Relationships between Boolean Function and Symmetric Group GIEE

Relationships between Boolean Function and Symmetric Group GIEE, NTU Alcom Lab Presenter: 陳炳元

Outline • Boolean space and boolean functions • Basic Group Theory • Symmetric Group • Relationships between symmetric group and boolean functions

Boolean space and boolean functions • The set of n-tube vectors Vn={ =(a 1, a 2, …, an) | ai GF(2), i= 1, 2, …n} is a boolean space, where GF(2) is a Galois field. Clearly, all the vectors in Vn are binary Sequences. Property: A boolean space Vn contains 2 n vectors.

Boolean space and boolean functions • A boolean function is defined on Vn by the mapping f: Vn V • • There are several ways to represent a boolean function: by a polynomial; by a binary sequence; by a (0, 1) sequence. Here we use the polynomial respresentation to discuss boolean function. Let a a a x =x 1 x 2 xn denote a monomial on Vn. 1 2 n

Boolean space and boolean functions • Every boolean function on Vn is a linear combination of monomials • f(x)= x c = 0 or 1, where the sign denote addition(XOR). Vn

Boolean space and boolean functions • Definition 1: • Let f be a function on Vn. If as x runs through all vectors in Vn, f(x)=1 is true 2 n-1 times f(x)=1, then the function f is said to be balanced. • Definition 2: • Let f be a function on Vn. The nonlinearity (denote by Nf) of the function f is defined by minimum Hamming distance from f to all affine functioons over Vn. i. e. Nf=min d(f, ), on Vn.

Boolean space and boolean functions • Definition 3: Let f be a boolean function on Vn. If for a vector Vn the function f(x) f(x ) is balanced, then the function f is said to have propagation criteria with respect to the vector . • Note: Vn, 0 wt( ) k f(x) f(x ) is balanced , then f(x) has propagation criteria of degree k denote PC(k). If k=1, denote SAC

Outline • Boolean space and boolean functions • Basic Group Theory • Symmetric Group • Relationships between symmetric group and boolean functions

Basic Group Theory • Definition 5: A group G is a set together with a binary operation : Gx. G G satisfying: • (1) a, b, c G, we have a (b c)=(a b) c • (2) e G a e=e a=a, a G • (3) a G a-1 G a a-1= a-1 a=e A group is said abelian if a b=b a a, b G NOTE: For simplicity, we will denote ab for a b

Basic Group Theory • Definition 6: Let G be a group, a subset H G is a subgroup if H is a group by using the binary operation of G, denote H G • Homomprphism: f : G H is a function between groups that respects the structure of groups. That is, a function satisfying f(xy) = f(x)f(y).

Basic Group Theory • Kernel: The kernel of f, denote ker(f), is defined to be {x G | f(x)=e. H}. • Lagrange Theorem: The order of a subgroup of a finite group is a divisor of the group.

Outline • Boolean space and boolean functions • Basic Group Theory • Symmetric Group • Relationships between symmetric group and boolean functions

Permutation of a Set • Let A be the set { a 1, a 2, …, an }. A permutation on A is a function f: A A that is both injective and serjective. • The set of all permutations on A is denoted by Sn • A permutation is represented by a matrix :

Examples of Permutation • Let A be the set { 1, 2, 3, 4, 5 } f and g are elements of S 5

Product of Permutations • The product of f and g is the composition function f。g

Cycles : A special kind of Permutation • An element f of Sn is a cycle (r-cycle) if there exists such that Cycles will be written simply as (i 1, i 2, . . . , ir) Example : = (1, 3, 4)

Permutations and cycles • Every permutation can be written as a product of disjoint cycles. For example We have 1 3 2 8 1 4 6 4 5 7 9 5 We can easily verify that f = (1, 3, 2, 8)(4, 6)(5, 7, 9)

Transpositions : A special kind of cycles • A 2 -cycle such as (3, 7) is called a transposition • Every cycle can be written as a product of transposition : i 2) (i 1, i 2, . . . , ir) = (i 1, ir)(i 1, ir-1). . . (i 1, i 3)(i 1, For example, (1, 3, 2, 4) = (1, 4)(1, 2)(1, 3)

Permutations and transpositions • Since every permutation can be expressed as a product of (disjoint) cycles, every permutation can be expressed as a product of transpositions For example,

Application • (7, 8) is a transposition in S 9 • In fact,

Application • We may as well use this table to represent the transposition (7, 8) 1 2 3 4 5 6 8 7 9

Application • So = 1 2 3 4 5 6 8 7 9

Application • What is the transposition representation after switching 7 and 9? ( Should be (? , ? )(7, 8) ) 1 2 3 4 5 6 8 7 9

Application • What is the transposition representation of the table? • YES, it is (7, 9)(7, 8) Because : (7, 9)(7, 8) = (7, 8, 9) 1 2 3 4 5 6 8 9 7

Symmetric Group • NOTE: 。 • (Sn , 。) is a group, It is called symmetric group • The order of Sn is n! • Theorem: Let G be a finite group, H is a subset of G H G if and only if a, b G, ab G

Outline • Boolean space and boolean functions • Basic Group Theory • Symmetric Group • Relationships between symmetric group and boolean functions

Relationships between symmetric group and boolean functions • Lemma 1: • Let f be a boolean function on Vn, Hf ={ Sn | f(x)=f(x)} then Hf is a subgroup of Sn Pf: 顯然e Hf 1, 2 Hf , ( 1 2 )f= 1( 2 f)= 1 f=f and ( 1 2 )f= 2( 1 f)= 2 f=f then we have 1 2= 1 2 and ( 1 2 )f=f Therefore it is a subgroup of Sn

Relationships between symmetric group and boolean functions • Lemma 2: Let f be a boolean function on Vn, [ if(x)]。 [ jf(x)]= ( i j )f(x)= kf(x) • Definition: