Calculating the inverse for a Rijndael s-box [duplicate]

Question

I want to calculate a Rijndael S-box but something went wrong. How can I calculate the inverse? In the following is my approach:

given: $f(x)=x^8+x^4+x^3+x+1 \rightarrow$ binary: 100011011

0x12 = 10010

---

calculation:

ggT(100011011,10010):    
 (x^8+x^4+x^3+x+1)/(x^4+x)=x^4+x+1    
-(x^8+x^5)
 __________
      x^5+x^4+x^3+x+1
    -(x^5+x^2)
     __________
         x^4+x^3+x^2+x+1
       -(x^4+x)
        __________
         x^3+x^2+1          => 1101=100011011+10010*10011

ggT(10010,1101):    
 (x^4+x)/(x^3+x^2+1)=x+1    
-(x^4+x^3+x)
 __________
      x^3
    -(x^3+x^2+1)
      ____________
           x^2+1            => 101=10010+1101*11

ggT(1101,101):    
     (x^3+x^2+1)/(x^2+1)=x+1    
    -(x^3+x)
     __________
          x^2+x+1 
         -(x^2+1)
         _________
               x             => 10=1101+101*11

ggT(101,10):    
    (x^2+1)/(x)=x
   -(x^2)
    __________
         1                    => 1=101+10*10

1 = 101+10*10
  = (10010+1101*11)+10(1101+101*11)
  = (10010+11(100011011+10010*10011))+10((100011011+10010*10011)+11(10010+1101*11)))
  = (10010+11(100011011+10010*10011))+10((100011011+10010*10011)+11(10010+(100011011+10010*10011)*11)))  



a = 100011011    b = 10010

  = (b+11(a+10011b))+10((a+10011b)+11(b+11(a+10011b)))
  = b+11a+11*10011b+10(a+10011b+11(b+11a+11*10011b))
  = b+11a+11*10011b+10(a+10011b+11b+11*11a+11*11*10011b)
  = b+11a+11*10011b+10a+10*10011b+10*11+10*11*11a+10*11*1110011b
  = b(1+11*10011+10*10011+10*11+10*11*11*10011)
  = b(110111100) -> 9 bit but i need 8 bit