The Advanced Encryption standard (AES) is a symmetric-key block cipher published by NIST in 2001.
To provide Security, AES uses four types of transformations:
- SubBytes
- ShiftRows
- MixCloumns
- AddRoundKey
Mix Column transformation operates at the column level.The bytes in the state column and constants matrix are interpreted as 8-bit words (or polynomials) with coefficients in GF(2)
- Multiplication of bytes is done in GF(2^8) with modulus (x^8+x^4+x^3+x+1)
- Addition is the same as XORing of 8 bit words
- In AES a byte that can be treat as a single entity ,it represented in Hex form.
Example 8 bit 10001100 represent in 8A in Hex
[ 02 * d4 ] + [ 03 * bf ] + [01 * 5d ] + [01 * 30 ]
convert each Hex data into Binary Example:
- convert binary to equivalent polynomial
- Example: 11010100 -> x^7 + x^6 + x^4 +x^2
t1=[ 02 * d4 ]
=[00000010] * [11010100]
=convert binary to equivalent polynomial
=x * (x7 + x6 + x4 + x2 )
t1= x8 + x7 + x5 + x3
t2 =[ 03 * bf ]
=[00000011] * [10111111]
=(x+1) * (x7 + x5 + x4 + x3 + x2 + x +1)
t2=(x8+x6+x5+x4+x3+x2+x+x7+x5+x4+x3+x2+x+1)
in GF(2) Addition is performed with XOR so same polynomial degree is cancelled out.
t2 =x8 + x6 + x7 +1
=[00000011] * [10111111]
=(x+1) * (x7 + x5 + x4 + x3 + x2 + x +1)
t2=(x8+x6+x5+x4+x3+x2+x+x7+x5+x4+x3+x2+x+1)
in GF(2) Addition is performed with XOR so same polynomial degree is cancelled out.
t2 =x8 + x6 + x7 +1
t3=[01 * 5d ]
=[00000001] * [01011101]
=1 * (x6 + x4 + x3 +x2 +1)
t3=x6 + x4 + x3 +x2 +1
=[00000001] * [01011101]
=1 * (x6 + x4 + x3 +x2 +1)
t3=x6 + x4 + x3 +x2 +1
t4=[01 * 30 ]
=[0000 0001] * [0011 0000]
=1 * (x5 + x4)
t4= x5 + x4
final ans= t1+ t2 + t3 + t4
= x8 + x7 + x5 + x3 +x8 +x7 +1 +x6 +x6 +x4 +x3 +x2 +1 +x5 +x4
in GF(2) Addition is performed with XOR so same polynomial degree is cancelled out.
=x2
=0000 0100
