How can linear cyrptanalysis break a DES cryptosystem?

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

Linear cryptanalysis is a potent cryptanalytic attack method that applies linear approximations to the action of a cryptographic algorithm. It is particularly effective against block ciphers such as the Data Encryption Standard (DES). To understand how linear cryptanalysis can break a DES cryptosystem, it is essential to consider the mechanics of both DES and the principles of linear cryptanalysis.

1. Overview of DES:

The Data Encryption Standard (DES) is a symmetric-key algorithm for the encryption of digital data. It operates on 64-bit blocks of data and uses a 56-bit key. DES is based on a Feistel structure, which divides the input block into two halves and processes them through multiple rounds of permutation and substitution. Each round involves the following steps:
– Expansion: The right half of the block is expanded from 32 bits to 48 bits.
– Key Mixing: The expanded right half is XORed with a round key derived from the main key.
– Substitution: The result is divided into eight 6-bit segments, each of which is passed through a corresponding S-box, producing a 4-bit output.
– Permutation: The 32-bit output from the S-boxes is permuted according to a predefined table.

The left and right halves are then swapped, and the process repeats for 16 rounds. The final permutation produces the encrypted output.

2. Principles of Linear Cryptanalysis:

Linear cryptanalysis, developed by Mitsuru Matsui in the early 1990s, seeks to exploit linear relationships between plaintext, ciphertext, and key bits. The core idea is to find linear approximations that hold with a certain probability for the non-linear components of the cipher, such as S-boxes in DES. The steps involved in linear cryptanalysis are as follows:
– Identify linear approximations: Determine linear equations that approximate the behavior of the S-boxes. These approximations are of the form P[i] \oplus C[j] \oplus K[k] = 0, where P is the plaintext bit, C is the ciphertext bit, and K is the key bit.
– Calculate bias: Measure the probability that the linear approximation holds. The bias is the deviation from 0.5 (the probability of a random guess being correct).
– Collect data: Encrypt a large number of plaintexts to gather sufficient data for statistical analysis.
– Analyze data: Use the collected data to identify the most likely key bits based on the observed biases.

3. Applying Linear Cryptanalysis to DES:

The application of linear cryptanalysis to DES involves several stages. Let's break down the process:

3.1. Identifying Linear Approximations:

The first step is to identify linear approximations for the S-boxes used in DES. Each S-box in DES has 6 input bits and 4 output bits. The goal is to find linear equations that approximate the S-box outputs with a certain bias. For example, consider an S-box with input bits x_1, x_2, x_3, x_4, x_5, x_6 and output bits y_1, y_2, y_3, y_4. A linear approximation might be of the form:

    \[ x_1 \oplus x_2 \oplus y_3 = 0 \]

This equation indicates that the XOR of the input bits x_1 and x_2 and the output bit y_3 is equal to 0 with a certain probability.

3.2. Calculating Bias:

Once linear approximations are identified, the next step is to calculate the bias for each approximation. The bias is the difference between the probability that the approximation holds and 0.5. For instance, if the approximation x_1 \oplus x_2 \oplus y_3 = 0 holds with a probability of 0.75, the bias is:

    \[ \epsilon = 0.75 - 0.5 = 0.25 \]

3.3. Collecting Data:

To perform linear cryptanalysis on DES, a large number of plaintext-ciphertext pairs are required. The number of pairs needed depends on the bias of the linear approximation. Generally, the more pairs collected, the higher the accuracy of the analysis. For DES, several thousand or even millions of pairs might be necessary.

3.4. Analyzing Data:

With the collected data, the next step is to analyze the plaintext-ciphertext pairs to determine the most likely key bits. This involves the following sub-steps:
– For each plaintext-ciphertext pair, compute the values of the linear approximations.
– Count the number of times each approximation holds.
– Use the counts to estimate the probabilities and compare them to the expected probabilities based on the biases.

4. Example of Linear Cryptanalysis on DES:

Consider a simplified example where we have identified a linear approximation for the first S-box in DES. Suppose the approximation is:

    \[ P[1] \oplus C[3] \oplus K[5] = 0 \]

Here, P[1] is the first bit of the plaintext, C[3] is the third bit of the ciphertext, and K[5] is the fifth bit of the key. The steps to break DES using this approximation are as follows:

4.1. Data Collection:

Encrypt a large number of plaintexts using DES and collect the corresponding ciphertexts. For simplicity, let's assume we have collected 1,000,000 plaintext-ciphertext pairs.

4.2. Analyzing Data:

For each plaintext-ciphertext pair, compute the value of P[1] \oplus C[3]. If the result is 0, increment a counter; otherwise, decrement the counter. The counter value will indicate the bias of the approximation.

4.3. Estimating Key Bits:

Based on the counter value, estimate the probability that the approximation holds. Compare this probability to the expected probability based on the bias. If the observed probability is significantly higher or lower than 0.5, it suggests that the key bit K[5] is likely to be 0 or 1, respectively.

5. Practical Considerations:

While the example above is simplified, practical linear cryptanalysis on DES involves more complex approximations and multiple S-boxes. Additionally, the attack requires significant computational resources to collect and analyze the data. However, with enough data and computational power, linear cryptanalysis can effectively reduce the key space and make it feasible to recover the key.

6. Countermeasures:

To mitigate the risk of linear cryptanalysis, modern cryptographic algorithms are designed with resistance to such attacks in mind. Techniques such as increasing the number of rounds, using larger key sizes, and incorporating more complex S-boxes can enhance the security of block ciphers. For DES, the introduction of Triple DES (3DES) provides a higher level of security by applying the DES algorithm three times with different keys.

Other recent questions and answers regarding Data Encryption Standard (DES) - Key schedule and decryption:

View more questions and answers in Data Encryption Standard (DES) - Key schedule and decryption

More questions and answers:

Tagged under: , , , , ,