COMMUNICATION THEORY OF SECRECY SYSTEMS 



693 



— if D is measured in bits per letter. This hcliavior is shown in Fig. 7, to- 

 gether with the approximating curves. 



By a similar argument the equivocation of message can be calculated. 

 It is 



He{M) = RoX for RoN « He{K) 

 He(M) = He(K) for RoN » He{K) 

 He{M) = He(K) - cp(N) for RoN ^ He{K) 



where <p(N) is the function shown in Fig. 7 with .V scale reduced by factor 



of ^ . Thus, IIe{M) rises linearly with slope Rq , until it nearly intersects 

 Ro 



H (K)»l 



H(K) 

 ND(DIGITS) 



Fig. 7 — Equivocation for random cipher 



HtK>2 



the He(K) line. After a rounded transition it follows the He(K) curve down. 

 It will be seen from Fig. 7 that the equivocation curves approach zero 

 rather sharply. Thus we may, with but little ambiguity, speak of a point at 

 which the solution becomes unique. This number of letters will be called 

 the unicity distance. For the random cipher it is approximately H{K)/D. 



15. Application to Standard Ciphers 



Most of the standard ciphers involve rather complicated enciphering and 

 deciphering operations. Furthermore, the statistical structure of natural 

 languages is extremely involved. It is therefore reasonable to assume that 

 the formulas derived for the random cipher may be applied in such cases. 

 It is necessary, however, to apply certain corrections in some cases. The 

 main points to be observed are the following: 



