AES Round Steps

The AES algorithm encrypts data by executing 4 different operations; modifying gradually the state matrix until the ciphertext is obtained. The operations are: AddRoundKey, SubBytes, ShiftRows, MixColumns and an extra operation called the KeySchdule which is in charge to expand the original key for each AES round. This operations are going to be executed N times depending on the key length.

AddRoundKey: This transformation applies a bitwise XOR between the elements of the State array and the elements of the RoundKey. The RoundKey is obtained from the cipher key by computing the KeySchedule which will be explained later. The State and the RoundKey are of the same size where “a” is the current state, “k” the round key and “b” is the next state. AddRoundKey is denoted by:

SubBytes: This step provides non linearity byte substitution; it operates independently on each byte of the state matrix using a look up table, called the Rijndael S-box. The design ofthe S-box is thought to be resistant to attacks made by differential and linear cryptanalysis and algebraic manipulation, and has been extensively analyzed during the competition to verify its security. It is constructed by the following two transformations:

1. Compute the multiplicative inverse in the finite field GF(2^8)
2. Apply a bitwise affine transformation over GF(2), which is a polynomial multiplication by a constant matrix using the inversed byte, followed by a XOR operation with a constant byte

The constant matrix and the constant byte are provided by the AES Standard as described in FIPS 197. On image 2 we can see the constant matrix and the constant byte, where b0….b7 is the multiplicative inverse as a vector

After doing the previous steps we will obtain the Rijndael S-Box. Image 4 shows the S-Box generated; afterwards, each byte of the state table is updated. For example if state(i,j)={3D} the new value is determined by the intersection of the row with value 3 and the column with value D, providing an output of state'(i,j)=27.

ShiftRows: This operation shifts each row of the state cyclically to the left depending on the row index. First row is not shifted, second row is shifted one position, third row two positions and finally the fourth row is shifted three positions to the left.

MixColumns: The MixColumns step along with the ShiftRows step is the primary source of diffusion in AES. MixColumns performs a column by column transformation, treating each column of the state as polynomials over GF (2^8). Each polynomial or column is then multiplied with a fixed polynomial:

this latter polynomial is expressed as a fixed matrix, the operation can be written as matrix multiplication as follows:

Each column of the state is then multiplied by the fixed polynomial expressed as a matrix and and the next state is obtained

KeySchedule: The AES algorithm takes the cipher key and performs a key schedule also referred to as a key expansion to generate keys for each AES round. The number of round keys necessary to encrypt one block of information is related to the key length because this will determine the number of rounds. For example, a key length of 128 bits will require in total 11 round keys, 1 for the initial round, 9 for standard rounds and 1 for the final round. The key schedule is therefore a method to extend the cipher key, where it can be seen as an array of 32 bit columns numbered from 0 to 43. The i-th column of the cipher key k matrix is denoted by Wi. The first four columns are filled with the given cipher key

Columns in position that are multiple of 4 are calculated by:

• Applying a rotation operation and SubBytes transformation to the previous column W(i-1)
• Adding this result to the column 4 positions earlier W(i-4) plus a round constant Rcon.

The reminding 3 columns are calculated by adding the previous column W(i-1) with the column 4 positions earlier W(i-4)