Boolean Functions and their Applications Loen, Norway, June 17–22, 2018 Journey into differential and graph theoretical properties of (generalized) Boolean function Pante St˘ anic˘ a ( includes joint work with T. Martinsen, W. Meidl, A. Pott, C. Riera, . Solé ) P Department of Applied Mathematics Naval Postgraduate School Monterey, CA 93943, USA; pstanica@nps.edu Pante Stanica Differential and graph theoretical properties
The objects of the investigation: (Generalized) Boolean functions I Boolean function f : F n 2 → F 2 Generalized Boolean function f : F n 2 → Z q ( q ≥ 2); its set GB q n ; when q = 2, B n ; (Generalized) Walsh-Hadamard transform : 2 π i H ( q ) ζ f ( x ) � q ; (use W f , if q = 2) ( − 1 ) u · x , ζ q = e ( u ) = q f x ∈ F n 2 � f ( x )( − 1 ) u · x Fourier transform: F f ( u ) = x ∈ F n 2 Let 2 k − 1 < q ≤ 2 k . Then GB q n ∋ f ← → { a i } 0 ≤ i ≤ k − 1 ⊂ B n : f ( x ) = a 0 ( x ) + 2 a 1 ( x ) + · · · + 2 k − 1 a k − 1 ( x ) , ∀ x ∈ F n 2 . Pante Stanica Differential and graph theoretical properties
Characterizing generalized bent f : F n 2 → Z 2 k f : GB q n is generalized bent ( gbent ) if |H f ( u ) | = 2 n / 2 , ∀ u . Theorem (Various Authors 2015–’17) Let f ( x ) = a 0 ( x ) + 2 a 1 ( x ) + · · · + 2 k − 2 a k − 2 ( x ) + 2 k − 1 a k − 1 ( x ) be a function in GB 2 k n , k > 1 , a i ∈ B n , 0 ≤ i ≤ k − 1 , and ˜ f ∈ a k − 1 ⊕ � a 0 , a 1 , . . . a k − 2 � . Then f is gbent iff ˜ f is bent (n even), respectively, semibent (n odd), with an (explicit) extra condition on the Walsh-Hadamard coeff. Pante Stanica Differential and graph theoretical properties
Differential properties of generalized Boolean functions I 2 is a linear structure of f ∈ GB q u ∈ F n n if the derivative D u f ( x ) := f ( x ⊕ u ) − f ( x ) = c ∈ Z q constant, for all x ∈ F n 2 . Let S f = { x ∈ F n 2 | H f ( x ) � = 0 } � = ∅ (gen.WH support) Theorem (Martinsen–Meidl–Pott–S., 2018) Let f ∈ GB 2 k n , with f ( x ) = � k − 1 i = 0 2 i a i ( x ) , a i ∈ B n . The following are equivalent: ( i ) a is a linear structure for f. ( ii ) a is a linear structure for a i , s.t. a i ( a ) = a i ( 0 ) , 0 ≤ i < k − 1 . ( iii ) a satisfies ζ f ( a ) − f ( 0 ) = ( − 1 ) a · w , for all w ∈ S f . Pante Stanica Differential and graph theoretical properties
Differential properties of generalized Boolean functions II We say that f ∈ GB 2 k n satisfies the ( generalized ) propagation criterion of order ℓ (1 ≤ ℓ ≤ n ), gPC ( ℓ ) , iff the x ∈ V n ζ f ( x ) − f ( x ⊕ v ) = 0, for all autocorrelation C f ( v ) = � vectors v ∈ F n 2 of weight 0 < wt ( v ) ≤ ℓ . f is gbent ⇐ ⇒ gPC ( n ) . Theorem (Martinsen–Meidl–Pott–S., 2018) Let f ∈ GB 2 k n , and A ( w ) = ( D w f ) − 1 ( j ) = { x | f ( x ⊕ w ) − f ( x ) = j } . j Then f is gPC ( ℓ ) if and only if, for 1 ≤ wt ( w ) ≤ ℓ , j + 2 k − 1 | , ∀ 0 ≤ j ≤ 2 k − 1 − 1 . | A ( 0 ) 0 | = 2 n , | A ( 0 ) | = 0 , | A ( w ) | = | A ( w ) j j Pante Stanica Differential and graph theoretical properties
Can one "visualize” some cryptographic properties of a Boolean function? Cayley graph of f : F n 2 → F 2 , G f = ( F n 2 , E f ) , E f = { ( w , u ) ∈ F n 2 × F n 2 : f ( w ⊕ u ) = 1 } . Adjacency matrix A f = { a i , j } , a i , j := f ( i ⊕ j ) (where i is the binary representation as an n -bit vector of the index i ); Spectrum of G f is the set of eigenvalues of A f ( G f ). Cayley graph G f has eigenvalues λ i = W f ( i ) , ∀ i . Pante Stanica Differential and graph theoretical properties
Cayley graph example: f ( x 1 , x 2 , x 3 ) = x 1 x 2 ⊕ x 1 x 3 ⊕ x 3 Pante Stanica Differential and graph theoretical properties
Strongly regular graphs A graph is regular of degree r (or r -regular) if every vertex has degree r ; The Cayley graph of a Boolean function is always a regular graph of degree wt ( f ) . We say that an r -regular graph G with v vertices is a strongly regular graph (SRG) with parameters ( v , r , e , d ) if ∃ integers e , d ≥ 0 s.t. for all vertices u , v : the number of vertices adjacent to both u , v is e if u , v are adjacent, the number of vertices adjacent to both u , v is d if u , v are nonadjacent. We assume throughout that G f is connected (in fact, one can show that all connected components of G f are isomorphic). Pante Stanica Differential and graph theoretical properties
Bernasconi-Codenotti correspondence Shrikhande & Bhagwandas ’65: A connected r -regular graph is strongly regular iff ∃ exactly three distinct eigenvalues λ 0 = r , λ 1 , λ 2 (also, e = r + λ 1 λ 2 + λ 1 + λ 2 , d = r + λ 1 λ 2 ). The parameters satisfy r ( r − e − 1 ) = d ( v − r − 1 ) . The adjacency matrix A satisfies ( J is the all 1 matrix) A 2 = ( d − e ) A + ( r − e ) I + eJ . Bernasconi-Codenotti correspondence: Bent functions exactly correspond to strongly regular graphs with e = d . Pante Stanica Differential and graph theoretical properties
P .J. Cameron: “ Strongly regular graphs lie on the cusp between highly structured and unstructured. For example, there is a unique srg with parameters ( 36 , 10 , 4 , 2 ) , but there are 32548 non-isomorphic srg with parameters ( 36 , 15 , 6 , 6 ) . In light of this, it will be difficult to develop a theory of random strongly regular graphs! ” Pante Stanica Differential and graph theoretical properties
Plateaued functions and their Cayley graphs f ∈ GB 2 k n is called s-plateaued if |H f ( u ) | ∈ { 0 , 2 ( n + s ) / 2 } for all u ∈ F n 2 . For k = 1: s = 0 ( n even), f is bent; if s = 1 ( n odd), or s = 2 ( n even), we call f semibent. Advantages: they can be balanced and highly nonlinear with no linear structures. In general, the spectrum of the Cayley graph of an s -plateaued f : F n 2 → F 2 will be 4-valued (so, not srg!): if n + s 2 } , then the the WH transform of f takes values in { 0 , ± 2 n + s 2 − 1 } ; Fourier transform of f takes values in { wt ( f ) , 0 , ± 2 Pante Stanica Differential and graph theoretical properties
Cayley graphs of plateaued Boolean functions: example Cayley graph of the semibent f ( x ) = x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 4 x 5 ⊕ x 2 x 3 x 5 ⊕ x 3 x 4 x 5 30 5 2 25 20 23 11 16 3 28 8 19 12 31 15 24 13 22 6 17 10 29 1 26 21 18 14 9 4 27 32 7 Pante Stanica Differential and graph theoretical properties
Cayley graphs of plateaued Boolean functions with wt ( f ) = 2 ( n + s − 2 ) / 2 There is one case when we do obtain an srg: Theorem (Riera–Solé–S. 2018) If f : F n 2 → F 2 is s-plateaued and wt ( f ) = 2 ( n + s − 2 ) / 2 , then G f (if connected) is the complete bipartite graph between supp ( f ) and supp ( f ) (if disconnected, it is a union of complete bipartite graphs). Moreover, G f is strongly regular with 0 , 2 ( n + s − 2 ) / 2 � � ( e , d ) = . Pante Stanica Differential and graph theoretical properties
Strongly walk-regular graphs van Dam and Omidi: G is strongly ℓ -walk-regular of parameters ( σ ℓ , µ ℓ , ν ℓ ) if there are σ ℓ , µ ℓ , ν ℓ walks of length ℓ between every two adjacent, every two non-adjacent, and every two identical vertices, respectively. Every strongly regular graph of parameters ( v , r , e , d ) is a strongly 2-walk-regular graph with parameters ( e , d , r ) . Pante Stanica Differential and graph theoretical properties
Cayley graphs of plateaued Boolean functions with wt ( f ) � = 2 ( n + s − 2 ) / 2 Theorem (Riera–Solé–S. 2018) Let f : F n 2 → F 2 be a Boolean function, and assume that G f is connected and r := wt ( f ) � = 2 ( n + s − 2 ) / 2 . Then, f is s-plateaued (with 4 -valued spectra) if and only if G f is strongly 3 -walk-regular of parameters ( σ, µ = ν ) = ( 2 − n r 3 + 2 n + s − 2 − 2 s − 2 r , 2 − n r 3 − 2 s − 2 r ) . 1 In fact, we showed that it is ℓ -walk regular for all odd ℓ , and found the parameters explicitly. Go2OpenQues Pante Stanica Differential and graph theoretical properties
Generalized Boolean and their Cayley graphs I For f ∈ GB q n , (gen.) Cayley graph G f : V n vertices; ( u , v ) edge of (multiplicative) weight ζ f ( u ⊕ v ) (additively f ( u ⊕ v ) ). 1 1 1 1 10 3 1 - ⅈ 1 ⅈ 1 ⅈ 4 1 1 - ⅈ - 1 16 - ⅈ 1 - ⅈ - 1 - ⅈ ⅈ 1 ⅈ - 1 - ⅈ 1 ⅈ 1 1 1 - ⅈ 1 1 - ⅈ - ⅈ - 1 ⅈ ⅈ 1 1 1 - ⅈ - 1 ⅈ 1 - 1 ⅈ 13 6 ⅈ 7 15 ⅈ - 1 - 1 1 1 1 ⅈ - ⅈ 1 - ⅈ 1 ⅈ - ⅈ ⅈ ⅈ 1 1 1 - ⅈ ⅈ - ⅈ ⅈ 1 - ⅈ 1 - ⅈ ⅈ ⅈ 1 - ⅈ 1 ⅈ 1 1 - 1 ⅈ - ⅈ 1 - 1 - ⅈ ⅈ ⅈ ⅈ 1 - ⅈ - 1 1 - 1 1 ⅈ 5 2 14 1 - ⅈ - 1 - ⅈ 11 1 - ⅈ 1 1 - ⅈ ⅈ - ⅈ - ⅈ 1 - 1 - ⅈ 1 - ⅈ ⅈ - ⅈ ⅈ 1 ⅈ 1 ⅈ 1 1 1 - 1 1 1 ⅈ - ⅈ 1 1 9 12 - 1 8 ⅈ - ⅈ 1 1 Pante Stanica Differential and graph theoretical properties 1
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