Bent functions, difference sets and strongly regular graphs - PowerPoint PPT Presentation

Bent functions, difference sets and strongly regular graphs Wilfried Meidl Sabancı University December 1, 2013

◮ Bent Functions, Definition, Properties ◮ Bent Functions and ◮ Difference Sets ◮ Strongly Regular Graphs ◮ A Construction of Bent Functions ◮ Interpretation with Difference Sets ◮ Graph Interpretation

Walsh (Fourier) Transform Definition p : a prime f : V n − → F p For each b ∈ V n , � ǫ f ( x ) − < b , x > � , ǫ p = e 2 π i / p . f ( b ) = p x ∈ V n Remark For V n = F n p , < b , x > = b · x , for V n = F p n , < b , x > = Tr n ( bx ).

Walsh (Fourier) Transform Definition p : a prime f : V n − → F p For each b ∈ V n , � ǫ f ( x ) − < b , x > � , ǫ p = e 2 π i / p . f ( b ) = p x ∈ V n Remark For V n = F n p , < b , x > = b · x , for V n = F p n , < b , x > = Tr n ( bx ). Definition f ( b ) | = p n / 2 for all b ∈ V n ⇒ f is a bent function. | �

Walsh (Fourier) Transform Definition p : a prime f : V n − → F p For each b ∈ V n , � ǫ f ( x ) − < b , x > � , ǫ p = e 2 π i / p . f ( b ) = p x ∈ V n Remark For V n = F n p , < b , x > = b · x , for V n = F p n , < b , x > = Tr n ( bx ). Definition f ( b ) | = p n / 2 for all b ∈ V n ⇒ f is a bent function. Alternatively, | � f : V n − → F p is bent if and only if the derivative of f in direction a D a f ( x ) = f ( x + a ) − f ( x ) is balanced for all a ∈ V n , a � = 0.

Walsh coefficients � f ( b ) ⋄ For Boolean bent functions � f ( b ) = ± 2 n / 2 . ⋄ (Kumar-Scholz-Welch 1985) For p -ary bent functions, � ± p n / 2 ǫ f ∗ ( b ) : n even or n odd and p ≡ 1 mod 4 � p f ( b ) = ± ip n / 2 ǫ f ∗ ( b ) n odd and p ≡ 3 mod 4 , : p for a function f ∗ : V n → F p , the so called dual function of f .

Regularity of Bent Functions Let f : V n → F p be a bent function. Then � f ( b ) = ζ p n / 2 ǫ f ∗ ( b ) , for all b ∈ V n . p ζ can only be ± 1 or ± i . ⋄ f is called regular if for all b ∈ V n , ζ = 1. ⋄ f is called weakly regular if, for all b ∈ V n , ζ is fixed. ⋄ If ζ changes with b then f is called not weakly regular.

Plateaued Functions, Partially Bent Functions Definition n + s f : V n → F p is called s-plateaued if, for all b ∈ V n , | � f ( b ) | = p 2 or 0.

Plateaued Functions, Partially Bent Functions Definition n + s f : V n → F p is called s-plateaued if, for all b ∈ V n , | � f ( b ) | = p 2 or 0. f : V n → F p is called partially bent if, for all a ∈ V n , D a f ( x ) is balanced or constant.

Plateaued Functions, Partially Bent Functions Definition n + s f : V n → F p is called s-plateaued if, for all b ∈ V n , | � f ( b ) | = p 2 or 0. f : V n → F p is called partially bent if, for all a ∈ V n , D a f ( x ) is balanced or constant. Fact: The set of elements a ∈ V n for which D a f ( x ) is constant is a subspace of V n , the linear space Λ of f . Partially bent functions are s -plateaued, s is the dimension of Λ. We call f then s-partially bent.

Boolean Bent Functions and Difference Sets Recall: Let G be a finite (abelian) group of order ν . A subset D of G of cardinality k is called a ( ν, k , λ )-difference set in G if every element g ∈ G , different from the identity, can be written as d 1 − d 2 , d 1 , d 2 ∈ D , in exactly λ different ways. Hadamard difference set in elementary abelian 2-group: ( ν, k , λ ) = (2 n , 2 n − 1 ± 2 n 2 − 1 , 2 n − 2 ± 2 n 2 − 1 ). Theorem A Boolean function f : F n 2 → F 2 is a bent function if and only if D = { x ∈ F n 2 | f ( x ) = 1 } is a Hadamard difference set in F n 2 .

Bent Functions and Relative Difference Sets Let G be a group of order mn and let N be a subgroup of order n . A k -subset R of G is called an ( m , n , k , λ )-relative difference set in G relative to N if every element g ∈ G \ N can be represented in exactly λ ways in the form r 1 − r 2 , r 1 , r 2 ∈ R , and no non-identity element in N has such a representation. Theorem For a function f : F n p → F p let R = { ( x , f ( x )) | x ∈ F n p } ⊂ F n p × F p . The set R is a ( p n , p , p n , p n − 1 )-relative difference set in F n p × F p (relative to F p ) if and only if f is a bent function.

Bent functions and strongly regular graphs For a function f : F n p → F p , p odd, let { x ∈ F n D 0 = p | f ( x ) = 0 } , { x ∈ F n p | f ( x ) is a nonzero square in F p } , D S = { x ∈ F n D N = p | f ( x ) is a nonsquare F p } .

Bent functions and strongly regular graphs For a function f : F n p → F p , p odd, let { x ∈ F n D 0 = p | f ( x ) = 0 } , { x ∈ F n p | f ( x ) is a nonzero square in F p } , D S = { x ∈ F n D N = p | f ( x ) is a nonsquare F p } . Theorem (Yin Tan et al. 2010/2011) For an odd prime p let f : F n p → F p be a weakly regular bent function in even dimension n , with f (0) = 0, for which there exists a constant k with gcd( k − 1 , p − 1) = 1 such that for all t ∈ F p f ( tx ) = t k f ( x ) . Then the Cayley graphs of the sets D 0 \ { 0 } , D S , D N are strongly regular graphs.

Bent functions and strongly regular graphs For a function f : F n p → F p , p odd, let { x ∈ F n D 0 = p | f ( x ) = 0 } , { x ∈ F n p | f ( x ) is a nonzero square in F p } , D S = { x ∈ F n D N = p | f ( x ) is a nonsquare F p } . Theorem (Yin Tan et al. 2010/2011) For an odd prime p let f : F n p → F p be a weakly regular bent function in even dimension n , with f (0) = 0, for which there exists a constant k with gcd( k − 1 , p − 1) = 1 such that for all t ∈ F p f ( tx ) = t k f ( x ) . Then the Cayley graphs of the sets D 0 \ { 0 } , D S , D N are strongly regular graphs. Vertices: Elements of F n p . The vertices x , y are adjacent if f ( x − y ) ∈ D 0 \ { 0 } ( f ( x − y ) ∈ D S , f ( x − y ) ∈ D N ).

A construction of bent functions Theorem (C ¸e¸ smelio˘ glu, McGuire, M. 2012) For each y = ( y 1 , y 2 , . . . , y s ) ∈ F s p , let f y ( x ) : F m p → F p be an s-plateaued function. If supp ( � f y ) ∩ supp ( � f ¯ y ) = ∅ for y ∈ F s y, then the function F ( x , y 1 , y 2 , . . . , y s ) from F m + s y , ¯ p , y � = ¯ p to F p defined by F ( x , y 1 , y 2 , . . . , y s ) = f y 1 , y 2 ,..., y s ( x ) is bent.

A construction of bent functions Theorem (C ¸e¸ smelio˘ glu, McGuire, M. 2012) For each y = ( y 1 , y 2 , . . . , y s ) ∈ F s p , let f y ( x ) : F m p → F p be an s-plateaued function. If supp ( � f y ) ∩ supp ( � f ¯ y ) = ∅ for y ∈ F s y, then the function F ( x , y 1 , y 2 , . . . , y s ) from F m + s y , ¯ p , y � = ¯ p to F p defined by F ( x , y 1 , y 2 , . . . , y s ) = f y 1 , y 2 ,..., y s ( x ) is bent. For p = 2, s = 1 (Leander, McGuire 2009; Charpin et. al. 2005) F ( x , y ) = yf 1 ( x ) + ( y + 1) f 0 ( x ) , i.e. � f 0 ( x ) : y = 0 , F ( x , y ) = f 1 ( x ) : y = 1 .

Proof For a ∈ F m p , b ∈ F s p , and putting y = ( y 1 , . . . , y s ), the Walsh transform � F of F at ( a , b ) is � � � ǫ F ( x , y ) − a · x − b · y ǫ F ( x , y ) − a · x � ǫ − b · y F ( a , b ) = = p p p x ∈ F m p , y ∈ F s y ∈ F s x ∈ F m p p p � � � ǫ f y ( x ) − a · x � ǫ − b · y ǫ − b · y = = f y ( a ) . p p p y ∈ F s x ∈ F m y ∈ F s p p p p belongs to the support of exactly one � As each a ∈ F m f y , y ∈ F s p , � � � � m + s �� � = | ǫ − b · y � 2 . for this y we have F ( a , b ) f y ( a ) | = p � p

Special case Let f : F n p → F p be a bent function. Then f seen as a function from F n p × F s p to F p , is s -partially bent with linear space F s p .

Special case Let f : F n p → F p be a bent function. Then f seen as a function from F n p × F s p to F p , is s -partially bent with linear space F s p . If { f y : y ∈ F s p } is a set of bent functions from F n p to F p then the set of functions in m = n + s variables p } is a set of p s s -partially { f y ( x ) + x n +1 y 1 + · · · + x n + s y s : y ∈ F s bent functions with Walsh transforms with pairwise disjoint supports.

Special case Let f : F n p → F p be a bent function. Then f seen as a function from F n p × F s p to F p , is s -partially bent with linear space F s p . If { f y : y ∈ F s p } is a set of bent functions from F n p to F p then the set of functions in m = n + s variables p } is a set of p s s -partially { f y ( x ) + x n +1 y 1 + · · · + x n + s y s : y ∈ F s bent functions with Walsh transforms with pairwise disjoint supports. With x = ( x 1 , . . . , x n ), ¯ x = ( x n +1 , . . . , x n + s ), the function F ( x , ¯ x , y ) = f y ( x ) + x n +1 y 1 + · · · + x n + s y s := g ( y 1 ,..., y s ) ( x , ¯ x ) is an example for the construction of a bent function.

Applications ◮ Construction of infinite classes of not weakly regular bent functions (C ¸e¸ smelio˘ glu, McGuire, M., JCTA. 2012)

Applications ◮ Construction of infinite classes of not weakly regular bent functions (C ¸e¸ smelio˘ glu, McGuire, M., JCTA. 2012) ◮ Bent functions (ternary) of maximal algebraic degree (C ¸e¸ smelio˘ glu, M., IEEE Trans. Inform. Theory 2012, DCC 2013)