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Mathematics in Code: Why Frequency Analysis is so Important for Code Breaking
In the world of information security, cryptography has always played a vital role, and the technology for cracking passwords is constantly evolving. Frequency analysis, this ancient and powerful technique, is not only an application of mathematics, but also reveals the mystery of language through in-depth research on the characteristics of text. Whether in ancient times or modern times, frequency analysis has always been one of the key tools for hackers to decipher encrypted messages.
Frequency analysis is the study of the frequencies of letters in ciphertext, an aid in breaking classical codes.
The basic principle of frequency analysis is that text in any particular language has a specific distribution pattern in the frequency of occurrence of letters and letter combinations. Taking English as an example, the letters E, T, A and O are the most common letters, while Z, Q, X and J are extremely rare. Therefore, the letters that appear frequently in the ciphertext may correspond to these common letters after analysis.
Leavitt, Egypt, and Greatness are at the center of cryptographic processing, and frequency analysis is undoubtedly the key.
In the context of a simple substitution cipher, the ciphertext letter corresponding to each plaintext letter is fixed, so the frequency of a particular letter directly indicates the information of the plaintext letter. For example, if the letter X appears significantly more frequently than other letters in the ciphertext, then it is likely to represent the English letter E, which is one of the most frequent letters. However, T and A are just as common, which makes initial identification difficult, so hackers need to think and try more.
In addition to the frequency of individual letters, analysis of bigrams (i.e., two-syllable combinations) and trigrams (i.e., three-syllable combinations) is equally important. Statistics show that the letters Q and U almost always appear together, which provides encryptors with a wealth of information to help crack the code.
To a large extent, frequency analysis techniques can reflect the inherent rules in language, so hackers can use this feature to decipher encrypted information.
Specific Examples
Suppose a hacker named Eve intercepts a password and knows that the message is encrypted using a simple substitution cipher, so she begins to perform frequency analysis. Based on the statistics of the ciphertext letters, she found that the letter I appeared most frequently, while XL and XLI were the most common double and triple letter groups. Based on these observations, Eve began to guess the correspondence between the letters, such as guessing
X~t, L~h and I~e.
As more letters were identified, Eve was able to further infer more letters, and eventually successfully cracked the code, and the entire text became clearly readable. However, it is a process, and incorrect assumptions are common, sometimes requiring backtracking on poor choices.
During the cracking process, the variability of statistical data may lead to errors in initial guesses, which is a challenge that hackers must face.
Combined with modern computer technology, the accuracy and efficiency of frequency analysis have made a qualitative leap. Nowadays, with professional software, hackers can complete complex letter statistics and analysis in a few seconds, no longer relying on tedious manual calculations. This adds an urgent reminder to the reality of information leakage:
Conclusion
With the advent of the digital age, traditional encryption methods seem to have gradually lost their effectiveness in protecting sensitive information. However, frequency analysis technology still plays an unparalleled role in data protection. So, as individual users, how can we increase our awareness of encryption technology to protect our messages from being trapped in easily cracked codes?
