Breaking Cryptographic Ciphers: How Algorithms Determine the Correct Key

Abstract: When a computer algorithm attempts to break a cryptographic cipher, especially in scenarios where the encrypted text (ciphertext) is known but the original text (plaintext) is not, the process can be challenging. This article explores the mechanisms behind the guessing of keys in the context of symmetric ciphers and discusses the feasibility of correctly decrypting ciphertext given only its format. Through the application of cryptographic principles and understanding bijective functions, this piece provides a detailed insight into the methods and challenges involved.

Introduction

Cryptography plays a crucial role in securing our digital communications and data. A cryptographic cipher, such as AES (Advanced Encryption Standard), uses a secret key to transform plaintext into ciphertext. A reverse process, known as decryption, involves the use of the same key to revert the ciphertext back to plaintext. The core challenge in cryptography lies in determining the correct key, especially when the plaintext is unknown.

Key Determination Mechanism

Perfect Knowledge of Plaintext and Ciphertext

When both the plaintext and the ciphertext are known, the task of determining the key becomes significantly easier. A simple comparison can be made to find the key that decrypts the ciphertext to the plaintext or encrypts the plaintext to the ciphertext. However, this approach is not feasible in many real-world scenarios due to the vast number of possible keys.

Limited Knowledge of Plaintext Format

In scenarios where the plaintext format is known, such as plain text, XML files, JPEG images, or Word documents, the decryption process becomes more manageable. For instance, if the decrypted result resembles a valid English text or follows a specific file structure, it increases the likelihood that the key is correct. However, a false positive can still occur, where the decrypted text may appear valid but is not the intended plaintext.

Deciphering with No Plaintext Knowledge

When the plaintext format is unknown, the task is almost impossible. The plaintext could be arbitrary binary data, making it difficult to identify the correct key. Even if a key produces a valid format, there is a very low probability that the format is the intended plaintext.

Mathematical Underpinnings

Bijective Functions and Key Space

Modern symmetric ciphers, such as AES, employ bijective functions to map input to output within a specific key size. For AES-128, the key space consists of all possible 128-bit keys, which is a bijective function from {0, 1}^128 to {0, 1}^128. The total number of such functions is given by 2^{128}!. To cover all possible permutations, a key of at least 4.3 cdot 10^{40} bits would be required.

Real-World Key Sizes

Practically, key sizes like 128 or 256 bits are used, which significantly reduce the complexity of the key space. Despite this, the security of these ciphers lies in the enormous number of possible keys, making brute-force attacks impractically time-consuming.

Practical Approaches to Key Discovery

Pattern Recognition and Human Verification

When the plaintext format is known, decryption algorithms can look for common words or structures within the decrypted text. For example, if the plaintext is supposed to be English text, words like "the," "have," "order," and "action" can be used as clues. If the decrypted text contains these words, it is more likely that the key is correct. However, false positives can still occur, and a human must verify the meaningfulness of the decrypted text.

Historical Lessons from WW2

The historical decryption efforts, such as the successful decryption of German Enigma machine ciphers, highlight the importance of regularly changing keys. Without this, once a key is discovered, it can be used to decrypt many communications. This underscores the necessity of renewing keys periodically to maintain the security of cryptographic systems.

Conclusion

The process of breaking cryptographic ciphers, especially when plaintext is unknown, is complex and computationally intensive. The success of such efforts relies on guessing the key, which is facilitated by known plaintext formats, the volume of the key space, and the application of bijective functions. Understanding these principles helps in assessing the security of different cryptographic systems and the importance of key management in cryptography.

References

[1] Schneier, B. (2015). .Cryptography Engineering: Design Principles and Practical Applications. Wiley.

[2] Stallings, W. (2019). Cryptography and Network Security: Principles and Practice. Pearson.