Can DES be broken by differential cryptanalysis?
Differential cryptanalysis is a form of cryptanalysis applicable primarily to block ciphers, which involves analyzing the effect of particular differences in input pairs on the differences at the output. This method was introduced by Eli Biham and Adi Shamir in the late 1980s and has since become a fundamental tool in the cryptanalyst's toolkit. The Data Encryption Standard (DES), a symmetric-key algorithm for the encryption of digital data, has been one of the primary subjects of differential cryptanalysis.
The DES algorithm, designed by IBM in the early 1970s and adopted as a federal standard in 1977 by the National Institute of Standards and Technology (NIST), operates on 64-bit blocks of data using a 56-bit key. DES employs 16 rounds of a Feistel network, where each round consists of a series of substitutions and permutations determined by the key schedule.
To understand whether DES can be broken by differential cryptanalysis, it is important to consider the specifics of both the DES structure and the principles of differential cryptanalysis.
DES Structure and Key Schedule
DES is built upon a combination of substitution and permutation operations. Each round of DES involves the following steps:
1. Expansion (E-box): The 32-bit half-block is expanded to 48 bits using an expansion permutation.
2. Key Mixing: The expanded half-block is XORed with a round key derived from the main key using a key schedule.
3. Substitution (S-box): The 48-bit result is divided into eight 6-bit segments, each of which is passed through a corresponding S-box to produce a 4-bit output. The eight 4-bit outputs are concatenated to form a 32-bit block.
4. Permutation (P-box): The 32-bit block is permuted using a fixed permutation table.
5. XOR and Swap: The permuted block is XORed with the other half of the data block, and the halves are swapped.
The key schedule generates a series of 16 round keys, each 48 bits in length, from the original 56-bit key. This process involves permuted choice operations and left circular shifts.
Differential Cryptanalysis Principles
Differential cryptanalysis involves studying how differences in plaintext pairs propagate through the cipher to produce differences in the ciphertext. The basic idea is to select pairs of plaintexts that have a specific difference, encrypt them, and analyze the resulting ciphertext differences. By observing how these differences evolve through the rounds of the cipher, an attacker can infer information about the key.
Key concepts in differential cryptanalysis include:
– Differential: The XOR difference between two values. For example, if 
and 
are two plaintexts, their differential is 
.
– Characteristic: A sequence of differences that describes how an input differential propagates through the rounds of the cipher.
– Probability: The likelihood that a given input differential will produce a specific output differential after a certain number of rounds.
Application to DES
DES was specifically designed to resist differential cryptanalysis, which was not publicly known at the time of its design but was understood by IBM and the NSA. The S-boxes in DES were carefully chosen to minimize the probability of differential characteristics, making differential cryptanalysis more difficult.
However, Biham and Shamir demonstrated that DES is not immune to differential cryptanalysis. They showed that, although the full 16-round DES is resistant to practical differential attacks, reduced-round versions of the cipher are vulnerable. Specifically, they developed attacks on DES with fewer than 16 rounds.
For instance, an attack on an 8-round DES can be conducted with a complexity of about 
chosen plaintexts, which is significantly more efficient than a brute-force attack on the full key space. For the full 16-round DES, the complexity of differential cryptanalysis is much higher, making it impractical with the computational resources available at the time of their research.
Example of Differential Cryptanalysis on DES
To illustrate how differential cryptanalysis works, consider a simplified example with a reduced-round DES:
1. Choose a Differential: Select a specific input differential 
. For simplicity, assume affects only a few bits.
2. Generate Plaintext Pairs: Generate a large number of plaintext pairs 
such that 
.
3. Encrypt Plaintext Pairs: Encrypt each pair to obtain ciphertexts 
.
4. Analyze Output Differentials: Calculate the output differential 
for each pair.
5. Identify Characteristics: Identify patterns in the output differentials that suggest specific characteristics of the cipher's internal structure.
6. Infer Key Information: Use the identified characteristics to infer information about the round keys, and ultimately the main key.
Practical Considerations
While differential cryptanalysis is a powerful tool, its practical application to DES requires significant computational resources and a large number of chosen plaintexts. Modern cryptographic practices have evolved to use more complex and secure algorithms, such as the Advanced Encryption Standard (AES), which are designed to resist not only differential cryptanalysis but also a wide range of other cryptanalytic attacks.
Differential cryptanalysis is a well-established technique in the field of cryptography that can be used to analyze and, in some cases, break block ciphers like DES. While the full 16-round DES is resistant to practical differential attacks, reduced-round versions of DES are vulnerable. The design of DES, particularly its S-boxes, reflects an awareness of differential cryptanalysis principles, demonstrating the importance of careful cryptographic design in ensuring security.
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