COMMUNICATION THEORY OF SECRECY SYSTEMS 681 



into a given cryptogram E is equal to that of all keys transforming Mj 

 into the same E, for all 1/, , Mj and E. 



Now there must be as many E's as there are M's since, for a fixed i, Ti 

 gives a one-to-one correspondence between all the If' s and some of the E's. 

 For perfect secrecy Pm{E) = P{E) 5^ for any of these £'s and any M. 

 Hence there is at least one key transforming any M into any of these jE's. 

 But all the keys from a fixed M to diflferent £'s must be different, and 

 therefore the number of (liferent keys is at least as great as the number of M's. 

 It is possible to obtain perfect secrecy with only this number of keys, as 



Fig. 5 — Perfect system. 



one shows by the following example: Let the Mi be numbered 1 to « and 

 the Ei the same, and using n keys let 



TiMj = Es 



where s = i -\- j (Mod n). In this case we see that Pe(M) = - = P(E) 



and we have perfect secrecy. x\n example is shown in Fig. 5 with 5 = 

 i -{- j - 1 (Mod 5). 



Perfect systems in which the number of cryptograms, the number of 

 messages, and the number of keys are all equal are characterized by the 

 properties that (1) each M is connected to each E by exactly one line, (2) 

 all keys are equally likely. Thus the matrix representation of the system 

 is a "Latin square." 



In MTC it was shown that information may be conveniently measured 

 by means of entropy. If we have a set of possibilities with probabilities 

 Pi , p2, ' ■ ■ , pn , the entropy H is given by : 



H - -Zpilog/'i. 



