CryptoBook
  • CryptoBook
  • Book Plan
  • Style Guide
    • Sample Page
  • Contributors
  • Fundamentals
    • Mathematical Notation
    • Division and Greatest common divisor
      • Euclidean Algorithm
    • Modular Arithmetic
      • Theorems of Wilson, Euler, and Fermat
        • Fermat's Little Theorem in Detail
        • Euler's Theorem in Detail
      • Quadratic Residues
    • Continued Fractions
  • Number Theory
  • Ideals
  • Polynomials With Shared Roots
  • Integer Factorization
    • Pollard rho
    • Sieves
  • Abstract algebra
    • Groups
      • Another take on groups
      • Discrete Log Problem
    • Rings
    • Fields
    • Polynomials
  • Elliptic Curves
    • Untitled
  • Lattices
    • Introduction
    • LLL reduction
      • Gram-Schmidt Orthogonalization
      • Lagrange's algorithm
      • LLL reduction
    • Lattice reduction
      • Minkowski reduced
      • HKZ reduced
      • LLL reduced
    • Applications
      • Coppersmith algorithm
      • Extensions of Coppersmith algorithm
    • Hard lattice problems
    • Lattices of interest
    • Cryptographic lattice problems
      • Short integer solutions (SIS)
      • Learning with errors (LWE)
      • Ring-LWE
      • NTRU
    • Interactive fun
    • Resources and notations
  • Asymmetric Cryptography
  • RSA
    • Proof of correctness
    • RSA application
    • Low Private Component Attacks
      • Wiener's Attack
      • Boneh-Durfee Attack
    • Common Modulus Attack
    • Recovering the Modulus
  • Diffie-Hellman
    • MITM
  • Elliptic Curve Cryptography
  • Symmetric Cryptography
    • Encryption
    • The One Time Pad
    • AES
      • Rijndael Finite Field
      • Round Transformations
  • Hashes
    • Introduction / overview
    • The Birthday paradox / attack
  • Isogeny Based Cryptography
    • Introduction to Isogeny Cryptography
    • Isogenies
    • Isogeny and Ramanujan Graphs
  • Appendices
    • Sets and Functions
    • Probability Theory
Powered by GitBook
On this page

Was this helpful?

Export as PDF
  1. Symmetric Cryptography

AES

Advanced Encryption Standard

PreviousThe One Time PadNextRijndael Finite Field

Last updated 4 years ago

Was this helpful?

Introduction

The Advanced Encryption Standard most known as AES is one of the most used ciphers nowadays. Created by Vinent Rijmen and Joan Daemen under the name Rijndael, it won the NIST competition that resulted in its standardization in 2001 to replace older algorithms such as DES (and its variant 3DES). In fact, it is six times faster than 3DES.

AES encrypts a block of 16 bytes only at a time, though ciphertexts tend to be much longer. To accomodate this, cipherexts are cut in blocks of 16 bytes using an operating mode [see future section on mode]. We only focus on the encryption of a single block.

The array of 16 bytes(p0,…,p15)(p_0,\ldots,p_{15})(p0​,…,p15​)are arranged from up to bottom, column by column in4×44 \times 44×4matrix. During the encryption, the state of this matrix changes and results in a 16-bytes ciphertext(c0,…,c15)(c_0,\ldots,c_{15})(c0​,…,c15​)whose output can be read following the same ordering:

A key is involved and three sizes are possible: 128, 192, or 256 bits. Depending of the size, there are a few differences which will be explained later. For now, it is sufficient to know that round keys are derived from this master key.

Our interest is to look at what goes inside the transformation between the plaintext and the ciphertext. Basically, there are four operations on the state matrix, each important for the security of AES:

  • AK: add round key;

  • SR: shift row;

  • SB: substitution box;

  • MC: MixColumn.

All these operations are executed a several number of times in what are called rounds to mix the plaintext enough. A look on the flow of an encryption is given in the figure below.

Two particular cases can be noticed:

  • the first round is preceded by an additional AK;

  • last round is missing MC.

The number of rounds NR is different depending on the master key length:

Key length

Number of rounds

128

10

192

12

256

14

AES encryption.
Rounds of AES.