 1, a 3 = α 17 6 ad b = α 4 i (1). From Theorem 3.1, we have a hyper-bet fuctio of the form Tr 1 1 (x m 1 ) + Tr 1 1 (α6 17 x 3(m 1) ) + Tr 4 1(x 1 ). Take R = {1}, a 1 = 1 ad b = β i (1). From Theorem 3.0, we have a hyper-bet fuctio of the form Tr 1 1 (x m 1 ) + Tr 4 1(βx 1 ), where β is a primitive -th root of uity i F 16. Fially, take d =, R = {1} ad a 1 = α6 11 i (). From Propositio 3.19, we have a hyper-bet fuctio of the form Tr 1 1 (α6 11 x ( 1) ) + Tr 4 1(bx 1 ), where b F 16 \{0, 1, β, β, β 3, β 4 }.

9 9 V. CONCLUSION I this paper, we cosider a ew class of Boolea fuctios with multiple trace terms D. With some restrictios, we preset the characterizatio of hyper-betess of fuctios i D. We give a lik betwee hyper-bet fuctios of D ad some character sums ivolvig Dickso polyomials. Further, we relate hyper-betess of fuctios i D to some equatios o weights of Boolea fuctios ivolvig Dickso polyomials. This characterizatio of hyper-betess of fuctios i D provides more hyper-bet fuctios ad eriches the theory of hyper-bet fuctios. Naturally, further study o the characterizatio is to ivestigate the hyper-betess i other cases, such as (1) a r F m ad b F 16 \{b (b+1)(b 4 +b+1) = 0}; ()for some of r, a r F \ F m. ACKNOWLEDGMENT Baocheg Wag ad Yixia Yag ackowledge support from Natioal Sciece Foudatio of Chia Iovative Grat ( ), the CAS/SAFEA Iteratioal Partership Program for Creative Research Teams. 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