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

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={=(a1,a2,…,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 2n 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 x=x1x2xn denote a monomial on Vn. a1 a2 an

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 Definition1: Let f be a function on Vn. If as x runs through all vectors in Vn, f(x)=1 is true 2n-1 times f(x)=1, then the function f is said to be balanced. Definition2: 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 Definition3: 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 (1)a,b,cG, we have a(bc)=(ab)c Definition5:A group G is a set together with a binary operation : GxG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 Definition6: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)=eH}. Lagrange Theorem: The order of a subgroup of a finite group is a divisor of the 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 { a1, a2, …, 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 S5

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 (i1, i2, ... , 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 : (i1, i2, ... , ir) = (i1, ir)(i1, ir-1) ... (i1, i3)(i1, i2) 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 S9 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) 1 2 3 4 5 6 8 9 7 Because : (7,9)(7,8) = (7, 8, 9)

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 Lemma1: 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 Lemma2: Let f be a boolean function on Vn, [if(x)]。 [jf(x)]= (ij )f(x)= kf(x) Definition: