Today I want to write about AES cryptography algorithm also known as "Rijndael". AES stands for Advanced Encryption Standard and its developed by U.S. National Institute of Standards (NIST) in 2001
The first following video will clearly shows how AES algorithm works,
In my point of view the hardest part is the Mix Column part, here i will give you more detailed information to how to solve this part.
For example if you have this two table (on the left and center) and want to calculate the result (on the right):
.

* 

= 

S’0,c = ({02}.63) ⊕ ({03}.C0) ⊕ FE ⊕ 9C
S’1,c = 63 ⊕ ({02}.C0) ⊕ ({03}.FE) ⊕ 9C
S’1,c = 63 ⊕ ({02}.C0) ⊕ ({03}.FE) ⊕ 9C
S’2,c = 63 ⊕ C0 ⊕ ({02}.FE) ⊕ ({03}.9C)
S’3,c = ({03}.63) ⊕ C0 ⊕ FE ⊕ ({02}.9C)
so basically for calculation of S'0,c we need to do four multiplies :
 one multiply by 2
 one multiply by 1
 one multiply by 1
 one multiply by 3
before starting describing the details i suggest you to use windows calculator for your calculation, after running the calculator go to "View" menu and choose "Programmer". It will help you to get hexadecimal or binary or calculating the xor and left shift!
 For those that are multiply by 1 you just need to leave them alone! (for example in S'0,c calculation you just leave FE or 9C )
 For calculating those that are multiplied by 2 you need to do the following (example ({02}.63)):
*** as we miss a zero in left shift in our example so we don't need to perform the last stage and the result of above example is 11000110
 convert it to binarey (for example 63 = 1100011 )
 in case the result is less than 8bits like above example add 0 bits on the left side until it become 8bits 01100011.
 now do one left shif on 01100011 so the result will be 11000110 ( all the times a zero will come in from right side)
 If in previous stage you lost a "zero" from left just skip this stage but if you lose a "one" from left side you have to xor the result from stage3 to 00011011 (this is a fixed number! and not related to this example . So again if you lost a 1 on left shift you have to XOR the result of left shift with 00011011 )
 For calculating those that are multiplied by 3 you need to do the following (example ({03}.C0) ):
example: ({03}.C0)
 its the same as calculation for multiply by two but has a small stage at the end and that stage is you have to XOR the result with the original value binary.
C0=11000000 (original value in bin)
11000000 << = 1000 0000 ( we do a left shift and we lost a 1 from left side)
10000000 ⊕ 00011011 = 10011011 (now we will xor with fix number of 00011011 because in previous stage it lost a 1 on the left)
10011011 ⊕ 1100 0000 = 01011011 ( now we xor the result with the original value that we had at the first)
so final result is 01011011
now we need to XOR the result of multiplications...
FE= 11111110
9C= 10011100
1100 0110 ⊕ 0101 1011 ⊕ 1111 1110 ⊕ 1001 1100 = 1111 1111 = FF
and below is another example:
And later if you want to double check you results you can check the AES algorithm results in any round by following tool(round 1, round2 , ... round10), This program will ask for a state of key and state of plain text and in result it will calculate the result in all rounds and show in the charts.
Click here to Download the tool