Finding Probable Frequency Sums to Reduce the Key Space of Homophonic Substitution Ciphers

Publication date: July 2020 Clearly, brute-force decryption attempts are an inefficient means of decryption. Fortunately, patterns in the distribution of letter frequencies in languages may be used to immediately identify which cipher characters are likely to represent which plaintext characters. For example, the letter ‘e’ is the most frequent letter in English text (Friedman, 1938), therefore the most frequent cipher character in a given monoalphabetic substitution cipher text likely represents the letter ‘e’. This frequency analysis approach may be repeated for all character frequencies in the cipher text for complete decryption. Homophonic substitution ciphers circumvent frequency analysis by mapping each character in a plaintext alphabet to multiple cipher characters (e.g. a {e, !}, b {~, q}, c {z, X}, etc.). In this way, the ciphertext alphabet is greater than the plaintext alphabet, and the frequency distribution of ciphertext characters does not immediately resemble that of the underlying plaintext language. For a cipher of this kind with two unique homophones mapped to each character of the plaintext English alphabet, there are ~1.2 × 1060 possible keys. A machine capable of one trillion decryption attempts per second would take ~3.8 × 1040 years to brute-force the entire key space. Decrypting such a cipher exhaustively in a reasonable amount of time is far beyond the current technological grasp of cryptanalysts (Diffie and Hellman, 1977), and consequently many homophonic substitution ciphers remain unsolved (Kahn, 2005). Arguably, the strongest decryption method for homophonic substitution ciphers is the hill-climbing algorithm (Dhavare et al., 2013), wherein a parent key is generated and used to decrypt the ciphertext, and the fitness of this decryption attempt is measured. The key is then modified, and another decryption attempt is made with this modified key. If the fitness of the modified attempt is better than the initial attempt, the modified key is carried forward; otherwise, INTRODUCTION Substitution ciphers are a form of encryption whereby each character in a plaintext message is substituted for a corresponding cipher character. These cryptograms have existed in various forms for millennia (Whitman and Herbert, 2017) and, for some time, were secure means of exchanging data and information due to the nature of their encryption (Singh, 2000). For a plaintext alphabet of 26 characters like the English language, a monoalphabetic substitution cipher key which maps each of these to different singular cipher characters (e.g. a e, b h, c z, etc.) is concealed by the vast size of the key space, or the number of possible mixed alphabet cipher keys. In this case, the key space is 26! ≈ 4 × 1026. A single machine capable of one billion decryption attempts per second would take ~13 billion years to brute-force every solution of such a cipher, which implies trying every possible key exhaustively. This number is comparable to the age of the observable universe (Planck Collaboration et al., 2016). Finding Probable Frequency Sums to Reduce the Key Space of Homophonic Substitution Ciphers