Advanced Encryption Standard (AES)

Lecturer:

Prof. Dr. Michael Eichberg

Version:

2023-10-19

Based on:

Cryptography and Network Security - Principles and Practice, 8th Edition, William Stallings

Finite Field Arithmetic (Recap)

• A field is a set in which we can do addition, subtraction, multiplication, and division without leaving the set.

• Division is defined with the following rule: \(a/b = a(b^{-1})\).

• An example of a finite field (one with a finite number of elements) is the set \(Z_p\) consisting of all the integers \(\lbrace 0,1,....,p-1 \rbrace\), where p is a prime number and in which arithmetic is carried out modulo \(p\).

Finite Field Arithmetic (Recap)

• For convenience and for implementation efficiency, we would like to work with integers that fit exactly into a given number of bits with no wasted bit patterns.

  • Integers in the range 0 through \(2^n – 1\), which fit into an n-bit word.

• If one of the operations used in the algorithm is division, then we need to work in arithmetic defined over a field.

  • Division requires that each nonzero element has a multiplicative inverse.

• The set of such integers, \(Z_{2^n}\), using modular arithmetic, is not a field!

  • For example, the integer 2 has no multiplicative inverse in \(Z_{2^n}\), that is, there is no integer b, such that \(2b\; mod\; 2^n = 1\)

• A finite field containing \(2^n\) elements is referred to as \(GF(2^n)\).

Finite Field Arithmetic for AES

• In the Advanced Encryption Standard (AES) all operations are performed on 8-bit bytes

• The arithmetic operations of addition, multiplication, and division are performed over the finite field \(GF(2^8)\)

  AES uses the irreducible polynomial \(m(x) = x^8 + x^4 + x^3 +x +1\)

AES Key Elements

• AES uses a fixed block size of 128 bits.

• AES operates on a 4x4 column-major order array of 16 bytes/128 bits: \(b_0,b_1,\dots,b_{15}\) termed the state:

  \begin{equation*} \begin{bmatrix}b_{0}&b_{4}&b_{8}&b_{12}\\b_{1}&b_{5}&b_{9}&b_{13}\\b_{2}&b_{6}&b_{10}&b_{14}\\b_{3}&b_{7}&b_{11}&b_{15}\end{bmatrix} \end{equation*}

AES Encryption Process

AES Parameters

AES Encryption and Decryption Process

AES Detailed Structure

• Processes the entire data block as a single matrix during each round using substitutions and permutation.

• The key that is provided as input is expanded into an array of forty-four 32-bit words, \(w[i]\) if 128 bits are used for the keysize.

• The cipher begins and ends with an AddRoundKey stage.

• Can view the cipher as alternating operations of XOR encryption (AddRoundKey) of a block, followed by scrambling of the block (the other three stages), followed by XOR encryption, and so on.

• Each stage is easily reversible.

• The decryption algorithm makes use of the expanded key in reverse order, however the decryption algorithm is not identical to the encryption algorithm.

• State is the same for both encryption and decryption.

• Final round of both encryption and decryption consists of only three stages.

AES Uses Four Different Stages

Substitute bytes:

uses an S-box to perform a byte-by-byte substitution of the block

ShiftRows:

is a simple permutation.

MixColumns:

is a substitution that makes use of arithmetic over \(GF(2^8)\).

AddRoundKey:

is a simple bitwise XOR of the current block with a portion of the expanded key.

AES Substitute byte transformation

AES S-box

Each individual byte of State is mapped into a new byte in the following way: The leftmost 4 bits of the byte are used as a row value and the rightmost 4 bits are used as a column vlaue. These two values serve as indexes into the S-box.

AES Inverse S-box

Example: The (hex)value 0xA3 (x=A and y=3) is mapped by the S-box to the (hex)value 0x0A. The inverse S-box maps the value 0x0A (x=0 and y=A) back to the original value.

S-Box Rationale

• The S-box is designed to be resistant to known cryptanalytic attacks.

• The Rijndael developers sought a design that has a low correlation between input bits and output bits and the property that the output is not a linear mathematical function of the input.

• The nonlinearity is due to the use of the multiplicative inverse in the construction of the S-box.

Shift Row Transformation

Shift Row Transformation - Rationale

• More substantial than it may first appear!

• The State, as well as the cipher input and output, is treated as an array of four 4-byte columns.

• On encryption, the first 4 bytes of the plaintext are copied to the first column of State, and so on.

• The round key is applied to State column by column.

• Thus, a row shift moves an individual byte from one column to another, which is a linear distance of a multiple of 4 bytes.

• Transformation ensures that the 4 bytes of one column are spread out to four different columns.

Mix Column Transformation

Inverse Mix Column Transformation

Mix Colum Transformation - Example

Mix Column Transformation - Rationale

• Coefficients of a matrix based on a linear code with maximal distance between code words ensures a good mixing among the bytes of each column.

• The mix column transformation combined with the shift row transformation ensures that after a few rounds all output bits depend on all input bits.

AddRoundKey Transformation

• The 128 bits of State are bitwise XORed with the 128 bits of the round key.

• Operation is viewed as a columnwise operation between the 4 bytes of a State column and one word of the round key.

• Can also be viewed as a byte-level operation.

Input for a Single AES Encryption Round

AES Key Expansion

• Takes as input a four-word (16 byte) key and produces a linear array of 44 words (176) bytes.

• This is sufficient to provide a four-word round key for the initial AddRoundKey stage and each of the 10 rounds of the cipher.

• Key is copied into the first four words of the expanded key.

• The remainder of the expanded key is filled in four words at a time.

• Each added word \(w[i]\) depends on the immediately preceding word, \(w[i – 1]\), and the word four positions back, \(w[i – 4]\)

• In three out of four cases a simple XOR is used.

• For a word whose position in the w array is a multiple of 4, a more complex function \(g\) is used.

AES Key Expansion - Visualized

AES Round Key Computation

AES Key Expansion - Example (Round 1)

Let's assume: \(w[3] = (67,20,46,75)\):

• \(g(w[3])\):

  • circular byte left shift of \(w[3]\): \((20,46,75,67)\)

  • byte substitution using s-box: \((B7,5A,9D,85)\)

  • adding round constant \((01,00,00,00)\) gives: \(g(w[3]) = (B6,5A,9D,85)\)

• \(w[4] = w[0] \oplus g(w[3]) = (E2,32,FC,F1)\)

• \(w[5] = w[4] \oplus w[1] = (91,12,91,88)\)

• \(w[6] = w[5] \oplus w[2] = (B1,59,E4,E6)\)

• \(w[7] = w[6] \oplus w[3] = (D6,79,A2,93)\)

• First roundkey is: \(w[4] || w[5] || w[6] || w[7]\)

AES Key Expansion - Rationale

• The Rijndael developers designed the expansion key algorithm to be resistant to known cryptanalytic attacks

• Inclusion of a round-dependent round constant eliminates the symmetry between the ways in which round keys are generated in different rounds

Avalanche Effect in AES: Change in Plaintext

Avalanche Effect in AES: Change in Key

Equivalent Inverse Cipher

AES decryption cipher is not identical to the encryption cipher.

• The sequence of transformations differs although the form of the key schedules is the same.

• Has the disadvantage that two separate software or firmware modules are needed for applications that require both encryption and decryption.

Two separate changes are needed to bring the decryption structure in line with the encryption structure:

1. The first two stages of the decryption round need to be interchanged.

2. The second two stages of the decryption round need to be interchanged.

Interchanging InvShiftRows and InvSubBytes

• InvShiftRows affects the sequence of bytes in State but does not alter byte contents and does not depend on byte contents to perform its transformation

• InvSubBytes affects the contents of bytes in State but does not alter byte sequence and does not depend on byte sequence to perform its transformation

Interchanging AddRoundKey and InvMixColumns

• The transformations AddRoundKey and InvMixColumns do not alter the sequence of bytes in State.

• If we view the key as a sequence of words, then both AddRoundKey and InvMixColumns operate on State one column at a time.

• These two operations are linear with respect to the column input.

  That is, for a given State \(S_i\) and a given round key \(w_j\):

  \begin{equation*} InvMixColumns(S_i \oplus w_j) = InvMixColumns(S_i) \oplus InvMixColumns(w_j) \end{equation*}

Equivalent Inverse Cipher

Implementation Aspects

AES can be implemented very efficiently on an 8-bit processor:

AddRoundKey:

is a bytewise XOR operation.

ShiftRows:

is a simple byte-shifting operation.

SubBytes:

operates at the byte level and only requires a table of 256 bytes.

MixColumns:

requires matrix multiplication in the field \(GF(2^8)\), which means that all operations are carried out on bytes.

Implementation Aspects

Can be efficiently implemented on a 32-bit processor:

• Redefine steps to use 32-bit words

• Can precompute 4 tables of 256-words

• Then each column in each round can be computed using 4 table lookups + 4 XORs

• At a cost of 4Kb to store tables

• Designers believe this very efficient implementation was a key factor in its selection as the AES cipher