New classes of generalized bent functions Bimal Mandal Department - PowerPoint PPT Presentation

New classes of generalized bent functions Bimal Mandal Department of Mathematics Indian Institute of Technology Roorkee Roorkee, India This is a joint work with Pantelimon St˘ anic˘ a and Sugata Gangopadhyay Boolean Functions and Their Application (BFA) Norway, July 3-8, 2017 July 7, 2017

Outlines ❑ Introduction ❑ Subspace sum of a generalized Boolean function and their properties 0 and C p classes of bent functions ❑ Construction of D p , D p ❑ Existence and nonexistence of C p class of bent functions ❑ References 2 / 28

Generalized Boolean functions ❑ F p = { 0 , 1 , . . . , p − 1 } is a field of characteristic p . ❑ F p n is an extension field of degree n over F p . ❑ F n p = { x = ( x 1 , . . . , x n ) : x i ∈ F p } is a vector space over F p . ❑ It can be checked that x = ( x 1 , x 2 , . . . , x n ) �− → x 1 a 1 + x 2 a 2 + · · · + x n a n is an F p -vector space isomorphism from F n p . ❑ Any function f : F n p − → F p is said to be a generalized Boolean function . ❑ B p n = the set of all generalized Boolean functions on n variables. ❑ ❚r n 1 : F p n − → F p is defined by 1 ( x ) = x + x p + x p 2 + . . . + x p n − 1 . ❚r n 3 / 28

Continue ❑ Any f ∈ B p n can be uniquely expressed as � n � � � x a i f ( x 1 , x 2 , . . . , x n ) = µ a . i a =( a 1 ,..., a n ) ∈ F n i = 1 p ❑ The algebraic degree of f ∈ B p n is defined as � n � � deg( f ) = max a i : µ a � = 0 . a ∈ F n p i = 1 4 / 28

Continue ❑ The generalized Walsh–Hadamard transform of f ∈ B p n at a ∈ F n p is defined as � ζ f ( x ) − a · x . H f ( a ) = x ∈ F n p ❑ f ∈ B p n is called generalized bent function if n 2 for all a ∈ F n |H f ( a ) | = p p . ❑ f ∈ B p n is a generalized bent function if for all 0 � = a ∈ F n p � ζ f ( x + a ) − f ( x ) = 0 . x ∈ F n p ❑ The derivative of f ∈ B p n with respect to a ∈ F n p is defined as D a f ( x ) = f ( x + a ) − f ( x ) for all x ∈ F n p . 5 / 28

Continue ❑ A class of bent functions is complete if it is globally invariant under the action of the general affine group and under the addition of affine functions. ❑ Ω f = ( f ( x 0 ) , f ( x 1 ) , . . . , f ( x p n − 1 )) , f ∈ B p n . ❑ R p ( r , n ) = the set of all codewords Ω f , where f ∈ B p n with deg( f ) ≤ r and 0 ≤ r ≤ n ( p − 1 ) . ❑ Let A be a group algebra of F n p over the field F p . An element x ∈ A can be expressed as � x g X g , where x g ∈ F p . x = g ∈ F n p 6 / 28

Continue ❑ ψ : A − → F p is defined by � � x g X g �− x = → x g for all x ∈ A . g ∈ F n g F n p p ❑ P = { x ∈ A : ψ ( x ) = 0 } = { x ∈ A : � p x g = 0 } is the maximal g ∈ F n ideal of A . ❑ A = P 0 ⊃ P ⊃ P 2 ⊃ . . . ⊃ P n ( p − 1 ) = F p . ❑ P i P j = P i + j and P n ( p − 1 )+ 1 = { 0 } . n can be identified with the codeword Ω f = � ❑ f ∈ B p p f ( g ) X g . g ∈ F n 7 / 28

Preliminary ❑ For any elements a , b ∈ F n p , we have [Carlet, Eurocrypt’93] � � ζ f ( x ) − b · x = p dim E − n ˜ 2 ζ a · b f ( x ) − a · x , ζ x ∈− a + E x ∈ b + E ⊥ 2 πı p is the p th complex root of unity. where ζ = e ❑ The function f : Z n q × Z n q → Z q of the form [Carlet, Eurocrypt’93] f ( x , y ) = x · π ( y ) + q 2 φ E ( x , y ) is bent, where x · π ( y ) = 0 for all ( x , y ) ∈ E . 8 / 28

Subspace sum of a function k � ❑ V = � a 1 , · · · , a k � = { a ∈ F n p : a = c i a i , c i ∈ F p , 1 ≤ i ≤ k } . i = 1 ❑ Subspace sum of f ∈ B p n with respect to V is defined as � f ( x + u ) for all x ∈ F n S V f ( x ) = p . u ∈ V Example Let f ∈ B 3 n and V = � a � . Then S V f ( x ) = f ( x + 2 a ) + f ( x + a ) + f ( x ) . Remark Let i ∈ { 0 , 1 , . . . , p − 1 } and V = � a � , 0 � = a ∈ F n p . Then S V f ( x ) = S V f ( x + ia ) . (1) 9 / 28

The k -th derivative Lemma Let k ≤ p be a positive integer and f ∈ B p n . Then for any a ∈ F n p k � k � � ( − 1 ) i f ( x + ( k − i ) a ) for all x ∈ F n D a D a . . . D a f ( x ) = (2) p . i � �� � i = 0 k − times 10 / 28

The k -th derivative and the subspace sum I Theorem Let V = � a � and f ∈ B p f ( x ) for all x ∈ F n n . Then S V f ( x ) = D a D a . . . D a p . � �� � ( p − 1 ) − times Furthermore, for any r ∈ { 0 , 1 , 2 , . . . , p − 1 } f ( x ) for all x ∈ F n r S V f ( x ) = D ra D a . . . D a p . � �� � ( p − 2 ) − times Example Let f ∈ B p n and V = � a � . Then S V f ( x ) = f ( x + 2 a ) + f ( x + a ) + f ( x ) = D a D a f ( x ) 2 S V f ( x ) = 2 D a D a f ( x ) = D 2 a D a f ( x ) = D a D 2 a f ( x ) . 11 / 28

More on k -th derivative and the subspace sum Theorem Let V be a k-dimensional subspace of F n p generated by a 1 , a 2 , . . . , a k and f ∈ B p n . Then S V f ( x ) = D a 1 . . . D a 1 . . . D a k . . . D a k f ( x ) . � �� � � �� � ( p − 1 ) − times ( p − 1 ) − times 12 / 28

Codes and the subspace sum Proposition Let V be a k-dimensional subspace of F n p generated by a 1 , a 2 , . . . , a k . Let f ∈ B p n be any function of degree r and h ( x ) = S V f ( x ) , x ∈ F n p . �� v ∈ V X v � Then Ω f is the associated codeword of S V f, that is, �� � X v Ω h = Ω f . v ∈ V Proposition p and f ∈ B p Let V be a k-dimensional subspace of F n n of degree r. Then the degree of S V f is less than or equal to r − k ( p − 1 ) . In particular, the subspace sum of f with respect to any 1 -dimensional subspace of F n p has degree at most r − p + 1 . 13 / 28

Affine equivalence of subspace sums Theorem Let f ∈ B p n and S k [ f ] denote the multiset of subspace sum of f with p . If f , h ∈ B p respect to each k-dimensional subspace of F n n are affine equivalent, then so are S k [ f ] and S k [ h ] . Precisely, if a nonsingular affine transformation A (operating on F n p ) map f onto h, then it also maps S k [ f ] onto S k [ h ] . Corollary If P is any affine invariant for B p n , then f − → P {S k [ f ] } is also an affine invariant for B p n . 14 / 28

Maiorana-McFarland and subspace sums Theorem Let m = 2 n and f be a generalized Maiorana–McFarland bent function defined as f ( x , y ) = x · π ( y ) + g ( y ) . Then there exists an n-dimensional subspace E of F n p × F n p such that 1. the subspace sum of f with respect to any one dimensional subspaces of E is 0 if p is odd. 2. the subspace sum of f with respect to any two dimensional subspaces of E is 0 if p = 2 . 15 / 28

Some examples Fact (Helleseth et al., Fact 1 ) Any ternary function f from F 3 6 to F 3 , defined by f ( x ) = Tr 6 1 ( α 7 x 98 ) , (3) where α is a primitive element of F 3 6 , is bent and not weakly regular bent. Theorem The function f defined as in Equation (3) does not belong to the complete M p class. Proof. Let V = � a � , where a ∈ F ∗ 3 6 . If S V f ( x ) = 0 for all x ∈ F 3 6 , then 2 α 7 a 3 4 + 3 2 = 0, which is a contradiction. 16 / 28

0 and C p I Construction of D p , D p Theorem Let E = E 1 × E 2 , E 1 , E 2 ⊆ F n p with dim E 1 + dim E 2 = n and ǫ ∈ F p . The generalized Boolean function f on F n p × F n p of the form f ( x , y ) = x · π ( y ) + ǫφ E ( x , y ) (4) is a regular bent, where π is a permutation polynomial over F n p such that π ( E 2 ) = E ⊥ 1 . Remark The set of all the functions f defined as in Equation (4) is denoted by D p and the dual of f is ˜ f ( x , y ) = y · π − 1 ( x ) + ǫφ E ⊥ ( x , y ) . 17 / 28

0 and C p II Construction of D p , D p Lemma Let n = 2 t and p be an odd prime. Then for all x = ( x 1 , x 2 , · · · , x n ) , y = ( y 1 , y 2 , · · · , y n ) ∈ F n p , p − 1 n � � φ E 0 ( x , y ) = ( x i − j ) , i = 1 j = 1 where E 0 = { 0 } × F n p . 18 / 28

0 and C p III Construction of D p , D p Proof. If x = 0, then p − 1 n n � � � ( p − 1 )! = 1 = (( p − 1 )!) n = (( p − 1 )!) 2 t . ( 0 − j ) = i = 1 j = 1 i = 1 We know that ( p − 1 )! ≡ − 1 (mod p ) . ( Wilson’s Theorem) The generalized Boolean function f on F n p × F n p of the form p − 1 n � � f ( x , y ) = x · π ( y ) + ǫφ E 0 ( x , y ) = x · π ( y ) + ǫ ( x i − j ) (5) i = 1 j = 1 is a regular bent, where E 0 = { 0 } × F n p . 19 / 28

0 and C p IV Construction of D p , D p Remark The set of all the functions f defined as in Equation (5) is denoted by D p 0 ( D p 0 ⊂ D p ). If f ∈ D p 0 is an m variables, then m ≡ 0 (mod 4 ) . Theorem 0 and D p are not included in the class M p . Further, the In general, D p class M p is in general not included in D p 0 and D p classes. 20 / 28

0 and C p V Construction of D p , D p Proof. ( D p �→ M p ) x · ( π ( y ) − π 1 ( y )) = ǫ ( φ E ( x , y ) − φ E ( 0 , y )) . f ( x , y ) = x · ψ ( y ) + g ( y ) = x · ψ 1 ( y ) + ǫφ E ( x , y ) ( M p �→ D p ) g ( y ) = ǫφ E ( 0 , y ) ∈ { 0 , ǫ } for all y ∈ F n and p . 21 / 28

0 and C p VI Construction of D p , D p Theorem Let L be any linear subspace of F n p and π be any permutation on F n p such that for any element λ of F n p , the set π − 1 ( λ + L ) is a flat. Then the function f on F n p × F n p : f ( x , y ) = x · π ( y ) + ǫφ L ⊥ ( x ) , (6) where ǫ ∈ F p , is a generalized bent Boolean function. Remark ❑ The class of bent functions defined as in Equation (6) will be denoted by C p . ❑ In general C p is not included in the M p class. 22 / 28