How does linear cryptanalysis work?

Linear cryptanalysis is a method used to analyze and break symmetric-key block ciphers by exploiting linear approximations to the encryption process. This technique, introduced by Mitsuru Matsui in the early 1990s, leverages the fact that the relationship between plaintext, ciphertext, and key bits can often be approximated linearly, even in complex ciphers.

Here's a breakdown of how linear cryptanalysis works:

1. **Linear Approximation**:

   - **Approximate Linear Relation**: Identify a linear approximation that approximates the behavior of the cipher. This involves finding a linear equation that relates certain bits of the plaintext, ciphertext, and key. For example, you might have a relation like \( P_1 \oplus P_2 \oplus K_1 \approx C_1 \oplus C_2 \), where \( \oplus \) denotes XOR.

   - **Linear Approximation Table**: Construct a table of these linear approximations to evaluate how well they fit the actual encryption process.

2. **Collect Data**:

   - **Plaintext-Ciphertext Pairs**: Gather a large number of plaintext-ciphertext pairs encrypted under the same key. The more pairs you have, the more accurate your approximation can be.

   - **Apply Linear Approximation**: For each pair, use the linear approximation to determine whether the approximation holds. This involves computing the left and right sides of the linear equation and comparing them.

3. **Statistical Analysis**:

   - **Estimate Key Bits**: Analyze the collected data to determine which key bits fit the linear approximation. This involves statistical analysis to identify patterns or biases in the approximation that are indicative of the actual key bits.

   - **Key Recovery**: Once you have estimated the most likely key bits, you can use them to attempt to recover the full key or narrow down the key space.

4. **Optimization**:

   - **Refinement**: Improve the approximation by refining the linear equations or increasing the number of plaintext-ciphertext pairs. This step helps to increase the accuracy of the key recovery.

### Key Points:

- **Linear Approximations**: The effectiveness of linear cryptanalysis depends on finding good linear approximations that can reliably predict the behavior of the cipher.

- **Complexity**: Linear cryptanalysis can significantly reduce the complexity of breaking a cipher compared to brute force attacks. It works well against ciphers with weaker key schedules or those that have certain linear properties.

- **Applicability**: Linear cryptanalysis has been notably effective against ciphers like DES (Data Encryption Standard), where linear approximations can reveal vulnerabilities.

Overall, linear cryptanalysis is a powerful technique that exploits linear relationships within the encryption process to recover secret keys, and its effectiveness depends on the ability to find and exploit such relationships.