Between linear and differential cryptanalysis which is efficient for breaking DES?

/ / Published in Cybersecurity, EITC/IS/CCF Classical Cryptography Fundamentals, DES block cipher cryptosystem, Data Encryption Standard (DES) - Key schedule and decryption

The Data Encryption Standard (DES) is a symmetric-key algorithm for the encryption of digital data. Developed in the 1970s, DES became a widely adopted encryption standard. However, with the advancement of computational power and cryptanalytic techniques, DES has been subject to various forms of cryptanalysis, among which linear and differential cryptanalysis are particularly notable.

Linear Cryptanalysis

Linear cryptanalysis, introduced by Mitsuru Matsui in 1993, is a known-plaintext attack. The primary goal of this technique is to find a linear approximation to describe the behavior of the block cipher. This approximation is a linear equation that correlates plaintext bits, ciphertext bits, and key bits. The basic idea is to exploit linear relations that approximate the non-linear parts of the cipher, such as the S-boxes in DES.

To conduct linear cryptanalysis on DES, one typically follows these steps:

1. Identify Linear Approximations: Determine linear equations that approximate the behavior of the S-boxes. For DES, a common approximation involves finding linear expressions that hold with a certain probability, usually slightly better than random guessing (50%).

2. Collect Data: Gather a large number of plaintext-ciphertext pairs. The effectiveness of linear cryptanalysis depends heavily on the amount of data available. For DES, several thousand pairs may be required.

3. Count Biases: Evaluate the linear approximations by counting the number of times the approximations hold true for the collected data. The bias, or deviation from the expected 50%, provides insights into the key bits.

4. Key Determination: Use the observed biases to form hypotheses about the key bits. This step often involves solving a system of linear equations or employing statistical techniques to identify the most likely key.

An example of a linear approximation for DES might involve an equation like:

    \[ P_i \oplus P_j \oplus C_k \oplus K_m = 0 \]

where P_i, P_j are plaintext bits, C_k is a ciphertext bit, and K_m is a key bit. The goal is to find such approximations that hold with a probability significantly different from 0.5.

Differential Cryptanalysis

Differential cryptanalysis, developed by Eli Biham and Adi Shamir in the late 1980s, is another powerful cryptanalytic technique. This method focuses on the differences between pairs of plaintexts and their corresponding ciphertexts. It is a chosen-plaintext attack, meaning the attacker can choose pairs of plaintexts with specific differences and analyze the resulting ciphertext differences to gain information about the key.

The procedure for differential cryptanalysis on DES typically involves:

1. Define Differences: Select pairs of plaintexts with a specific difference, often denoted as \Delta P. The difference can be an XOR operation between the plaintext pairs.

2. Analyze Differential Characteristics: Determine how these differences propagate through the cipher. For DES, this involves studying how differences in the plaintext affect differences in the ciphertext after several rounds of encryption.

3. Collect Pairs: Encrypt a large number of plaintext pairs with the chosen differences and observe the resulting ciphertext differences. This step requires a significant amount of data, often in the range of thousands of pairs.

4. Identify Key Bits: Use the observed differential patterns to infer information about the key. This step involves analyzing the propagation of differences through the S-boxes and other components of the cipher to identify likely key candidates.

An example of a differential characteristic might involve observing how a difference in one bit of the plaintext propagates through the S-boxes and affects the ciphertext. If a particular difference pattern occurs with a higher probability than expected, it can provide clues about the key bits.

Efficiency Comparison

When comparing the efficiency of linear and differential cryptanalysis for breaking DES, several factors need to be considered:

1. Data Complexity: Both techniques require a substantial amount of data, but differential cryptanalysis often requires fewer plaintext-ciphertext pairs than linear cryptanalysis. This is because differential cryptanalysis can exploit specific patterns in the differences, which may be more pronounced and easier to detect.

2. Computational Complexity: The computational effort involved in analyzing the data and identifying key bits can vary. Linear cryptanalysis often involves solving linear equations and performing statistical analysis, which can be computationally intensive. Differential cryptanalysis, on the other hand, may involve more straightforward bitwise operations and difference propagation analysis.

3. Success Probability: The probability of successfully recovering the key depends on the quality of the approximations or differential characteristics used. Linear cryptanalysis relies on finding good linear approximations, which may not always be possible. Differential cryptanalysis, with its focus on difference propagation, may have a higher success rate for certain ciphers, including DES.

4. Implementation Complexity: The practical implementation of these attacks can vary in complexity. Linear cryptanalysis requires careful statistical analysis and hypothesis testing, which can be challenging to implement correctly. Differential cryptanalysis, while still complex, may be more straightforward in terms of the steps involved.

Historical Context and Practical Considerations

Historically, differential cryptanalysis was the first major cryptanalytic technique to pose a serious threat to DES. When Biham and Shamir introduced their method, it demonstrated that DES was not as secure as previously believed. Although DES was designed with some resistance to differential cryptanalysis in mind, the attack still proved effective under certain conditions.

Linear cryptanalysis, introduced later, provided another powerful tool for cryptanalysts. Matsui's work showed that DES could be broken with a significant amount of known plaintexts, further highlighting the vulnerabilities of the cipher.

In practice, both techniques have been used to analyze DES and other block ciphers. The choice of which method to use often depends on the specific context and the resources available. For instance, if an attacker has access to chosen plaintexts, differential cryptanalysis may be more practical. If only known plaintexts are available, linear cryptanalysis might be the preferred approach.

Example Scenarios

To illustrate the application of these techniques, consider the following scenarios:

1. Linear Cryptanalysis Example:
– An attacker has access to 2^43 known plaintext-ciphertext pairs encrypted with DES.
– The attacker identifies a linear approximation that holds with a probability of 0.51.
– By analyzing the data and counting the occurrences of the approximation, the attacker can form hypotheses about the key bits.
– After several iterations and statistical analysis, the attacker narrows down the key space and eventually recovers the full key.

2. Differential Cryptanalysis Example:
– An attacker can choose pairs of plaintexts with a specific difference \Delta P.
– The attacker encrypts 2^47 such pairs and observes the resulting ciphertext differences.
– By analyzing the differential characteristics and how they propagate through the S-boxes, the attacker identifies patterns that suggest certain key bits.
– Using these patterns, the attacker narrows down the key space and eventually recovers the full key.

Both linear and differential cryptanalysis are powerful techniques for breaking DES, each with its own strengths and weaknesses. Differential cryptanalysis is often more efficient in terms of data requirements and success probability, making it a preferred choice in many scenarios. However, linear cryptanalysis also provides valuable insights and can be effective with sufficient data and computational resources. The choice of technique depends on the specific context, the type of data available, and the attacker's goals.

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