COMMUMCATION TUEORV OF SECRECY SYSTEMS 677 



etc. are in the same residue class. It is obvious in this case that these crypto- 

 grams are essentially equivalent. All that is of importance in a simple sub- 

 stitution with random key is tlie pallern of letter repetitions, the actual 

 letters being dummy variables. Indeed we might dispense with them en- 

 tirely, indicating the pattern of repetitions in E as follows: 



This notation describes the residue class but eliminates all information as 

 to tlie specific member of the class. Thus it leaves precisely that information 

 which is cryptanalytically pertinent. This is related to one method of attack- 

 ing simple substitution ciphers — the method of pattern words. 



In the Caesar type cipher only the first differences mod 26 of the crypto- 

 gram are significant. Two cryptograms with the same Aei are in the same 

 residue class. One breaks this cipher by the simple process of writing down 

 the 26 members of the message residue class and picking out the one which 

 makes sense. 



The \'igenere of period d with random key is another example of a pure 

 cipher. Here the message residue class consists of all sequences with the 

 same first differences as the cryptogram, for letters separated by distance d. 

 For d = 3 the residue class is defined by 



nio — nii = 62 — e^ 

 mz — me = 63 — 66 

 mi — nir = 6i — ey 



where E = e^ , 62 , • • • is the cryptogram and mi , W2 , • • • is any M in the 

 corresponding residue class. 



In the transposition cipher of period d with random key, the residue class 

 consists of all arrangements of the e, in which no e, is moved out of its block 

 of length d, and any two e, at a distance d remain at this distance. This is 

 used in breaking these cijihers as follows: The cr^-ptogram is written in 

 successive blocks of length d, one under another as below (d — 5) : 



61 62 63 6i 66 



66 67 es 69 6ia 

 en 612 ■ 



