AES MixColumns() Procedure Algorithm

A detailed description of the MixColumns() procedure algorithm is provided. The MixColumns() procedure performs a matrix multiplication of a given 'state' with a static matrix. The MixColumns() procedure is used in the AES encryption process.

The MixColumns() - The MixColumns() procedure performs a matrix multiplication of a given 'state' with a static matrix. The MixColumns() procedure is key procedure used in the AES encryption process.

Here is the algorithm that the MixColumns() procedure should follow:

Procedure Name: 
   MixColumns(state)

Input:
   state: A 4x4 byte array

Algorithm:
   |     |   |0x02 0x03 0x01 0x01|   |     |
   |     |   |0x01 0x02 0x03 0x01|   |     |
   |state| = |0x01 0x01 0x02 0x03| ● |state|
   |     |   |0x03 0x01 0x01 0x02|   |     |

Note that "●" represents a matrix multiplication defined as:
   |a1|   |0x02 0x03 0x01 0x01|   |b1|
   |a2|   |0x01 0x02 0x03 0x01|   |b2|
   |a3| = |0x01 0x01 0x02 0x03| ● |b3|
   |a4|   |0x03 0x01 0x01 0x02|   |b4|

is equivalent to: 
   a1 = 0x02●b1 ⊕ 0x03●b2 ⊕ 0x01●b3 ⊕ 0x01●b4
   a2 = 0x01●b1 ⊕ 0x02●b2 ⊕ 0x03●b3 ⊕ 0x01●b4
   a3 = 0x01●b1 ⊕ 0x01●b2 ⊕ 0x02●b3 ⊕ 0x03●b4
   a4 = 0x03●b1 ⊕ 0x01●b2 ⊕ 0x01●b3 ⊕ 0x02●b4

where "⊕" is the binary XOR operation, 
and "●" is Rijndael's GF (Galois Field) multiplication.

Rijndael's GF multiplication used above can be defined as:

If we express bytes "a" and "b" in a polynomial form:
   a = 2^7*a7⊕2^6*a6⊕2^5*a5⊕2^4*a4⊕2^3*a3⊕2^2*a2⊕2^1*a1⊕2^0*a0
   b = 2^7*b7⊕2^6*b6⊕2^5*b5⊕2^4*b4⊕2^3*b3⊕2^2*b2⊕2^1*b1⊕2^0*b0
then:
   c = a ● b = 
    ((2^7*a7⊕2^6*a6⊕2^5*a5⊕2^4*a4⊕2^3*a3⊕2^2*a2⊕2^1*a1⊕2^0*a0)*2^7*b7
   ⊕ (2^7*a7⊕2^6*a6⊕2^5*a5⊕2^4*a4⊕2^3*a3⊕2^2*a2⊕2^1*a1⊕2^0*a0)*2^6*b6
   ⊕ (2^7*a7⊕2^6*a6⊕2^5*a5⊕2^4*a4⊕2^3*a3⊕2^2*a2⊕2^1*a1⊕2^0*a0)*2^5*b5
   ⊕ (2^7*a7⊕2^6*a6⊕2^5*a5⊕2^4*a4⊕2^3*a3⊕2^2*a2⊕2^1*a1⊕2^0*a0)*2^4*b4
   ⊕ (2^7*a7⊕2^6*a6⊕2^5*a5⊕2^4*a4⊕2^3*a3⊕2^2*a2⊕2^1*a1⊕2^0*a0)*2^3*b3
   ⊕ (2^7*a7⊕2^6*a6⊕2^5*a5⊕2^4*a4⊕2^3*a3⊕2^2*a2⊕2^1*a1⊕2^0*a0)*2^2*b2
   ⊕ (2^7*a7⊕2^6*a6⊕2^5*a5⊕2^4*a4⊕2^3*a3⊕2^2*a2⊕2^1*a1⊕2^0*a0)*2^1*b1
   ⊕ (2^7*a7⊕2^6*a6⊕2^5*a5⊕2^4*a4⊕2^3*a3⊕2^2*a2⊕2^1*a1⊕2^0*a0)*2^0*b0
     ) ⓜ 0x011b

Now we have only one last operation to understand, the GF modulo, "ⓜ", which can be defined as:

If we say: 
    c = a ⓜ b
then should have:
    a = b●d ⊕ c
    c < b

Here is an example of Rijndael's GF multiplication:

   0x57 ● 0x83 
   = (2^6 ⊕ 2^4 ⊕ 2^2 ⊕ 2^1 ⊕ 2^0) ● (2^7 ⊕ 2^1 ⊕ 2^0)     
   = ((2^13 ⊕ 2^11 ⊕ 2^9 ⊕ 2^8 ⊕ 2^7) 
   ⊕  (2^7 ⊕ 2^5 ⊕ 2^3 ⊕ 2^2 ⊕ 2^1) 
   ⊕  (2^6 ⊕ 2^4 ⊕ 2^2 ⊕ 2^1 ⊕ 2^0) 
     ) ⓜ 0x011b
   = ((2^13 ⊕ 2^11 ⊕ 2^9 ⊕ 2^8 ⊕ 2^6 ⊕ 2^5 ⊕ 2^4 ⊕ 2^3 ⊕ 2^0)
     ) ⓜ 0x011b
   = 0x2b79 ⓜ 0x011b
   = 0xc1

The last step can approved by: 
   0x2b79 
   = 0x011b●0x28 ⊕ 0xc1
   = (2^8 ⊕ 2^4 ⊕ 2^3 ⊕ 2^1 ⊕ 2^0)●(2^5 ⊕ 2^3) ⊕ (2^7 ⊕ 2^6 ⊕ 2^1)
   = (2^13 ⊕ 2^9 ⊕ 2^8 ⊕ 2^6 ⊕ 2^5) ⊕ (2^11 ⊕ 2^7 ⊕ 2^6 ⊕ 2^4 ⊕ 2^3)
     ⊕ (2^7 ⊕ 2^6 ⊕ 2^0)
   = 2^13 ⊕ 2^11 ⊕ 2^9 ⊕ 2^8 ⊕ 2^6 ⊕ 2^5 ⊕ 2^4 ⊕ 2^3 ⊕ 2^0
   = 0x2b79 

Here is a demonstration on how to calculate 0x2b79 ⓜ 0x011b manually in binary mode:

Express the dividend in binary format:
   0x2b79 = 0b10101101111001

Express the divisor in binary format:
   0x011b = 0b100011011

Do the manual calculation:
                       101000 <- quotient
    divisor  ----------------
   100011011 | 10101101111001 <- dividend
             ⊕ 100011011
             ----------------
                 100000011001
               ⊕ 100011011
               --------------
                     11000001 <- remainder

Convert the remainder to hexadecimal format:
   0b11000001 = 0xc1

We get the result:
   0x2b79 ⓜ 0x011b = 0xc1

Last update: 2015.

Table of Contents

 About This Book

 Cryptography Terminology

 Cryptography Basic Concepts

Introduction to AES (Advanced Encryption Standard)

 What Is AES (Advanced Encryption Standard)?

 AES, or Rijndael, Encryption Algorithm

 AES Key Schedule Algorithm

 AES Key Schedule Example

AES MixColumns() Procedure Algorithm

 Example Vector of AES Encryption

 AES Standard Decryption Algorithm

 AES Equivalent Decryption Algorithm

 Introduction to DES Algorithm

 DES Algorithm - Illustrated with Java Programs

 DES Algorithm Java Implementation

 DES Algorithm - Java Implementation in JDK JCE

 DES Encryption Operation Modes

 DES in Stream Cipher Modes

 PHP Implementation of DES - mcrypt

 Blowfish - 8-Byte Block Cipher

 Secret Key Generation and Management

 Cipher - Secret Key Encryption and Decryption

 Introduction of RSA Algorithm

 RSA Implementation using java.math.BigInteger Class

 Introduction of DSA (Digital Signature Algorithm)

 Java Default Implementation of DSA

 Private key and Public Key Pair Generation

 PKCS#8/X.509 Private/Public Encoding Standards

 Cipher - Public Key Encryption and Decryption

 MD5 Mesasge Digest Algorithm

 SHA1 Mesasge Digest Algorithm

 OpenSSL Introduction and Installation

 OpenSSL Generating and Managing RSA Keys

 OpenSSL Managing Certificates

 OpenSSL Generating and Signing CSR

 OpenSSL Validating Certificate Path

 "keytool" and "keystore" from JDK

 "OpenSSL" Signing CSR Generated by "keytool"

 Migrating Keys from "keystore" to "OpenSSL" Key Files

 Certificate X.509 Standard and DER/PEM Formats

 Migrating Keys from "OpenSSL" Key Files to "keystore"

 Using Certificates in IE (Internet Explorer)

 Using Certificates in Firefox

 Using Certificates in Google Chrome

 Outdated Tutorials

 References

 PDF Printing Version