Higher order nonlinearity of some cryptographic functions

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1 International Journal of Computational and Applied Mathematics. ISSN Volume 1, Number (017), pp Research India Publications Higher order nonlinearity of some cryptographic functions Deep Singh 1 and Amit Paul Department of Mathematics, Central University of Jammu, Samba, India. Abstract The security of various cryptosystems is strongly related to the higher order nonlinearity of cryptographic functions. This paper investigates some cryptographic functions with good nd and 4th order nonlinearities. Firstly, we tighten the lower bounds on nd order nonlinearity for the function φ λ (u) = Tr1 n (λup ) with p = s + s + 1,λ F s and n = 7s. Further, we give lower bounds for 4th order nonlinearity of 10-variable partial spreads: φ(u) = Tr 10 1 (λu 10 1 ), λ F 10. AMS subject classification: Keywords: Boolean functions, higher-order nonlinearity, trace functions, Kasami functions, Walsh-Hadamard transform. 1. Introduction Boolean functions are considered to be the building blocks in the design of several symmetric key cryptosystems. Let φ : F n F be a Boolean function on n-unknowns. The rth order nonlinearity nl r (φ), 0<r n of φ is the minimum Hamming distance of φ from the functions of degree r (when r = 1, it becomes nl(φ), the first order nonlinearity). The collection of different values of nl r (φ) for 1 r n 1 is nonlinearity profile for φ. The rth order nonlinearity nl r (φ) is a natural generalization of first order nonlinearity of φ which is important for prevention of affine approximation attacks [1, 1, 13]. The best upper bound on nl r (φ) in [6] is asymptotically equivalent to 15 nl r (φ) = n 1 (1 + ) r n + O(n r ). 1 corresponding author

2 D. Singh and A. Paul For rth order nonlinearity (r > 1) of Boolean functions, we do not have an algorithm unlike the first order nonlinearity. The best algorithm presented in [8] for the case r = for n 11 and up to n = 13 for some functions. Cryptographer feels that there is a need to obtain theoretical bounds of higher order nonlinearities of Boolean functions which are satisfied for all values of n. The rth order bent functions with lower bound n r 3 (r + 5) are presented in [13]. Carlet et al. [5] in 006 derived the lower bounds on rth order nonlinearities of Boolean functions by means of algebraic immunity, the bounds were further improved by Carlet [3]. In [4], Carlet presented recursive approach for rth order nonlinearity. He obtained lower bounds of nonlinearity profiles for the Kasami functions, Welch functions, inverse functions. Using the Carlet s recursive approach various authors [11, 14, 18, 0] have obtained the bounds on the second order nonlinearities of some functions. In this article, we deduce lower bounds on nd order nonlinearity of functions φ λ (u) = Tr1 n (λup ) with p = s + s + 1, λ F s and n = 7s. Further, we obtain lower bounds on 4th order nonlinearity of 10-variable monomial partial spreads: φ(u) = Tr (λu 1 ), λ F 10.. Preliminaries Let F n be the n degree extension field of F. The set of all units of F n is denoted by F n. A function φ : F n F is called n-variable Boolean function. Suppose B n is the collection of all Boolean functions such that cardinality B n = n. The support of φ B n is defined as supp(φ) ={u F n : φ(u) = 1}. The Hamming weight of φ B n is defined as wt(φ) = supp(φ). The Hamming distance between two Boolean function h, κ B n is d(h, κ) = {α F n : h(α) = κ(α)}. The algebraic normal form of φ B n is φ(u 1,u,...,u n ) = α J u j, j J J {1,,...,n} where α J F and the terms j J u j are monomials. The maximum degree of the monomial with nonzero coefficient is algebraic degree of φ. For any subfield F t of F n(obviously t n), the function the function Trt n : F n F t defined by Trt n (u) = u + ut + u t + +u (n 1)t is called a trace function. For t = 1, Tr1 n (u) = u + u + u + +u n 1 is absolute trace function. The derivative of φ B n along α F n is given by D α φ(u) = φ(u) + φ(u+ α) for all u F n. If W = v 1,...,v m is a t-dimensional subspace in F n then D W φ(u) = D v1 D vm φ(u), for all u F n is t-th order derivative of φ along W. The Walsh Hadamard Transform of φ B n is defined as W φ (α) = ( 1) φ(u)+trn 1 (αu), α F n u F n

3 Higher order nonlinearity of some cryptographic functions 3 The sequence of Walsh coefficients of φ is Walsh Hadamard spectrum (WHS) of φ. The minimum Hamming distance of φ B n from affine functions is nonlinearity of φ given as nl(φ) = n 1 1 max W φ (α). α F n Parseval s identity Wφ (α) = n, implies that nl(φ) n 1 n 1. The α F n function with maximum possible nonlinearity is called bent function [17] and exists only for n-even. Rothaus [17] in 1976 proved that for even n maximum possible nonlinearity of n-variable Boolean functions is n 1 n 1. Let W be a vector space of dimension n over F q, a field of characteristic. A map Q : W F q is a quadratic form on W if 1. Q(mu) = m Q(u) m F q, u W.. B(u, v) = Q(u) + Q(v) + Q(0) + Q(u + v) is bilinear on W. The kernel of B(u, v) denoted by E Q is the subspace of W and is defined as E Q ={u W : B(u, v) = 0 v W}. Lemma.1. [] Suppose W be a vector space of dimension n over F q, a field of characteristic. For a quadratic form Q on W, the dimension of both W and kernel of B(u, v) possess same parity. Lemma.. [] Suppose φ B n is quadratic. The kernel E φ is E φ ={u F n : D α φ = constant}. Lemma.3. [16] If φ B n is quadratic, then the WHT of φ is only linked with the kernel of φ. Lemma.4. [4] Suppose r<nand φ B n, then nl r (φ) 1 max nl r 1 (D α φ). α F n Lemma.5. [4] Suppose r<nand φ B n, then nl r (φ) n 1 1 n nl r 1 (D α φ). α F n In terms of higher-order derivative, for every positive integer l<r. nl r (φ) n 1 1 n nl r l (D α1 D αl φ). α 1 F n α F n α l F n

4 4 D. Singh and A. Paul Lemma.6. [4] Suppose r < n and φ B n. Also suppose for some nonnegative integers L and θ, and for 0 = α F n, we have Then nl r 1 (D α φ) n 1 L θ. (.1) nl r (φ) n 1 1 ( n 1)L θ+1 + n n 1 L n+θ 1. (.) 3. Main results This section presents lower bounds on higher order nonlinearities of some cryptographic functions. First, we provide bounds on nd order nonlinearities, further, in Subsection 3.1, we discuss 4th order nonlinearities. Theorem 3.1. Let φ λ (u) = Tr1 n (λup ) with p = s + s + 1, n= 7s, λ F s. Then dimension of kernel of bilinear form of D α (φ λ (u)) is either s or 5s. Proof. The derivative D α (φ λ (u)) with respect to α F n is D α φ λ (u) = φ λ (u + α) + φ λ (u) = Tr1 n (λ(u + α)s + s +1 ) + Tr1 n (λus + s +1 ) = Tr1 n (λ(αus + s + α s u s +1 + α s u s +1 + α s +1 u s +α s +1 u s + α s + s u + α s + s +1 )) quadratic. The WHS of D α φ λ (u) is equivalent to that of g λ (u), where g λ (u) is obtained by eliminating linear and constant terms in D α φ λ (u) as g λ (u) can also be written as g λ (u) = Tr n 1 (λ(αus + s + α s u s +1 + α s u s +1 )), g λ (u) = Tr n 1 (λαs u s +1 + (λ 6s α 6s + λα s )u s +1 ). Since s + 1 and s + 1 do not belongs to same cyclotomic coset. So, g λ (u) = 0 for any α F n. Since g λ(u) is a quadratic function. In the view of Lemma. and.3, we collect all those β s for which D β (g λ (u)) is constant. Now, D β (g λ (u)) = g λ (u + β) + g λ (u) = Tr1 n (λ(α(u + β)s + s + α s (u + β) s +1 + α s (u + β) s +1 )) +Tr1 n (λ(αus + s + α s u s +1 + α s u s +1 )) = Tr1 n (λ((αβs + α s β)u s + (αβ s + α s β)u s +(α s β s + α s β s )u)) +Tr n 1 (λ(αβs + s + α s β s +1 + α s β s +1 )).

5 Higher order nonlinearity of some cryptographic functions 5 Since u, α, β F n and λ F s. Using un = u, α n = α, β n = β,λ n = λ, we get D β (g λ (u)) = Tr n 1 (λu((α5s + α s )β 6s + α 6s β 5s + α s β s + (α 6s + α s )β s )) +Tr n 1 (λ(αβs + s + α s β s +1 + α s β s +1 )). Clearly, D β (g λ (u)) is equal to the constant if and only if (α 5s + α s )β 6s + α 6s β 5s + α s β s + (α 6s + α s )β s = 0. Raising power s th, we have (α 4s + α)β 5s + α 5s β 4s + αβ s + (α 5s + α s )β = 0, (3.1) n which is a s -polynomial. The polynomial L(u) = a i x qi with a i F q m,m>1is q polynomial over F q m. Let i=0 M(β) = (α 4s + α)β 5s + α 5s β 4s + αβ s + (α 5s + α s )β. The dimension of kernel of M(β) is lr, l = 0, 1, 4, 5. Now, quadratic form from F q 5 to F q (q = s ) is R(u) = Tr L E (λ(αus + s + α s u s +1 + α s u s +1 )), where L = F 7s and E = F s The roots of M(u) forms kernel of R(u). In fact, kernel of R(u) is the collection of β s where B(u) = 0 u with Since D b (G λ (x)) = Tr E F (B(u)),weget B(u) = R(u) + R(β) + R(u + β). B(u) = Tr L E (u(m(β))). Thus, R(u) and M(u) have same kernel. According to Lemma.1, R(u) has dimension of its kernel either 1 or 5 which implies either s or 5s is one of the root of M(u). Hence the dimension of the kernel of bilinear form of D α (φ λ (u)) is either s or 5s. Theorem 3.. Let φ λ (u) = Tr1 n (λup ) with p = s + s + 1, λ F s and n = 7s. Then nl (φ λ (u)) 7s 1 s 1 s ( 6s + 3s 1). Proof. From Theorem 3.1, dimension k of kernel of bilinear of D a (φ λ (u)) is either s or 5s. The nonlinearity of D α (φ λ (u)) i.e., nl(d α (φ λ (u))) is either n 1 1 n+s or n 1 1 n+5s. Therefore, we have max nl(d α (φ λ (u))) = n 1 1 α F n n+s.

6 6 D. Singh and A. Paul Now, Lemma.5 implies that nl (φ λ (u)) n 1 1 n α F n nl(d α φ λ (u)) = 7s s ( 7s s )( 7s 1 4s 1 ) = 7s 1 s 1 s ( 6s + 3s 1). Hence the result. Now with the help of Lemma.6, we improve the above results in the following theorem. Theorem 3.3. Let φ λ (u) = Tr1 n (λup ) with p = s + s + 1, λ F s and n = 7s. Then nl (φ λ (u)) 7s 1 s 4 4. Proof. From Theorem 3., we have max nl(d α (φ λ (u))) = n 1 1 α F n n+s. On comparing the above equation with equation (.1), we get L = 1 and θ = n + s. Thus, by (.), we obtain nl (φ λ (u)) n 1 3n+s 4 4 = 7s 1 s Lower bounds of 4th-order nonlinearity for monomial partial spread on 10-variables The monomial functions of the form f λ (x) = Tr n 1 (λx n 1 ), where λ F n are called monomial partial spreads on n-variables. For some values of λ these functions becomes PS type bent functions. For details we may refer to [, 7]. Dillon [7] has introduced an important class of Boolean functions called partial spreads. Suppose f B n,n= t. Consider a set {H i : i = 1,...,M} of subspaces of F n of dimension t, with H i H j = {0}, when i = j. The function f with is called a partial spreads (PS). supp(f ) = M i=0 H i

7 Higher order nonlinearity of some cryptographic functions 7 In the following theorem, we obtain lower bound for 4th order nonlinearity of monomial partial spreads on 10-variables: φ(u) = Tr 10 1 (λu 10 1 ) = Tr n 1 (λu31 ). Theorem 3.4. Let φ(u) = Tr (λu 1 ), for all u F n and λ F n. Then, we have nl 4 (φ λ ) 43. Proof. The derivative D α φ λ of φ λ along α F n is D α φ λ (u) = φ λ (u + α) + φ λ (u) = Tr1 n (λ(u + α) ) + Tr1 n (λu ) = Tr1 n (λ(αu α 4 u α 3 u α u α u )) + c(u), where c(u) is cubic function. The second derivative D β D α φ λ of φ λ along β F n (α = β) is D β D α φ λ (u) = φ λ (u + α + β) + φ λ (u + α) + φ λ (u + β) + φ λ (u) = Tr1 n [λ((αβ + βα )u (αβ 4 + βα 4 )u (αβ 3 + βα 3 )u (αβ + βα )u (α β + α β )u (α 3 β + α β 3 )u (α 3 β + α β 3 )u (α 4 β + α β 4 )u (α 4 β k + α k β 4 )u (α 4 β 3 + α 3 β 4 )u ++1 )]+q(u), where q(u) is a quadratic function. The third derivative D γ (D β D α φ λ ) of φ λ along γ F n (α = γ, β = γ)is D γ (D β D α φ λ (u)) = φ λ (u + β + α + γ)+ φ λ (u + β + α) + φ λ (u + α + γ)+ φ λ (u + α) +φ λ (u + β + γ)+ φ λ (u + β) + φ λ (u + γ)+ φ λ (u) = Tr n 1 [λ((αβ γ + βα γ + αβ γ + βα γ + α β γ + α β γ)u (αβ 4 γ 3 + βα γ 3 + αβ 3 γ 4 + βα 3 γ + α 3 β γ + α β 3 γ)u 4 + +(αβ 3 γ + α 3 βγ + αβ γ 3 + α βγ 3 + α 3 β γ + α β 3 γ)u 4 + +(αβ γ 4 + α βγ 4 + αβ 4 γ + α 4 βγ + α 4 β γ + α β 4 γ)u 3 + +(αβ γ 4 + α βγ 4 + αβ 4 γ + α 4 βγ + α 4 β γ + α β 4 γ)u 3 + +(αβ 4 γ 3 + α 4 βγ 3 + αβ 3 γ 4 + α 3 βγ 4 + α 4 β 3 γ + α 3 β 4 γ)u + +(α β γ 3 + α β γ 3 + α 3 β γ + α β 3 γ + α 3 β γ +α β 3 γ )u (α β γ 4 + α β γ 4 + α 4 β γ + α β 4 γ +α 4 β γ + α β 4 γ )u (α 3 β γ 4 + α β 3 γ 4 + α 4 β γ 3 +α β 4 γ 3 + α 4 β 3 γ + α 3 β 4 γ )u +1 + (α 3 β γ 4 + α β 3 γ 4 +α 4 β γ 3 + α β 4 γ 3 + α 4 β 3 γ + α 3 β 4 γ )u +1 )]+l(u).

8 8 D. Singh and A. Paul Since D γ (D β D α φ λ (u)) is quadratic. The WHS of D γ (D β D α φ λ ) is equivalent to the WHS of h λ (u) with h λ (u) = Tr n 1 [λ((αβ γ + βα γ + αβ γ + βα γ + α β γ + α β γ)u (αβ γ 3 + βα γ 3 + αβ 3 γ + βα 3 γ + α 3 β γ + α β 3 γ)u 4 + +(αβ 3 γ + α 3 βγ + αβ γ 3 + α βγ 3 + α 3 β γ + α β 3 γ)u 4 + +(αβ γ 4 + α βγ 4 + αβ 4 γ + α 4 βγ + α 4 β γ + α β 4 γ)u 3 + +(αβ γ 4 + α βγ 4 + αβ 4 γ + α 4 βγ + α 4 β γ + α β 4 γ)u 3 + +(αβ 4 γ 3 + α 4 βγ 3 + αβ 3 γ 4 + α 3 βγ 4 + α 4 β 3 γ + α 3 β 4 γ)u + +(α β γ 3 + α β γ 3 + α 3 β γ + α β 3 γ + α 3 β γ + α β 3 γ )u (α β γ 4 + α β γ 4 + α 4 β γ + α β 4 γ + α 4 β γ + α β 4 γ )u (α 3 β γ 4 + α β 3 γ 4 + α 4 β γ 3 + α β 4 γ 3 + α 4 β 3 γ + α 3 β 4 γ )u +1 +(α 3 β γ 4 + α β 3 γ 4 + α 4 β γ 3 + α β 4 γ 3 + α 4 β 3 γ + α 3 β 4 γ )u +1 )]. Let E hλ ={u F ( n ) : B(u, y) = 0 with y F n}, where B(u, y) is the bilinear form of h λ is given by B(u, y) = h λ (0) + h λ (u) + h λ (u) + h λ (u + y) B(u, y) = Tr1 n [λ(y4 {R 1 u 3 + R u + R 3 u + R 7 u}+y 3 {R 1 u 4 + R 4 u where +R 5 u + R 8 u}+y {R u 4 + R 4 u 3 + R 6 u + R 9 u}+y {R 3 u 4 +R 5 u 3 + R 6 u + R 10 u}+y{r 7 u 4 + R 8 u 3 + R 9 u + R 10 u })] = Tr1 n (yp (u)), R 1 = αβ γ + βα γ + αβ γ + βα γ + α β γ + α β γ R = αβ γ 3 + βα γ 3 + αβ 3 γ + βα 3 γ + α 3 β γ + α β 3 γ R 3 = αβ 3 γ + α 3 βγ + αβ γ 3 + α βγ 3 + α 3 β γ + α β 3 γ R 4 = αβ γ 4 + α βγ 4 + αβ 4 γ + α 4 βγ + α 4 β γ + α β 4 γ R 5 = αβ γ 4 + α βγ 4 + αβ 4 γ + α 4 βγ + α 4 β γ + α β 4 γ R 6 = αβ 4 γ 3 + α 4 βγ 3 + αβ 3 γ 4 + α 3 βγ 4 + α 4 β 3 γ + α 3 β 4 γ R 7 = α β γ 3 + α β γ 3 + α 3 β γ + α β 3 γ + α 3 β γ + α β 3 γ R 8 = α β γ 4 + α β γ 4 + α 4 β γ + α β 4 γ + α 4 β γ + α β 4 γ R 9 = α 3 β γ 4 + α β 3 γ 4 + α 4 β γ 3 + α β 4 γ 3 + α 4 β 3 γ + α 3 β 4 γ R 10 = α 3 β γ 4 + α β 3 γ 4 + α 4 β γ 3 + α β 4 γ 3 + α 4 β 3 γ + α 3 β 4 γ and P (u) = (λr 1 u 3 + λr u + λr 3 u + λr 7 u) n 4 +(λr 1 u 4 + λr 4 u + λr 5 u + λr 8 u) n 3 +(λr u 4 + λr 4 u 3 + λr 6 u + λr 9 u) n +(λr 3 u 4 + λr 5 u 3 + λr 6 u + λr 10 u) n 1 +(λr 7 u 4 + λr 8 u 3 + λr 9 u + R 10 u ).

9 Higher order nonlinearity of some cryptographic functions 9 Let L λ (u) = (P (u)) 4 = λ(r 1 x 3 + R x + R 3 x + R 7 x) + λ [R 1 x5 + R 4 x3 + R 5 x + R 8 x ] +λ [R x6 + R 4 x5 + R 6 x3 + R 9 x ]+λ 3 [R 3 3 x7 + R 3 5 x6 + R 3 6 x5 +R 3 10 x3 ]+λ 4 [R 4 7 x8 + R 7 8 x7 + R 7 9 x6 + R 7 10 x5 ]. (3.) L (λ) (u) is a linearized polynomial in u. The degree of L (λ) (u) is at most 8, this implies that k 6. The Walsh transform of D γ (D β D α φ λ ) at λ F 10 is Therefore, the nonlinearity of D β D α φ λ is W Dγ (D β D α φ λ )(λ) = 10+k nl(d γ (D β D α φ λ )) = 9 1 max W Dγ (D β D α φ λ )(λ) λ F = 56. From Lemma.4, we conclude that the 4th order nonlinearity of φ λ is Hence, nl 4 (φ λ ) 1 3 max nl(d α (D β D α φ λ )) α,β,γ F n nl 4 (φ λ ) 3. (3.3) Also, nl(d γ (D β D α φ λ )) 56, for all α, β, γ F 10 (α = β = γ).so there is a scope to improve the bound obtained in (3.3). Lemma.5 implies that nl 4 (φ λ ) nl(d β D α φ λ ) γ F 10 β F 10 α F 10 = 9 1 ( 10 1) ( 10 ) 0 ( 10 3).56 = = 43.

10 10 D. Singh and A. Paul 4. Conclusion The comparison of the results obtained in Theorem 3.3 with the results given by Iwatakurosawa [13], Singh [18] and general bounds i.e., nl (φ) n 3 [4] is provided in Table 1. It is observed that the results given by us in Theorem 3.3 are better than those given in [4, 13, 18]. Table 1: Comparison of results in Theorem 3.3 with the results obtained in [4, 13, 18] for n = 7s n,s 14, 1,3 8,4 35,5 4,6 Bounds in Theorem Bounds by Singh [18] Bounds by Iwata-kurosawa [13] General bounds by Carlet [4] Since there is always need of functions having good cryptographic properties, in particular, functions with good higher order nonlinearities are employed to prevent higher order approximation attacks. Therefore, we expect that the results in this paper will help in selecting good cryptographic functions. Acknowledgement The second author thanks to UGC, India for providing financial support through Rajiv Gandhi National Fellowship. References [1] Biham, E., and Shamir, A., 1991, Differential cryptyanalysis of DES-like cryptosystems, In Advances in cryptography CRYPTO 1990, Lecture Notes in Computer Science, Springer-Verlag, Vol. 537, pp. 1. [] Canteaut, A., Charpin, P., and Kyureghyan, G., 008, A new class of monomial bent functions, Finite Fields and Their Applications, Vol. 14, pp [3] Carlet, C., 006 On the higher order nonlinearities of algebraic immune functions, In CRYPTO 006, Lecture Notes in Computer Science, Springer-Verlag, Vol. 4117, pp [4] Carlet, C., 008, Recursive lower bounds on the nonlinearity profile of Boolean functions and their applications, IEEE Trans. Inform. Theory, Vol. 54 (3), pp [5] Carlet, C., Dalai, D. K., Gupta, K. C., and Maitra, S., 006, Algebraic immunity for cryptographically significant Boolean functions: Analysis and Construction, IEEE Trans. Inform. Theory, Vol. 5 (7), pp

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