How does differential cryptanalysis work?

Differential cryptanalysis is a powerful technique used in cryptanalysis to break cryptographic algorithms, particularly block ciphers. Here’s a detailed explanation of how it works:

Overview

Differential cryptanalysis involves analyzing the differences in input and output pairs of plaintexts and ciphertexts to deduce information about the key used in encryption. It was first publicly introduced by Eli Biham and Adi Shamir in the late 1980s and has since been a significant method for attacking block ciphers.

Detailed Explanation

1. Input and Output Differences:
- In differential cryptanalysis, the attacker studies how differences in plaintext inputs (ΔP) influence differences in ciphertext outputs (ΔC) with a certain probability.
- By observing these differences across multiple pairs of plaintext and ciphertext, the attacker attempts to deduce patterns that reveal information about the secret key.

2. Statistical Analysis:
- The attacker collects a large number of plaintext-ciphertext pairs and computes the expected differences in plaintext and ciphertext.
- Through statistical analysis, the attacker identifies patterns where specific differences in plaintext lead to predictable differences in ciphertext.

3. Constructing a Differential:
- A differential is a characteristic of the cipher that describes how differences in input propagate through the encryption process to produce differences in output.
- The attacker constructs differentials based on observed input-output pairs and uses these to infer information about the key.

4. Key Recovery:
- Once a significant differential pattern is identified, the attacker uses it to mount an attack on the cipher.
- By iterating through possible key values and checking if the differential holds, the attacker can eventually deduce the correct key used for encryption.

Example

Let's illustrate with a simplified example using a hypothetical block cipher:
- Suppose a differential analysis reveals that when a specific difference ΔP1 in plaintext leads to ΔC1 in ciphertext with a certain probability, it suggests a potential relationship influenced by the encryption key.
- By testing different key values and checking if the observed differential matches the expected probability, the attacker narrows down the possible keys until finding the correct one.

Effectiveness and Applications

Differential cryptanalysis has been particularly effective against early block ciphers like DES (Data Encryption Standard) due to their simpler structure and fewer rounds. Modern block ciphers, such as AES (Advanced Encryption Standard), are designed with resistance to differential cryptanalysis in mind, employing complex round structures and key schedules that significantly increase the difficulty of mounting such attacks.

In conclusion, differential cryptanalysis is a sophisticated method that leverages statistical analysis of input-output differentials to break cryptographic algorithms. It underscores the importance of robust cipher design and the continuous evolution of encryption techniques to withstand such attacks.