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Investigations on cubic rotation symmetric bent functions
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Electronic Notes in Discrete Mathematics 56 (2016) 15–19
www.elsevier.com/locate/endm
Investigations on cubic rotation symmetric bent functions Sugata Gangopadhyay Department of Computer Science and Engineering Indian Institute of Technology Roorkee, INDIA
Bhupendra Singh Secure Systems Division Centre for Artificial Intelligence and Robotics DRDO, Bangalore, INDIA
V. Vetrivel Department of Mathematics Indian Institute of Technology Madras, INDIA Abstract We identify an infinite class of cubic rotation symmetric bent functions and prove that these functions do not have affine derivatives. Some experimental results concerning rotation symmetric bent functions in 6, 8, 10 and 12 variables are also included. Keywords: Boolean functions, bent functions, rotation symmetry, affine equivalence.
1
Introduction
Let [n] = {i ∈ Z : 0 ≤ i ≤ n − 1} where Z is the set of integers, and Fn2 = {x = (x0 , x1 , . . . , xn−1 ) : xi ∈ F2 , for all i ∈ [n]} where F2 is the prime field of characteristic 2. Let Bn be the set of all Boolean functions from Fn2 to F2 . Let http://dx.doi.org/10.1016/j.endm.2016.11.003 1571-0653/© 2016 Elsevier B.V. All rights reserved.
16
S. Gangopadhyay et al. / Electronic Notes in Discrete Mathematics 56 (2016) 15–19
xi ∈ {0, 1} for i ∈ [n]. For k ∈ [n], define ρkn (xi ) =
xi+k if i + k ≤ n − 1 xi+k−n if i + k ≥ n
,
and ρkn (x1 , . . . , xn ) = (ρkn (x1 ), . . . , ρkn (xn )), Definition 1.1 A Boolean function f ∈ Bn is said to be rotation symmetric (RotS) if for any x = (x0 , . . . , xn−1 ) ∈ Fn2 , f (ρkn (x0 , . . . , xn−1 )) = f (x0 , . . . , xn−1 ) for any k ∈ [n]. The algebraic normal form of f ∈ Bn is f (x0 , . . . , xn−1 ) = a + i∈[n] ai xi + i,j∈[n],i=j aij xi xj + . . . + a01...(n−1) x0 . . . xn−1 where xi ’s are the variables and a, ai , aij , . . . ∈ F2 . The number of variables in any of the highest order product terms with nonzero coefficient is said to be the algebraic degree of f written deg(f ). The support of f is supp(f ) = {x ∈ Fn2 : f (x) = 0}. The cardinality of supp(f ) is the weight of f written as wt(f ) = |supp(f )|. The Hamming distance between two Boolean functions f, g ∈ Bn is d(f, g) = |{x ∈ Fn2 : f (x) = g(x)}|. The derivative of f ∈ Bn at a ∈ Fn2 , Da f , is defined by Da f (x) = f (x) + f (x + a) for all x ∈ Fn2 . Suppose V is a two dimensional subspace of Fn2 generated by the vectors a, b ∈ Fn2 . Then DV f (x) = Da,b f (x) = f (x) + f (x + a) + f (x + b) + f (x + a + b) is referred to as the second-derivative of f with respect to V . Given any f ∈ Bn , the multi-set S(f ) = [wt(DV f ) : all subspaces V ⊆ Fn2 , dim(V ) = 2] is referred to as the second-derivative spectrum of f . If g(x) = f (Ax + b) + u · x + ε where A ∈ GL(n, F2 ), b, u ∈ Fn2 and ε ∈ F2 , then f, g ∈ Bn are said to be affine equivalent. Gangopadhyay [3] demonstrated that if S(f ) = S(g) then f and g are affine inequivalent. We refer to S(f ) as the second-derivative spectrum of f . The nonlinearity and second-order nonlinearity of f ∈ Bn are nl(f ) = min{d(f, g) : g ∈ A(n)} and nl2 (f ) = min{d(f, g) : g ∈ Q(n)} where A(n) and Q(n) are the sets of functions of algebraic degree up to one (affine) and two, respectively.
2
Cubic rotation symmetric bents with no affine derivatives
The are exactly three affine inequivalent classes of cubic bent functions in 6 variables of which two contain RotS bents [2,3,5] which are f1 (x0 , . . . , x5 ) = x0 x1 x2 +x1 x2 x3 +x2 x3 x4 +x3 x4 x5 +x4 x5 x0 +x5 x0 x1 +x0 x2 x4 + x1 x3 x5 + x0 x3 + x1 x4 + x2 x5 f2 (x0 , . . . , x5 ) = x0 x2 x3 + x1 x3 x4 + x2 x4 x5 + x3 x5 x0 + x4 x0 x1 + x5 x1 x2 + x0 x2 + x1 x3 + x2 x4 + x3 x5 + x4 x0 + x5 x1 + x0 x3 + x1 x4 + x2 x5 .
S. Gangopadhyay et al. / Electronic Notes in Discrete Mathematics 56 (2016) 15–19
17
The function f2 has no affine derivative and have the maximum secondorder nonlinearity nl2 (f2 ) = 16 among all the cubic bents in B6 This points to the importance of cubic bent without affine derivative which have been studied by Hou [4] and Canteaut and Charpin [1]. 2.1
Cubic RotS bent functions without any affine derivative
We consider a subclass of a class of RotS bent functions constructed by Tang et al. [6] and prove that no function in it has affine derivative. Theorem 2.1 Suppose n = 2m where n, m ∈ Z+ and t ∈ [m] \ {0}, m ≡ 2t (mod n) such that m/ gcd(m, t) is odd. Then a RotS bent function of the form f (x) =
(xi xi+t xi+m + xi xi+t ) +
i∈[n]
xi xi+m
i∈[m]
has no affine derivative. Proof. Since the last two terms of f are quadratic it is enough to consider the derivative of the function g(x) = f (x) + i∈[n] xi xi+t + x∈[m] xi xi+m . Da g(x) =
(xi + ai )(xi+t + ai+t )(xi+m + ai+m ) +
i∈[n]
=
ai xi+t xi+m +
i∈[n]
xi ai+t xi+m +
i∈[n]
xi xi+t xi+m
i∈[n]
xi xi+t ai+m + L(x)
i∈[n]
where L(x) is a linear function of x. Let T1 = i∈[n] ai xi+t xi+m , T2 = i∈[n] xi ai+t xi+m , T3 = i∈[n] xi xi+t ai+m . Suppose that there exist i, j ∈ [n] such that xi+t xi+m = xj xj+m . Then either i+t≡j
(mod n), i + m ≡ j + m
(mod n)
(1)
(mod n).
(2)
or i+t≡j+m
(mod n)i + m ≡ j
Solving (1) we have t ≡ 0 (mod n). Equations (2) gives us i + t ≡ i + 2m (mod n), i.e., t ≡ 0 (mod n). Since t ∈ [m] \ {0}, this leads to a contradiction. Therefore, there is no quadratic monomial common between T1 and T2 . Suppose that there exist i, j ∈ [n] such that xi+t xi+m = xj xj+t . Then either i+t≡j
(mod n), i + m ≡ j + t
(mod n)
(3)
18
S. Gangopadhyay et al. / Electronic Notes in Discrete Mathematics 56 (2016) 15–19
or i+t≡j+t
(mod n), i + m ≡ j
(mod n).
(4)
By solving (3) and (4) we have m ≡ 2t (mod n) and m ≡ 0 (mod n), respectively. Since none of these conditions hold we conclude that there is no quadratic monomial common between T1 and T3 . If Da g(x) is affine, then i∈[n] ai xi+t xi+m + i∈[n] xi ai+t xi+m + i∈[n] xi xi+t ai+m = 0, for all n x = (x0 , . . . , xn−1 ) ∈ F2 , which implies that
ai xi+t xi+m = 0,
(5)
i∈[n]
for all x = (x0 , . . . , xn−1 ) ∈ Fn2 , since T1 shares no monomial with T2 and T3 . If there exist i, j ∈ [n], i = j such that xi+t xi+m = xj+t xj+m , then i+t≡j+m
(mod n), i + m ≡ j + t
(mod n).
(6)
Solving Equations (6) we obtain t ≡ 0 (mod m) which is a contradiction, since t ∈ [m] \ {0}. Thus, from (5) we have ai = 0, for all i ∈ [n]. This proves that, Da f is affine if and only if a = 0, that is, f has no affine derivative. 2
3
Second-derivative spectra of cubic RotS bent functions
We have classified the cubic RotS bent functions on n = 6, 8, 10, 12 variables with respect to their second-derivative spectra. From [2,3] functions belonging to distinct classes are definitely affine inequivalent. 3.1
Cubic Rots bent in n = 6 variables
For n = 6, we have 27 = 128 cubic rotation symmetric Boolean functions, out of these, 10 functions are rotation symmetric bent function having only two distinct second-derivative spectra. 3.2
Cubic Rots bent in n = 8 variables
For n = 8, we have 211 = 2048 cubic rotation symmetric Boolean functions, out of these, 56 functions are rotation symmetric bent function having only three distinct second-derivative spectra.
S. Gangopadhyay et al. / Electronic Notes in Discrete Mathematics 56 (2016) 15–19
3.3
19
Cubic Rots bent in n = 10 variables
For n = 10, we have 217 = 131072 cubic rotation symmetric Boolean functions, out of these, 1572 functions are rotation symmetric bent function having only six distinct second-derivative spectra. 3.4
Cubic Rots bent in n = 12 variables
For n = 12, we have 225 = 33554432 cubic rotation symmetric Boolean functions, out of these, 14320 functions are rotation symmetric bent function having only twelve distinct second-derivative spectra.
References [1] A. Canteaut and P. Charpin. Decomposing Bent Functions, IEEE Tran. Inform. Theory, 49(8) (2003) 2004–2019. [2] J. F. Dillon. Elementary Hadamard difference sets, in: proceedings of 6th S. E. Conference of Combinatorics, Graph Theory, and Computing, Utility Mathematics, Winnipeg, (1975), pp. 237–249. [3] S. Gangopadhyay, Affine inequivalence of cubic Maiorana–McFarland bent functions, Discrete Applied Mathematics, 161(7-8) (2013) 1141–1146. [4] X. D. Hou, Cubic bent functions, Disc. Math. 189 (1998) 149–161. [5] O. S. Rothaus, On bent functions, J. Combin. Theory, Ser. A 20 (1976), 300–305. [6] C. Tang, Y. Qi, Z. Zhou and C. Fan, Two infinite classes of rotation symmetric bent functions with simple representation, CoRR abs/1508.05674 (2015).
Electronic Notes in Discrete Mathematics 56 (2016) 15–19
www.elsevier.com/locate/endm
Investigations on cubic rotation symmetric bent functions Sugata Gangopadhyay Department of Computer Science and Engineering Indian Institute of Technology Roorkee, INDIA
Bhupendra Singh Secure Systems Division Centre for Artificial Intelligence and Robotics DRDO, Bangalore, INDIA
V. Vetrivel Department of Mathematics Indian Institute of Technology Madras, INDIA Abstract We identify an infinite class of cubic rotation symmetric bent functions and prove that these functions do not have affine derivatives. Some experimental results concerning rotation symmetric bent functions in 6, 8, 10 and 12 variables are also included. Keywords: Boolean functions, bent functions, rotation symmetry, affine equivalence.
1
Introduction
Let [n] = {i ∈ Z : 0 ≤ i ≤ n − 1} where Z is the set of integers, and Fn2 = {x = (x0 , x1 , . . . , xn−1 ) : xi ∈ F2 , for all i ∈ [n]} where F2 is the prime field of characteristic 2. Let Bn be the set of all Boolean functions from Fn2 to F2 . Let http://dx.doi.org/10.1016/j.endm.2016.11.003 1571-0653/© 2016 Elsevier B.V. All rights reserved.
16
S. Gangopadhyay et al. / Electronic Notes in Discrete Mathematics 56 (2016) 15–19
xi ∈ {0, 1} for i ∈ [n]. For k ∈ [n], define ρkn (xi ) =
xi+k if i + k ≤ n − 1 xi+k−n if i + k ≥ n
,
and ρkn (x1 , . . . , xn ) = (ρkn (x1 ), . . . , ρkn (xn )), Definition 1.1 A Boolean function f ∈ Bn is said to be rotation symmetric (RotS) if for any x = (x0 , . . . , xn−1 ) ∈ Fn2 , f (ρkn (x0 , . . . , xn−1 )) = f (x0 , . . . , xn−1 ) for any k ∈ [n]. The algebraic normal form of f ∈ Bn is f (x0 , . . . , xn−1 ) = a + i∈[n] ai xi + i,j∈[n],i=j aij xi xj + . . . + a01...(n−1) x0 . . . xn−1 where xi ’s are the variables and a, ai , aij , . . . ∈ F2 . The number of variables in any of the highest order product terms with nonzero coefficient is said to be the algebraic degree of f written deg(f ). The support of f is supp(f ) = {x ∈ Fn2 : f (x) = 0}. The cardinality of supp(f ) is the weight of f written as wt(f ) = |supp(f )|. The Hamming distance between two Boolean functions f, g ∈ Bn is d(f, g) = |{x ∈ Fn2 : f (x) = g(x)}|. The derivative of f ∈ Bn at a ∈ Fn2 , Da f , is defined by Da f (x) = f (x) + f (x + a) for all x ∈ Fn2 . Suppose V is a two dimensional subspace of Fn2 generated by the vectors a, b ∈ Fn2 . Then DV f (x) = Da,b f (x) = f (x) + f (x + a) + f (x + b) + f (x + a + b) is referred to as the second-derivative of f with respect to V . Given any f ∈ Bn , the multi-set S(f ) = [wt(DV f ) : all subspaces V ⊆ Fn2 , dim(V ) = 2] is referred to as the second-derivative spectrum of f . If g(x) = f (Ax + b) + u · x + ε where A ∈ GL(n, F2 ), b, u ∈ Fn2 and ε ∈ F2 , then f, g ∈ Bn are said to be affine equivalent. Gangopadhyay [3] demonstrated that if S(f ) = S(g) then f and g are affine inequivalent. We refer to S(f ) as the second-derivative spectrum of f . The nonlinearity and second-order nonlinearity of f ∈ Bn are nl(f ) = min{d(f, g) : g ∈ A(n)} and nl2 (f ) = min{d(f, g) : g ∈ Q(n)} where A(n) and Q(n) are the sets of functions of algebraic degree up to one (affine) and two, respectively.
2
Cubic rotation symmetric bents with no affine derivatives
The are exactly three affine inequivalent classes of cubic bent functions in 6 variables of which two contain RotS bents [2,3,5] which are f1 (x0 , . . . , x5 ) = x0 x1 x2 +x1 x2 x3 +x2 x3 x4 +x3 x4 x5 +x4 x5 x0 +x5 x0 x1 +x0 x2 x4 + x1 x3 x5 + x0 x3 + x1 x4 + x2 x5 f2 (x0 , . . . , x5 ) = x0 x2 x3 + x1 x3 x4 + x2 x4 x5 + x3 x5 x0 + x4 x0 x1 + x5 x1 x2 + x0 x2 + x1 x3 + x2 x4 + x3 x5 + x4 x0 + x5 x1 + x0 x3 + x1 x4 + x2 x5 .
S. Gangopadhyay et al. / Electronic Notes in Discrete Mathematics 56 (2016) 15–19
17
The function f2 has no affine derivative and have the maximum secondorder nonlinearity nl2 (f2 ) = 16 among all the cubic bents in B6 This points to the importance of cubic bent without affine derivative which have been studied by Hou [4] and Canteaut and Charpin [1]. 2.1
Cubic RotS bent functions without any affine derivative
We consider a subclass of a class of RotS bent functions constructed by Tang et al. [6] and prove that no function in it has affine derivative. Theorem 2.1 Suppose n = 2m where n, m ∈ Z+ and t ∈ [m] \ {0}, m ≡ 2t (mod n) such that m/ gcd(m, t) is odd. Then a RotS bent function of the form f (x) =
(xi xi+t xi+m + xi xi+t ) +
i∈[n]
xi xi+m
i∈[m]
has no affine derivative. Proof. Since the last two terms of f are quadratic it is enough to consider the derivative of the function g(x) = f (x) + i∈[n] xi xi+t + x∈[m] xi xi+m . Da g(x) =
(xi + ai )(xi+t + ai+t )(xi+m + ai+m ) +
i∈[n]
=
ai xi+t xi+m +
i∈[n]
xi ai+t xi+m +
i∈[n]
xi xi+t xi+m
i∈[n]
xi xi+t ai+m + L(x)
i∈[n]
where L(x) is a linear function of x. Let T1 = i∈[n] ai xi+t xi+m , T2 = i∈[n] xi ai+t xi+m , T3 = i∈[n] xi xi+t ai+m . Suppose that there exist i, j ∈ [n] such that xi+t xi+m = xj xj+m . Then either i+t≡j
(mod n), i + m ≡ j + m
(mod n)
(1)
(mod n).
(2)
or i+t≡j+m
(mod n)i + m ≡ j
Solving (1) we have t ≡ 0 (mod n). Equations (2) gives us i + t ≡ i + 2m (mod n), i.e., t ≡ 0 (mod n). Since t ∈ [m] \ {0}, this leads to a contradiction. Therefore, there is no quadratic monomial common between T1 and T2 . Suppose that there exist i, j ∈ [n] such that xi+t xi+m = xj xj+t . Then either i+t≡j
(mod n), i + m ≡ j + t
(mod n)
(3)
18
S. Gangopadhyay et al. / Electronic Notes in Discrete Mathematics 56 (2016) 15–19
or i+t≡j+t
(mod n), i + m ≡ j
(mod n).
(4)
By solving (3) and (4) we have m ≡ 2t (mod n) and m ≡ 0 (mod n), respectively. Since none of these conditions hold we conclude that there is no quadratic monomial common between T1 and T3 . If Da g(x) is affine, then i∈[n] ai xi+t xi+m + i∈[n] xi ai+t xi+m + i∈[n] xi xi+t ai+m = 0, for all n x = (x0 , . . . , xn−1 ) ∈ F2 , which implies that
ai xi+t xi+m = 0,
(5)
i∈[n]
for all x = (x0 , . . . , xn−1 ) ∈ Fn2 , since T1 shares no monomial with T2 and T3 . If there exist i, j ∈ [n], i = j such that xi+t xi+m = xj+t xj+m , then i+t≡j+m
(mod n), i + m ≡ j + t
(mod n).
(6)
Solving Equations (6) we obtain t ≡ 0 (mod m) which is a contradiction, since t ∈ [m] \ {0}. Thus, from (5) we have ai = 0, for all i ∈ [n]. This proves that, Da f is affine if and only if a = 0, that is, f has no affine derivative. 2
3
Second-derivative spectra of cubic RotS bent functions
We have classified the cubic RotS bent functions on n = 6, 8, 10, 12 variables with respect to their second-derivative spectra. From [2,3] functions belonging to distinct classes are definitely affine inequivalent. 3.1
Cubic Rots bent in n = 6 variables
For n = 6, we have 27 = 128 cubic rotation symmetric Boolean functions, out of these, 10 functions are rotation symmetric bent function having only two distinct second-derivative spectra. 3.2
Cubic Rots bent in n = 8 variables
For n = 8, we have 211 = 2048 cubic rotation symmetric Boolean functions, out of these, 56 functions are rotation symmetric bent function having only three distinct second-derivative spectra.
S. Gangopadhyay et al. / Electronic Notes in Discrete Mathematics 56 (2016) 15–19
3.3
19
Cubic Rots bent in n = 10 variables
For n = 10, we have 217 = 131072 cubic rotation symmetric Boolean functions, out of these, 1572 functions are rotation symmetric bent function having only six distinct second-derivative spectra. 3.4
Cubic Rots bent in n = 12 variables
For n = 12, we have 225 = 33554432 cubic rotation symmetric Boolean functions, out of these, 14320 functions are rotation symmetric bent function having only twelve distinct second-derivative spectra.
References [1] A. Canteaut and P. Charpin. Decomposing Bent Functions, IEEE Tran. Inform. Theory, 49(8) (2003) 2004–2019. [2] J. F. Dillon. Elementary Hadamard difference sets, in: proceedings of 6th S. E. Conference of Combinatorics, Graph Theory, and Computing, Utility Mathematics, Winnipeg, (1975), pp. 237–249. [3] S. Gangopadhyay, Affine inequivalence of cubic Maiorana–McFarland bent functions, Discrete Applied Mathematics, 161(7-8) (2013) 1141–1146. [4] X. D. Hou, Cubic bent functions, Disc. Math. 189 (1998) 149–161. [5] O. S. Rothaus, On bent functions, J. Combin. Theory, Ser. A 20 (1976), 300–305. [6] C. Tang, Y. Qi, Z. Zhou and C. Fan, Two infinite classes of rotation symmetric bent functions with simple representation, CoRR abs/1508.05674 (2015).