Polyalphabetic CIPHERS Linguistics 484

Summary The idea How to recognize: index of coincidence How many alphabets: Kasiski

The idea Remove the invariant that a plaintext letter always maps to the same cryptotext letter. Smooth out the frequency distribution, removing clues.

Monoalphabetic PlaintextCryptosystem Ciphertext

Polyalphabetic Plaintext AA Cryptosystem Ciphertext BB CC

Polyalphabetic Plaintext AA Cryptosystem Ciphertext BB CC

Polyalphabetic Plaintext AA Cryptosystem Ciphertext BB CC

Polyalphabetic Plaintext AA Cryptosystem Ciphertext BB CC

Polyalphabetic system Cryptosystem with several components. Systematic way of moving from one cryptosystem to the next.

Vigenère (simplified) Component ciphers are shift ciphers, using so called Direct Standard Alphabet You use each alphabet for one character, then move on.

Vigenère (simplified) ABCDEFGHIJKLMNOPQRSTUVWXYZ ABCDEFGHIJKLMNOPQRSTUVWXYZ BCDEFGHIJKLMNOPQRSTUVWXYZA CDEFGHIJKLMNOPQRSTUVWXYZAB ZABCDEFGHIJKLMNOPQRSTUVWXY

You and your friend agree a single letter key, say ‘S’. Encrypt the first letter with the ‘S’ alphabet, second with ‘T’ alphabet, and so on.

Vigenère (simplified) key=”s” ABCDEFGHIJKLMNOPQRSTUVWXYZ.... STUVWXYZABCDEFGHIJKLMNOPQR TUVWXYZABCDEFGHIJKLMNOPQRS UVWXYZABCDEFGHIJKLMNOPQRST VWXYZABCDEFGHIJKLMNOPQRSTU.. BOOKJWVP

Polyalphabetic system Cryptosystem with several components. Systematic way of moving from one cryptosystem to the next. But two weaknesses in simplified Vigenère. Direct standard alphabets. Breaking one character gets whole alphabet. Pattern of movement is too obvious.

Full Vigenère Use keyword to control jump between alphabets Pattern of movement no longer as obvious.

Vigenère key=”SYMBOL” ABCDEFGHIJKLMNOPQRSTUVWXYZ STUVWXYZABCDEFGHIJKLMNOPQR YZABCDEFGHIJKLMNOPQRSTUVWX MNOPQRSTUVWXYZABCDEFGHIJKL BCDEFGHIJKLMNOPQRSTUVWXYZA OPQRSTUVWXYZABCDEFGHIJKLMN LMNOPQRSTUVWXYZABCDEFGHIJK THE ATOMIC ENERGYL..

Exercise Encipher THE ATOMIC ENERGY with the keyword SYMBOL Decipher AVYUL HWLEE UCZLL LTYVI YOFJI ZSLNI knowing that the keyword is HOUSE

Vigenère key=”HOUSE” ABCDEFGHIJKLMNOPQRSTUVWXYZ HIJKLMNOPQRSTUVWXYZABCDEFG OPQRSTUVWXYZABCDEFGHIJKLMN UVWXYZABCDEFGHIJKLMNOPQRST STUVWXYZABCDEFGHIJKLMNOPQR EFGHIJKLMNOPQRSTUVWXYZABCD AVYUL TH...

How many alphabets? Index of co-incidence Babbage-Kasiski examination Once you have how many alphabets, use frequency analysis as for regular shift ciphers. Breaking Vigenère.

Index of co-incidence Based on arguments about probability. Intuition: measure roughness of frequency distribution Mathematical details follow

Roughness of distributions Smoothest distribution has each letter happening 1/26th of the time. Roughest has one letter happening 100% of the time Normal English has some uneveness, less smooth than totally uniform.

Index of co-incidence Get a frequency f[letter] for each letter. Multiply f[letter]*(f[letter]-1) to get number of co- incidences involving that letter. Add the results for all letters together. Divide by the number of co-incidences you would expect if all the letters were the same.

Index of co-incidence IC = sum(f[letter]*(f[letter]-1)) / N(N-1) / N(N-1)

Index of coincidence IC has a value of if the letters are evenly distributed, which is what you get if the polyalphabet uses many many alphabets It has a value of for English text, monoalphabetic encryptions of English text, many other things

Number of alphabets IC large0.038

Idea to quantify roughness Count the number of times a pair of letters drawn at random from the text happen to be the same. For the roughest possible, we always get the same letter, so a text of length N has N(N-1) repeats. For the smoothest possible, we get way fewer.

Babbage-Kasiski method Explained well in Code Book p Key idea: what does it mean if we find a sequence of repeated characters in a message that has been encoded using a repeated keyword.

Babbage-Kasiski method Key idea: what does it mean if we find a sequence of four or more repeated characters in a message that has been encoded using a repeated keyword. Most likely: a sequence of four or more English characters in the plaintext has been encoded twice starting from the same place in the repeating keyword. Less likely: it’s an accident, some other arrangement of English letters gives rise to a repeat by chance.

Repeats OK, if a repeat is due to the fact that the same thing is encoded twice in the same way, then the keyword must be used a whole number of times to get from one to the other. So, keep track of the spacing between repeats.

Repeats Nearly every repeat will have a spacing that divides evenly by the length of the keyword. So, break the spacings into factors and look for something that (almost) always turns up.

Repeats sequencespacingfactors EFIQ PSDLP55 WCXYM ETRL

Once you have how many alphabets, use frequency analysis as for regular shift ciphers. If there are five different alphabets, tally up characters 1,6,11,... into one table, 2,7,12,... into a second, 3,8,13,... into the third, and so on up to the fifth. The results will show the characteristic frequency pattern of a shifted alphabet (high A, E close to each other, low J,K next to each other, X,Y,Z all low, etc.) Breaking Vigenère.

Once you have how many alphabets, use frequency analysis as for regular shift ciphers. The results will show the characteristic frequency pattern of a shifted alphabet (high A, E close to each other, low J,K next to each other, X,Y,Z all low, etc.) See if the keyword is sensible. Might be an English word. Then plug in letters and check whether the message works out. Breaking Vigenère.