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Linear Cryptanalysis

Linear Cryptanalysis

A powerful cryptanalytic attack that uses linear approximations of non-linear components in block ciphers to recover key bits.

Linear Cryptanalysis

Linear cryptanalysis is a cryptanalysis technique developed by Mitsuru Matsui in 1993, originally designed to break the Data Encryption Standard cipher. This method represents a significant breakthrough in symmetric cryptography analysis.

Core Principles

The attack relies on finding linear approximations of the inherently non-linear functions within a cipher. These approximations take the form:

P[i1,i2,...,ia] ⊕ C[j1,j2,...,jb] ⊕ K[k1,k2,...,kc] = 0

Where:

  • P[i] represents input plaintext bits
  • C[j] represents output ciphertext bits
  • K[k] represents key bits
  • ⊕ denotes the XOR operation

Methodology

  1. Finding Bias

    • Identify linear approximations that hold with probability ≠ 0.5
    • The further this probability deviates from 0.5, the stronger the bias
    • Larger bias typically leads to more effective attacks

  2. Data Collection

    • Gather known plaintext-ciphertext pairs
    • The number of required pairs is proportional to 1/ε², where ε is the bias

  3. Key Recovery

    • Use statistical analysis to determine likely key bits
    • Piling-up lemma helps combine multiple approximations

Mathematical Foundation

The success of linear cryptanalysis relies on the probability theory and statistical analysis of bit patterns. The bias of an approximation is defined as:

ε = |Pr(equation holds) - 1/2|

Countermeasures

Modern cipher designs incorporate defenses against linear cryptanalysis:

Historical Impact

Linear cryptanalysis has influenced:

Applications

The technique has been applied to analyze:

Limitations

  • Requires large amounts of known plaintext
  • May be impractical against modern ciphers
  • Computational complexity can be prohibitive

See Also