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

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

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

4. 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.

5. 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 a an

6. 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

7. 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.

8. 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

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

11. 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

12. 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).

13. 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.

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

15. 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 :

16. Examples of Permutation
Let A be the set { 1, 2, 3, 4, 5 }
f and g are elements of S5

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

18. 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)

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

20. 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)

21. 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,

22. Application
(7, 8) is a transposition in S9
In fact,

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

24. Application
So = 1 2 3 4 5 6 8 7 9

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

26. 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)

27. 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

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

29. 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

30. 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: