In this paper we characterize (octal) bent generalized Boolean functions
defined on $\BBZ_2^n$ with values in $\BBZ_8$. Moreover, we propose several
constructions of such generalized bent functions for both $n$ even and $n$ odd

Martinsen, Thor

Stanica, Pante

English

arXiv

In this paper we characterize (octal) bent generalized Boolean functions defined on $\BBZ_2^n$ with values in $\BBZ_8$. Moreover, we propose several constructions of such generalized bent functions for both $n$ even and $n$ odd

Pantelimon Stanica

Thor Martinsen

Cryptology ePrint Archive

Octal Generalized Boolean FunctionsPantelimon Stănică, Thor Martinsen,1 Department of Applied Mathematics??Naval Postgraduate SchoolMonterey, CA 93943–5216, USA{tmartins,pstanica}@nps.eduAbstract. In this paper we characterize (octal) bent generalized Booleanfunctions defined on Zn2 with values in Z8. Moreover, we propose severalconstructions of such generalized bent functions for both n even and nodd.Keywords: Generalized Boolean functions; generalized bent functions; cross-corelation.1 IntroductionSeveral generalizations of Boolean functions have been proposed in the recentyears and the effect of the Walsh–Hadamard transform on them has been stud-ied [8, 12, 13]. The natural generalizations of bent functions in the Boolean case,namely generalized functions which have flat spectra with respect to the Walsh–Hadamard transform are of special interest.Let the set of integers, real numbers and complex numbers be denoted by Z,R and C, respectively. By Zr we denote the ring of integers modulo r. A functionfrom Zn2 to Z2 is said to be a Boolean function on n variables and the set of allsuch functions is denoted by Bn. A function from Zn2 to Zq (q a positive integer)is said to be a generalized Boolean function on n variables [13]. We denote theset of such functions by GBqn. We will consider these functions with an emphasison q = 8.Any element x ∈ Zn2 can be written as an n-tuple (xn, . . . , x1), where xi ∈ Z2for all i = 1, . . . , n. The addition over Z, R and C is denoted by ‘+’. The additionover Zn2 for all n ≥ 1, is denoted by ⊕. Addition modulo q is denoted by ‘+’and is understood from the context. If x = (xn, . . . , x1) and y = (yn, . . . , y1)are two elements of Zn2 , we define the scalar (or inner) product, by x · y =xnyn ⊕ · · · ⊕ x2y2 ⊕ x1y1. The cardinality of the set S is denoted by |S|. Ifz = a + b ı ∈ C, then |z| =√a2 + b2 denotes the absolute value of z, andz = a − b ı denotes the complex conjugate of z, where ı2 = −1, and a, b ∈ R.The conjugate of a bit b will also be denoted by b̄.?? T.M. is a Ph.D student in Applied Mathematics at the Naval Postgraduate School.The (normalized) Walsh–Hadamard transform of f ∈ Bn at any point u ∈ Zn2is defined byWf (u) = 2−n2∑x∈Zn2(−1)f(x)(−1)u·x.A function f ∈ Bn, where n is even, is a bent function if |Wf (u)| = 1 for allu ∈ Zn2 . In case n is even a function f ∈ Bn is said to be a semibent function ifand only if, |Wf (u)| ∈ {0,√2} for all u ∈ Zn2 .The sumCf,g(z) =∑x∈Zn2(−1)f(x)⊕g(x⊕z)is the crosscorrelation of f and g at z. The autocorrelation of f ∈ Bn at u ∈ Zn2is Cf,f (u) above, which we denote by Cf (u).Let ζ = e2πı/q be the q-primitive root of unity. The (generalized) Walsh–Hadamard transform of f ∈ GBqn at any point u ∈ Zn2 is the complex valuedfunction defined byHf (u) = 2−n2∑x∈Zn2ζf(x)(−1)u·x.A function f ∈ Bn is a generalized bent function (gbent, for short) if |Hf (u)| = 1for all u ∈ Zn2 .The sumCf,g(z) =∑x∈Zn2ζf(x)−g(x⊕z)is the crosscorrelation of f and g at z. The autocorrelation of f ∈ GBqn at u ∈ Zn2is Cf,f (u) above, which we denote by Cf (u).When 2h−1 < q ≤ 2h, given any f ∈ GBqn we associate a unique sequence ofBoolean functions ai ∈ Bn (i = 0, 1, . . . , h− 1) such thatf(x) = a0(x) + 2a1(x) + · · ·+ 2h−1ah−1(x), for all x ∈ Zn2 . (1)2 Properties of Walsh–Hadamard transform ongeneralized Boolean functionsWe gather in the current section several properties of the Walsh–Hadamardtransform and its generalized counterpart [6].Theorem 1 We have:(i) Let f ∈ Bn. Then, the inverse of the Walsh–Hadamard transform is(−1)f(y) = 2−n2∑u∈Zn2Wf (u)(−1)u·y.(ii) If f, g ∈ Bn, then ∑u∈Zn2Cf,g(u)(−1)u·x = 2nWf (x)Wg(x),Cf,g(u) =∑x∈Zn2Wf (x)Wg(x)(−1)u·x.(iii) Taking the particular case f = g we obtainCf (u) =∑x∈Zn2Wf (x)2(−1)u·x.(iv) A Boolean function f is bent if and only if Cf (u) = 0 at all nonzero pointsu ∈ Zn2 .(v) For any f ∈ Bn, the Parseval’s identity holds∑x∈Zn2Wf (x)2 = 2n.For more properties of these transforms and Boolean functions, the interestedreader can consult [1–3].The properties of the Walsh–Hadamard transform on generalized Booleanfunctions are similar to the Boolean function case.Theorem 2 We have:(i) Let f ∈ GBqn. The inverse of the Walsh–Hadamard transform is given byζf(y) = 2−n2∑u∈Zn2Hf (u)(−1)u·y.Further, Cf,g(u) = Cg,f (u), for all u ∈ Zn2 , which implies that Cf (u) is alwaysreal.(ii) If f, g ∈ GBqn, then ∑u∈Zn2Cf,g(u)(−1)u·x = 2nHf (x)Hg(x),Cf,g(u) =∑x∈Zn2Hf (x)Hg(x)(−1)u·x.(iii) Taking the particular case f = g we obtainCf (u) =∑x∈Zn2|Hf (x)|2(−1)u·x. (2)(iv) If f ∈ GBqn, then f is gbent if and only ifCf (u) ={2n if u = 0,0 if u 6= 0.(3)(v) Moreover, the (generalized) Parseval’s identity holds∑x∈Zn2|Hf (x)|2 = 2n. (4)3 The Walsh–Hadamard Transform on componentsLet ζ = e2πı/q be a q-primitive root of unity. Let f be written as f(x) =∑h−1i=0 ai(x)2i. For brevity, we use the notations ζi := ζ2i . It is easy to see that,for s ∈ Z2, we havezs =1 + (−1)s2+1− (−1)s2z, (5)and so, we have the identities ζai(x)i =12 (Ai +A′iζi), where Ai = 1 + (−1)ai(x)and A′i = 1− (−1)ai(x), Ī = {0, 1, . . . , h− 1} \ I.The Walsh–Hadamard coefficients of f are2n/2Hf (u) =∑xζf(x)(−1)u·x =∑xζ∑h−1i=0 ai(x)2i(−1)u·x=∑x(−1)u·xh−1∏i=0(ζ2i)ai(x)=∑x(−1)u·xh−1∏i=012(1 + (−1)ai(x) + (1− (−1)ai(x))ζi)= 2−h∑x(−1)u·x∑I⊆{0,...,h−1}∏i∈I,j∈ĪζiA′iAj= 2−h∑x(−1)u·x∑I⊆{0,...,h−1}ζ∑i∈I 2i ∏i∈I,j∈ĪA′iAj= 2−h∑x(−1)u·x∑I⊆{0,...,h−1}ζ∑i∈I 2i ∑J⊆I,K⊆Ī(−1)|J|(−1)∑j∈J aj(x)⊕∑k∈K ak(x)= 2−h∑I⊆{0,...,h−1}ζ∑i∈I 2i ∑J⊆I,K⊆Ī(−1)|J|∑x(−1)u·x(−1)∑`∈J∪K a`(x),and so, we obtain the next result.Theorem 3 The Walsh–Hadamard transform of f : Zn2 → Zq, 2h−1 < q ≤ 2h,where f(x) =∑h−1i=0 ai(x)2i, ai ∈ Bn is given byHf (u) = 2−h∑I⊆{0,...,h−1}ζ∑i∈I 2i ∑J⊆I,K⊆Ī(−1)|J|W∑`∈J∪K a`(x)(u).In the next section we will redo some of these calculations, for the particularcase q = 8, which will allow us to completely describe the generalized bentBoolean functions in that case.4 A Characterization of Generalized Bent Functions in Z8In this section we extend the result of Solé and Tokareva [13] to generalizedBoolean functions from Zn2 into Z8. Let ζ = e2πı/8 =√22 (1+ı) be the 8-primitiveroot of unity. Every function f : Zn2 → Z8 can be written asf(x) = a0(x) + a1(x)2 + a2(x)22, (6)where ai(x) are Boolean functions, and ‘+′ is the addition modulo 8. We provethe next lemma, which gives the connection between Walsh–Hadamard trans-forms of f and it components as in (6).Lemma 4 Let f ∈ GB8n as in (6). Then,4Hf (u) = α0Wa2(u) + α1Wa0⊕a2(u) + α2Wa1⊕a2(u) + α3Wa0⊕a1⊕a2(u),where α0 = 1 + (1 +√2)ı, α1 = 1 + (1−√2)ı, α2 = 1 +√2− ı, α3 = 1−√2− ı.Proof. We compute2n/2Hf (u) =∑x∈Zn2ζf(x)(−1)u·x=∑x∈Zn2ζa0(x)+aa(x)2+a2(x)22(−1)u·x (7)=∑x∈Zn2ζa0(x)ıa1(x)(−1)a2(x)⊕u·x.Use formula (5) with z = ı and z = ζ in equation (7), and obtain2n/2Hf (u) =∑x∈Zn2(1 + (−1)a0(x)2+1− (−1)a0(x)2ζ)·(1 + (−1)a1(x)2+1− (−1)a1(x)2ı)(−1)a2(x)⊕u·x=14∑x∈Zn2(−1)a2(x)⊕u·x(1 + (1 +√2)ı+ (1 + (1−√2)ı)(−1)a0(x)+(1 +√2− ı)(−1)a1(x) + (1−√2− ı)(−1)a0(x)(−1)a1(x))=14∑x∈Zn2(α0(−1)a2(x)⊕u·x + α1(−1)a0(x)⊕a2(x)⊕u·x+α2(−1)a1(x)⊕a2(x)⊕u·x + α3(−1)a0(x)⊕a1(x)⊕a2(x)⊕u·x),from which we derive our result. utCorollary 5 With the notations of the previous lemma, we have4√2|Hf (u)|2 = W 2−X2+2XY +Y 2−2WZ−Z2+√2(W 2+X2+Y 2+Z2), (8)where, we use for brevity, W := Wa2(u), X := Wa0⊕a2(u), Y := Wa1⊕a2(u),Z := Wa0⊕a1⊕a2(u).Proof. By replacing αi, ζ by their complex representations, the corollary followsin a rather straightforward, albeit tedious manner. utTheorem 6 Let f ∈ GB8n as in (6). Then:(i) If n is even, then f is generalized bent if and only if a2, a0⊕a2, a1⊕a2, a0⊕a1 ⊕ a2 are all bent, and (∗) Wa0⊕a2(u)Wa1⊕a2(u) = Wa2(u)Wa0⊕a1⊕a2(u),for all u ∈ Zn2 ;(ii) If n is odd, then f is generalized bent if and only if a2, a0 ⊕ a2, a1 ⊕ a2, a0 ⊕a1 ⊕ a2 are semi-bent with their values satisfying (∗).Proof. We use the W,X, Y, Z notations of Corollary 5. First, assume that a2, a0⊕a2, a1⊕a2, a0⊕a1⊕a2 are all bent (respectively, semi-bent). Then, replacing thecorresponding values of the Walsh–Hadamard transforms in equation (8) (andusing the imposed condition (*) on the Walsh–Hadamard coefficients) we obtain4√2|Hf (u)|2 = 4√2,and so, |Hf (u)| = 1, that is, f is gbent.Conversely, we assume that f is gbent, and so,4√2 = W 2 −X2 + 2XY + Y 2 − 2WZ − Z2 +√2(W 2 +X2 + Y 2 + Z2),which prompts the systemW 2 −X2 + 2XY + Y 2 − 2WZ − Z2 = 0 (9)W 2 +X2 + Y 2 + Z2 = 4. (10)We are looking for solutions in 2−n/2 Z (a subset of Q, if n is even or√2Q, if nis odd).We look at equation (10), initially, and apply Jacobi’s four squares theorem(see [7], for instance).Case (i). Let n = 2k be even. Thus, W,X, Y, Z are all rational (certainly, not all0). Write W = 2−n/2W ′, X = 2−n/2X ′, Y = 2−n/2Y ′, Z = 2−n/2Z ′, and replace(9) and (10) by the system in integersW ′2 −X ′2 + 2X ′Y ′ + Y ′2 − 2W ′Z ′ − Z ′2 = 0 (11)W ′2 +X ′2 + Y ′2 + Z ′2 = 22k+2. (12)Now, by Jacobi’s four-squares theorem, we know there are exactly 24 solutionsof (12), which are all variations in ± sign and order of (±2k,±2k,±2k,±2k) or(±2k+1, 0, 0, 0). Further, it is straightforward to check that among these 24 so-lutions, only the eight tuples (X ′, Y ′,W ′, Z ′) in the list below are also satisfyingequation (11),(−2k,−2k,−2k,−2k), (2k, 2k,−2k,−2k), (−2k,−2k, 2k, 2k), (−2k, 2k,−2k, 2k),(2k,−2k,−2k, 2k), (−2k, 2k, 2k,−2k), (2k,−2k, 2k,−2k), (2k, 2k, 2k, 2k).This implies that (X,Y,W,Z) ∈ 2−n/2Z4 are any of the following:(−1,−1,−1,−1), (1, 1,−1,−1), (−1,−1, 1, 1), (−1, 1,−1, 1),(1,−1,−1, 1), (−1, 1, 1,−1), (1,−1, 1,−1), (1, 1, 1, 1),(13)and (i) is shown (one can check easily that these solutions also satisfy condi-tion (∗)).Case (ii). Let n = 2k + 1 be odd. Then, at least one of X,Y,W,Z is nonzeroand belongs to√2Q). As before, write W = 2−n/2W ′, X = 2−n/2X ′, Y =2−n/2Y ′, Z = 2−n/2Z ′, and replace (9) and (10) by the system in integersW ′2 −X ′2 + 2X ′Y ′ + Y ′2 − 2W ′Z ′ − Z ′2 = 0 (14)W ′2 +X ′2 + Y ′2 + Z ′2 = 2 · 22k+2, (15)and so, by Jacobi’s four-squares theorem, equation (15) has exactly 24 solutions,which are all variations in ± sign and order of (±2k+1,±2k+1, 0, 0). Further,it is straightforward to check that among these 24 solutions, the eight tuples(X ′, Y ′,W ′, Z ′) in the list below are also satisfying equation (14),(0, 2k+1, 0, 2k+1), (0, 2k+1, 0,−2k+1), (0,−2k+1, 0, 2k+1), (0,−2k+1, 0,−2k+1)(2k+1, 0, 2k+1, 0), (2k+1, 0,−2k+1, 0, (−2k+1, 0, 2k+1, 0), (−2k+1, 0,−2k+1, 0).Thus, the solutions (X,Y,W,Z) to (9) and (10) are(0,√2, 0,√2), (0,√2, 0,−√2), (0,−√2, 0,√2), (0,−√2, 0,−√2),(√2, 0,√2, 0), (√2, 0,−√2, 0), (−√2, 0,√2, 0), (−√2, 0,−√2, 0),which also satisfy condition (*), and (ii) is shown. ut5 Constructions of generalized bent functions in Z8In this section we define several classes of generalized bent Boolean functions.Theorem 7 If f : Zn+22 → Z8 (n even) is given byf(x, y, z) = 4c(x) + (4a(x) + 2c(x) + 1)y + (4b(x) + 2c(x) + 1)z − 2yz,where a, b, c ∈ Bn such that all a, b, c, a⊕ c, b⊕ c and a⊕ b are bent satisfyingWa(x)Wb(x) +Wa⊕c(x)Wb⊕c(x) = −2Wa⊕b(x)Wc(x)), for all x ∈ Zn2 , (16)then f is gbent in GB8n+2.Proof. We compute the Walsh–Hadamard coefficients (using the fact that ζ =1√2(1 + ı) and ζ2 = ı)2(n+2)/2Hf (u, v, w) =∑(x,y,z)∈Zn+22ζf(x,y,z)(−1)u·x⊕vy⊕wz=∑x∈Zn2ζ4c(x)(−1)u·x·∑(y,z)∈Z22ζ(4a(x)+2c(x)+1)y+(4b(x)+2c(x)+1)z−2yz(−1)vy⊕wz=∑x∈Zn2(−1)c(x)⊕u·x ·(1 + (−1)v(−1)a(x)ıc(x)ζ+(−1)w(−1)b(x)ıc(x)ζ + (−1)a(x)⊕b(x)⊕c(x)⊕v⊕w).(17)Applying equation (5) with (z, s) = (ı, c(x)), that is,ic(x) =1 + (−1)c(x)2+1− (−1)c(x)2ı,we obtain2Hf (u, v, w) = Wc(u) +(−1)vζ2(Wa⊕c(u) +Wa(u) + ıWa⊕c(u)− ıWa(u))+(−1)wζ2(Wb⊕c(u) +Wb(u) + ıWb⊕c(u)− ıWb(u)) + (−1)v⊕wWa⊕b(u)= Wc(u) +(−1)v√2(Wa(u) + ıWa⊕c(u)) +(−1)w√2(Wb(u) + ıWb⊕c(u))+(−1)v⊕wWa⊕b(u).Therefore, the real and the imaginary parts of cHf (u, v, w) areRe(Hf (u, v, w)) = Wc(u) + (−1)v⊕wWa⊕b(u) +(−1)vWa(u) + (−1)wWb(u)√2,Im(Hf (u, v, w)) =(−1)vWa⊕c(u) + (−1)wWb⊕c(u)√2.and so,4|Hf (u, v, w)|2 =12(Wa(u)2 + Wb(u)2 + Wa⊕c(u)2 + Wb⊕c(u)2+2Wc(u)2 + 2Wa⊕b(u)2)+ (−1)v+w(Wa(u)Wb(u) + Wa⊕c(u)Wb⊕c(u) + 2Wc(u)Wa⊕b(u))+√2 ((−1)v(Wa(u)Wc(u) + Wb(u)Wa⊕b(u))+(−1)w(Wb(u)Wc(u) + Wa(u)Wa⊕b(u)))(18)Since a, b, c, a ⊕ c, b ⊕ c, a ⊕ b are all bent then |Wa(u)| = |Wb(u)| =|Wc(u)| = |Wa⊕b(u)| = |Wa⊕c(u)| = |Wb⊕c(u)| = 1. Further, from the im-posed conditions on these functions’ Walsh–Hadamard coefficients, we see thatWa(u)Wb(u)+Wa⊕c(u)Wb⊕c(u)+2Wc(u)Wa⊕b(u) = 0, and also Wa(u)Wc(u)+Wb(u)Wa⊕b(u) = 0, Wb(u)Wc(u)+Wa(u)Wa⊕b(u) = 0 (that is because if Wa(u)and Wb(u) have the same sign, then Wc(u),Wa⊕b have opposite signs; further,Wa(u) and Wb(u) have opposite signs, then Wc(u),Wa⊕b have the same sign).Using these equations, we get that 4|Hf (u, v, w)|2 = 4, and so, f is gbent.utRemark 8 It is rather straightforward to see that condition (16) has 16 solu-tions. More precisely, (Wa(x),Wb(x),Wa⊕c(x),Wb⊕c,Wa⊕b(x),Wc(x)) could beany of the following tuples:(−1,−1,−1,−1,−1, 1); (−1,−1,−1,−1, 1,−1);(−1, 1, 1, 1,−1, 1); (−1,−1, 1, 1, 1,−1);(−1, 1,−1, 1,−1,−1); (−1, 1,−1, 1, 1, 1);(−1, 1, 1,−1, 1,−1); (−1, 1, 1,−1, 1, 1);(1,−1,−1, 1,−1,−1); (1,−1,−1, 1, 1, 1);(1,−1, 1,−1, 1,−1); (1,−1, 1,−1, 1, 1);(1, 1,−1,−1,−1, 1); (1, 1,−1,−1, 1,−1);(1, 1, 1, 1,−1, 1); (1, 1, 1, 1, 1,−1).Theorem 9 If f : Zn+22 → Z8 (n even) is given byf ε(x, y, z) = 4c(x) + (4a(x) + 1)y + (4b(x) + 1)z + 2εyz, (19)where ε ∈ {1,−1}, a, b, c ∈ Bn such that all c, a⊕ c, b⊕ c and a⊕ b⊕ c are bent,withWa⊕c(u)Wb⊕c(u) + εWc(u)Wa⊕b⊕c(u) = 0, for all u ∈ Zn2 , (20)then f is gbent in GB8n+2.Proof. As in the proof of Theorem 7, we compute the Walsh–Hadamard coeffi-cients, obtaining2Hfε(u, v, w) = Wc(u) + (−1)vζWa⊕c(u) + (−1)wζWb⊕c(u)+(−1)v⊕wζ2+2εWa⊕b⊕c(u)= Wc(u)− ε(−1)v⊕wWa⊕b⊕c(u)+(−1)vWa⊕c(u) + (−1)wWb⊕c(u)√2+ı(−1)vWa⊕c(u) + (−1)wWb⊕c(u)√2,using the fact that ζ2+2ε = −ε, for ε ∈ {1,−1}. Taking the square of the complexnorm, we get4|Hfε(u, v, w)|2 = Wa⊕c(u)2 +Wb⊕c(u)2 +Wc(u)2 +Wa⊕b⊕c(u)2+2(−1)v+w (Wa⊕c(u)Wb⊕c(u) + εWc(u)Wa⊕b⊕c(u))+√2 ((−1)v(Wa⊕c(u)Wc(u) + εWb⊕c(u)Wa⊕b⊕c(u))+(−1)w(Wb⊕c(u)Wc(u) + εWa⊕c(u)Wa⊕b⊕c(u)))= 4,because c, a ⊕ c, b ⊕ c and a ⊕ b ⊕ c are all bent, so their Walsh–Hadamardcoefficients are 1 in absolute values, and equation (20) implies that the remainingcoefficients are all 0 (that can be seen by the following argument: if A,B,C,D ∈{±1}, and AB + CD = 0, then by multiplying by BC, we get AC + BD = 0,and by multiplying by AC we get BC +AD = 0).Therefore, |Hfε(u, v, w)|2 = 1, so f is gbent, and the theorem is proved. utRemark 10 It is rather easy to see that equation (20) has 8 solutions (as ex-pected, since there are four degrees of freedom and one constraint). Moreover,one can give plenty of concrete examples of functions a, b, c satisfying the condi-tions of our theorem. For example, if ε = −1, one could take in equation (19),a bent Boolean c, and a = b such that c ⊕ a is bent (for instance, if a = bare affine functions, that condition is immediate). Then, Wa⊕c(u)Wb⊕c(u) +εWc(u)Wa⊕b⊕c(u) = Wc⊕a(u)2 −Wc(u)2 = 0, and so, g as in our theorem isgbent.Theorem 11 Let f : Zn+12 → Z8 (n is even) be given byf(x, y) = 4c(x) + (4a(x) + 4c(x) + 2ε)y, (21)where ε ∈ {1,−1}. Then f is gbent in GB8n+1 if and only if a, c are bent in Bn.Moreover, if g is given byg(x, y) = 4c(x) + (4a(x) + 2c(x) + 2ε)y, (22)where ε ∈ {1,−1}, a, c ∈ Bn such that a, c, a⊕ c are all bent, then g is gbent inGB8n+1. Further, let h be given byh(x, y) = 4c(x) + (4a(x) + 2ε)y, (23)where ε ∈ {1,−1}. Then h is gbent in GB8n+1 if and only if c, a ⊕ c are bent inBn.Proof. We will show the first claim, since the proof of the remaining ones are ab-solutely similar. As in the proof of Theorem 7, the Walsh–Hadamard coefficientsat an arbitrary input (u, v) are√2Hf (u, v) = Wc(u) + ıε(−1)vWa(u) = Wc(u) + ε ı(−1)vWa(u),and so,2|Hf (u, v)|2 = Wc(u)2 +Wa(u)2.If a, c are bent, then |Wc(u)| = |Wa(u)| = 1, and so |Hf (u, v)| = 1, that is fis gbent. If f is gbent, then the equation Wc(u)2 + Wa(u)2 = 2 has as rationalsolutions only |Wc(u)| = |Wa(u)| = 1, and so, a, c are bent. utReferences1. C. Carlet, Boolean functions for cryptography and error correcting codes, In: Y.Crama, P. Hammer (eds.), Boolean Methods and Models, Cambridge Univ. Press,Cambridge. Available: http://www-roc.inria.fr/secret/Claude.Carlet/pubs.html.2. C. Carlet, Vectorial Boolean functions for cryptography, In: Y. Crama, P. Ham-mer (eds.), Boolean Methods and Models, Cambridge Univ. Press, Cambridge.Available: http://www-roc.inria.fr/secret/Claude.Carlet/pubs.html.3. T. W. Cusick, P. Stănică, Cryptographic Boolean functions and applications, El-sevier – Academic Press, 2009.4. J. F. Dillon, Elementary Hadamard difference sets, Proc. of Sixth S.E. Conferenceof Combinatorics, Graph Theory, and Computing, Congressus Numerantium No.XIV, Utilitas Math., Winnipeg 1975, 237–249.5. S. Gangopadhyay, B. K. Singh and P. Stănică, On Generalized Bent Functions,preprint.6. S. Gangopadhyay, B. K. Singh and P. Stănică, Generalized Bent Functions in theFramework of Generalized Partial Spreads, preprint.7. M. D. Hirschhorn, A simple proof of Jacobis foursquare theorem, Proc. Amer.Math. Soc., 101 (1987), 436–438.8. P. V. Kumar, R. A. Scholtz, and L. R. Welch, Generalized bent functions and theirproperties, J. Combin. Theory (A) 40 (1985), 90–107.9. F. J. MacWilliams, N. J. A. Sloane, The theory of error–correcting codes, North-Holland, Amsterdam, 1977.10. O. S. Rothaus, On bent functions, J. Combinatorial Theory Ser. A 20 (1976), 300–305.11. P. Sarkar, S. Maitra, Cross–Correlation Analysis of Cryptographically UsefulBoolean Functions and S-Boxes, Theory Comput. Systems 35 (2002), 39–57.12. K-U. Schmidt, Quaternary Constant-Amplitude Codes for Multicode CDMA, IEEEInternational Symposium on Information Theory, ISIT’2007 (Nice, France, June24–29, 2007), 2781–2785; available at http://arxiv.org/abs/cs.IT/0611162.13. P. Solé, N. Tokareva, Connections between Quaternary and Binary Bent Functions,http://eprint.iacr.org/2009/544.pdf.

Octal Bent Generalized Boolean Functions

Abstract

Similar works

Full text

Available Versions

Cryptology ePrint Archive