COMMUNICATION TIIPIORY OF SRCRKCV SYSTEMS 667 



where k, , I, , ■ ••, 5, in general have different perifxls. The period of their 

 sum, 



^, + /. + ■ • • +5, 



as in compound transposition, is the least common multiple of the indi\ idual 

 {)eriods. 



When the X'igenere is used with an unlimited key, never repeating, we 

 have the \'ernam system,^ with 



ei = nti + ki (mod 26) 



the ki being chosen at random and independently among 0, 1, • ■ ■, 25. If 

 the key is a meaningful text we have the "running key" cipher. 



4. Digram, Trigram, and N-gram substitution. 



Rather than substitute for letters one can substitute for digrams, tri- 

 grams, etc. General digram substitution requires a key consisting of a per- 

 mutation of the 26- digrams. It can be represented by a table in which the 

 row corresponds to the lirst letter of the digram and the column to the second 

 letter, entries in the table being the substitutes (usually also digrams). 



5. Single Mixed Alphabet Vigenere. 



This is a simple substitution followed by a Vigenere. 



Ci = finii) + ki 

 nii = f~'^(ei — ki) 



The "inverse" of this system is a Vigenere followed by simple substitution 



ei = ginii + ki) 

 tUi = g^^iei) — ki 



6. Matrix System? 



One method of ?/-gram substitution is to operate on successive «-grams 

 with a matrix having an inverse. The letters are assumed numbered from 

 to 25, making them elements of an algebraic ring. From the »-gram wi w> 

 • • • w„ of message, the matrix a^ gives an «-gram of cryptogram 



Ci = ^ aijnij / = \, • ' • , n 



7 = 1 



•> G. S. Vernam, "Cipher Printing Telegraph Systems for Secret Wire and Radio Tele- 

 graphic Communications," Journal American Institute of Electrical Engineers, v. XLV, 

 pp. 109-115, 1926. 



' See L. S. Hill, "Cryptography in an Algebraic Alphabet," American Matli. Monthly, 

 V. 36, No. 6, 1, 1929, pp. 306-312; also "Concerning Certain Linear Transformation 

 Apparatus of Cryptography," v. 38, No. 3, 1931, pp. 135-154. 



