Using this information we calculate the following If it were english, we would expect it to appear 162*0.082 = 13.284 times. To calculate the Chi-squared statistic for the ciphertext above, we see the letter A appears 18 times. This compares the letter counts in each decryption with what we would expect the counts to be if the text were english. To solve the Caesar cipher, we decipher the ciphertext with each of the 25 possible keys and calculate the Chi-squared statistic for each key. The cipher shown above is 162 characters, so we expect E to appear 162*0.127 = 20.57 times. To calculate the expected count just multiply the probability by the length of the ciphertext. If the letter E occurs with a proability of 0.127, we would expect it to occur 12.7 times in 100 characters. As a result we need to use the probabilities to calculate the expected count for each letter. The chi-squared statistic uses counts, not probabilities. We also know the probabilities of characters occuring in normal english text. Pualeapzzopmalkhjlyahpuubtilyvmwshjlzkvduaolhswohila Zpapzhafwlvmzbizapabapvujpwolypudopjolhjoslaalypuaolwsh Let us say we have a message enciphered using the Caesar cipher: aoljhlzhyjpwolypzvulvmaollhysplzaruvduhukzptwslzajpwoly The chi-squared statistic is a way for a computer to essentially perform this procedure. common letters in english are common in the ciphertext and rare english letters are rare in the ciphertext. This would occur when the shifted english frequencies line up with the ciphertext frequencies i.e. We could then compare them to the frequency distribution of english, and by shifting the two frequency distributions relative to one another we could find the shift that was used to encipher the plaintext. If we were to try solving a Caesar cipher by hand, a good first step would be to calculate the frequency distribution of the ciphertext characters. Javascript Chi-squared Calculator §Ĭhi-squared statistic against English distribution:Ĭhi-squared statistic against uniform distribution: We simply use the method to suggest a possible decryption. Unlike statisticians, we make no assumptions about the distribution of our data, and draw no conclusions about the significance of Ordinarily, statisticians use the chi-squared statistic for measuring the goodness of fit of data. This page will describe the use of the chi-squared statistic for cryptanalysis. Where C A is the count (not the probability) of letter A, and E A is the expected count of letter A. The formula for the chi-squared statistic is: If the two distributions are identical, the chi-squared statistic is 0, if the distributions are very different, some higher number will result. The Chi-squared Statistic is a measure of how similar two categorical probability distributions are.

Caesar cipher calculator