+ + Computing M ¨ o bius Transforms of Boolean Functions and Characterising Coincident Boolean Functions Josef Pieprzyk and Xian-Mo Zhang Department of Computing Macquarie University, Australia + 1
+ + Outline • The M¨ o bius Transform of a Boolean Func- tion f relates the truth table to its alge- braic normal form (ANF). • We compute the M¨ o bius Transforms of Boolean Functions in different methods, • We notice a special case when f is iden- tical with its M¨ o bius Transform. We call such a function coincident. • We characterise coincident Boolean Func- tions in different ways. + 2
+ + Brief Introduction to Boolean Functions • The vector space of n -tuples of elements from GF (2) is denoted by ( GF (2)) n . • A Boolean function f is a mapping from ( GF (2)) n to GF (2). We write f as f ( x ) or f ( x 1 , . . . , x n ) where x = ( x 1 , . . . , x n ). • We list all vectors in ( GF (2)) n as (0 , . . . , 0 , 0) = α 0 , (0 , . . . , 0 , 1) = α 1 , . . . , (1 , . . . , 1 , 1) = α 2 n − 1 and call α i the binary representation of integer i . • The truth table of a function f on ( GF (2)) n is a (0 , 1)-sequence defined by ( f ( α 0 ) , f ( α 1 ) , . . . , f ( α 2 n − 1 )), + 3
+ + Brief Introduction to Boolean Functions (Cont’d) • The Hamming weight of HW ( ξ ) is the number of nonzero coordinates of ξ . • In particular, if ξ represents the truth table of a function f , then HW ( ξ ) is called the Hamming weight of f , denoted by HW ( f ). + 4
+ + M ¨ o bius Transforms of Boolean Functions • The function f on ( GF (2)) n can be uniquely represented as f ( x 1 , . . . , x n ) = (1) α ∈ ( GF (2)) n g ( a 1 , . . . , a n ) x a 1 1 · · · x a n = � n where α = ( a 1 , . . . , a n ) and g is also a func- tion on ( GF (2)) n . • (1) is called the algebraic normal form (ANF) of f . • g is called the M¨ o bius transform of f , de- noted by g = µ ( f ). + 5
+ + Computing µ ( f ) by Matrix • Define 2 n × 2 n (0, 1)-matrix T n , such that the i th row of T n is the truth table of x a 1 1 · · · x a n where ( a 1 , . . . , a n ) is the binary n representation of the integer i . � � 1 1 • Theorem 1 T n satisfies : T 1 = and 0 1 � � T s − 1 T s − 1 T s = , where O 2 s − 1 denotes O 2 s − 1 T s − 1 the 2 s − 1 × 2 s − 1 zero matrix, s = 2 , 3 , . . . . • Lemma 1 T − 1 = T n . n + 6
+ + Computing µ ( f ) by Matrix (Cont’d) � � 1 1 • Example 1 T 1 = , 0 1 1 1 1 1 0 1 0 1 T 2 = and 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 T 3 = . 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 + 7
+ + Computing µ ( f ) by Matrix (Cont’d) • Theorem 2 The following are equivalent: (i) g = µ ( f ), (ii) f = µ ( g ), (iii) ( f ( α 0 ) , f ( α 1 ) , . . . , f ( α 2 n − 1 )) T n = ( g ( α 0 ) , g ( α g ( α 2 n − 1 )), (iv) ( g ( α 0 ) , g ( α 1 ) , . . . , g ( α 2 n − 1 )) T n = ( f ( α 0 ) , f ( α 1 f ( α 2 n − 1 )). • Example 2 Let f ( x 1 , x 2 , x 3 ) = 1 ⊕ x 2 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 2 x 3 . Then g = µ ( f ) has the truth table (10111001) and f has the truth table: (11010011). (10111001) T 3 = (11010011), (11010011) T 3 = (10111001). + 8
+ + Computing µ ( f ) by Polynomials • Define D α ( x ) = (1 ⊕ a 1 ⊕ x 1 ) · · · (1 ⊕ a n ⊕ x n ) where x = ( x 1 , . . . , x n ), α = ( a 1 , . . . , a n ). • It is known that � f ( x ) = f ( α ) D α ( x ) (2) α ∈ ( GF (2)) n • Lemma 2 (i) µ ( D α )( x ) = x a 1 1 · · · x a n n where α = ( a 1 , . . . , a n ), (ii) µ ( x a 1 1 · · · x a n n ) = D α ( x ). • Theorem 3 Set g = µ ( f ). Then f ( α ) x a 1 1 · · · x a n � µ ( f )( x ) = n α ∈ ( GF (2)) n + 9
+ + Computing µ ( f ) by Recursive Relations • It is known that f ( x ) = x 1 g ( y ) ⊕ h ( y ) where x = ( x 1 , . . . , x n ) and y = ( x 2 , . . . , x n ). • Theorem 4 µ ( f )( x ) = x 1 ( µ ( g )( y ) ⊕ µ ( h )( y )) ⊕ µ ( h )( y ). + 10
+ + Properties of µ ( f ) • Corollary 1 µ − 1 = µ . • Let P be a permutation on { 1 , . . . , n } . De- fine the function f P as f P ( x 1 , . . . , x n ) = f ( x P (1) , . . . , x P ( n ) ). • Theorem 5 µ ( f P ) = g P . • Note: P in Theorem 5 is a permutation on { 1 , . . . , n } but P cannot be extended to be a permutation on ( GF (2)) n . + 11
+ + Properties of µ ( f ) (Cont’d) • Theorem 6 deg ( f ) + deg ( µ ( f )) ≥ n . • Note: the lower bound in Theorem 6 can be reached. • Example 3 f ( x ) = (1 ⊕ x 1 ) · · · (1 ⊕ x n ). By Lemma 2, µ ( f ) is the constant one. Then deg ( f ) + deg ( µ ( f )) = n + 0 = n . + 12
+ + Concept of Coincident Boolean Functions • If f and g = µ ( f ) are identical, i.e., f = µ ( f ), Then f is called a coincident function on ( GF (2)) n . • Example 4 Set f ( x 1 , x 2 , x 3 , x 4 )= x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 2 ⊕ x 1 x 3 x 4 ⊕ x 1 x 2 x 4 ⊕ x 1 x 2 x 3 . Then the truth table of µ ( f ) is (0000011000011110). By computing, the truth table of f is also (0000011000011110). Then f is coinci- dent and µ ( f ) = f . • Theorem 7 Let ξ and η be the truth tables of f and g = µ ( f ). Then the following are equivalent: (i) f is coincident, (ii) g is coincident, (iii) ξT n = ξ , (iv) ηT n = η , (v) f and g are identical, (vi) ξ and η identical. + 13
+ + Characterisations and Constructions of Coincident Functions (by Matrix) • Set T ∗ n = T n ⊕ I 2 n , n = 1 , 2 , . . . . • Theorem 8 Let ξ and η be the truth ta- bles of f and g = µ ( f ) respectively. Then the following are equivalent: (i) f is coin- cident, (ii) g is coincident, (iii) ξT ∗ n = 0, (iv) ηT ∗ n = 0. • Theorem 9 f is coincident ⇐ ⇒ its truth table satisfies ( ζT ∗ n − 1 , ζ ). + 14
+ + Characterisations and Constructions of Coincident Functions (by Matrix)-Cont’d • Theorem 10 f is coincident ⇐ ⇒ its truth table ξ can be expressed as ξ = ηT ∗ n . • Theorem 11 f is coincident ⇐ ⇒ its truth table is a linear combination of rows of T ∗ n . + 15
+ + Characterisations and Constructions of Coincident Functions (by Matrix)-Cont’d 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 • Example 5 T ∗ 3 = . 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 • Consider f ( x 1 , x 2 , x 3 ) = x 2 x 3 ⊕ x 1 x 3 ⊕ x 1 x 2 x 3 . • By definition, f is coincident because f and µ ( f ) have the same truth table (00000111). • (00000111) T ∗ 3 = (00000000). By Theo- rem 8, f is coincident. • (00000111) = (01110000) T ∗ 3 . By Theo- rem 11, f is coincident. + 16
+ + Enumeration of Coincident Functions • Theorem 12 (1) T ∗ n has a rank 2 n − 1 , (ii) all the top 2 n − 1 rows of T ∗ n form a basis of rows of T ∗ n . • Theorem 13 f is coincident ⇐ ⇒ its truth table of f is a linear combination of top 2 n − 1 rows of T ∗ n . • Theorem 14 (i) There precisely exist 2 2 n − 1 coincident functions of n variables, (ii) they form 2 n − 1 - dimensional linear space. + 17
+ + Enumeration of Coincident Functions (Cont’d) 0 1 1 1 1 1 0 0 0 1 0 1 • Example 6 The top 4 rows of T ∗ 3 : 0 0 0 1 0 0 0 0 0 0 0 0 All (2 2 3 − 1 = 16) linear combinatios: (01111111), (00010101), (00010011), (00000001), (0000011 (00000110), (01101010), (00010100), (0110110 (01101011), (01111110), (01101100), (0111100 (01111001), (00010010), (00000000). • They have the ANFs: x 3 ⊕ x 2 ⊕ x 1 ⊕ x 2 x 3 ⊕ x 1 x 3 ⊕ x 1 x 2 ⊕ x 1 x 2 x 3 , x 2 x 3 ⊕ x 1 x 3 ⊕ x 1 x 2 x 3 , x 2 x 3 ⊕ x 1 x 2 ⊕ x 1 x 2 x 3 , x 1 x 2 x 3 , x 1 x 3 ⊕ x 1 x 2 ⊕ x 1 x 2 x 3 , x 1 x 3 ⊕ x 1 x 2 , x 3 ⊕ x 2 ⊕ x 1 ⊕ x 1 x 2 , x 2 x 3 ⊕ x 1 x 3 , x 3 ⊕ x 2 ⊕ x 1 ⊕ x 1 x 3 ⊕ x 1 x 2 x 3 , x 3 ⊕ x 2 ⊕ x 1 ⊕ x 1 x 2 ⊕ x 1 x 2 x 3 , x 3 ⊕ x 2 ⊕ x 1 ⊕ x 2 x 3 ⊕ x 1 x 3 ⊕ x 1 x 2 , x 3 ⊕ x 2 ⊕ x 1 ⊕ x 1 x 3 , x 3 ⊕ x 2 ⊕ x 1 ⊕ x 2 x 3 , x 3 ⊕ x 2 ⊕ x 1 ⊕ x 2 x 3 ⊕ x 1 x 2 x 3 , x 2 x 3 ⊕ x 1 x 2 , 0 + 18
+ + Characterisations and Constructions of Coincident Functions (by Polynomial) • Define a mapping Ψ as Ψ( f ) = h ⇐ ⇒ f ⊕ µ ( f ) = h . • Theorem 15 The following are equivalent: (i) h is coincident, (ii) h = Ψ( f ) or h = f ⊕ µ ( f ) for some f , (iii) Ψ( h ) = 0. • Lemma 3 D α ( x ) ⊕ x a 1 1 · · · x a n n is coincident. • Theorem 16 h is coincident ⇐ ⇒ if and only if h is a linear combination of all D α ( x ) ⊕ x a 1 1 · · · x a n n + 19
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