Basic Definitions 237A message such as
LEAVINGTODAY = 1104 00 2108 13 06 19 14 0300 24
is transmitted as
F(ll) F(4) F(0) F(21) F(8) F(13) F(6) F(19) F(14) F(3) F(0) F(24)
The computation is shown in Table 4.3.
Table4.3 Encryption Computation
F(0) = 3(0) + 5 (mod 26) = 5 F(3) = 3(3) + 5 (mod 26) = 14F(4) = 3(4) + 5 (mod 26) = 17 F(6) = 3(6) + 5 (mod 26) = 23
F(8) = 3(8) + 5 (mod 26) = 3 F(11) = 3(11) + 5 (mod 26) = 12
F(13) = 3(13) + 5 (mod 26) = 18 F(14) = 3(14) + 5 (mod 26) = 21
F(19) = 3(19) + 5 (mod 26) = 10 F(21) = 3(21) + 5 (mod 26) = 16
F(24) = 3(24) + 5 (mod 26) = 25
The message that is sent is
12 1705 1603 1823 1021140525The message is transformed into the following string of symbols:
MRFQDSXKVOFZ
The problem for the receiver is to know the inverse function and then apply it to
each of these two digit pairs to see the original message. The inverse for F(letter)
3(lettervalue) + 5 (mod 26) is a function of the same form-that is, G(lettervalue)
a (lettervalue) + b (mod 26) where a and b are determined as follows:
G o F (lettervalue) = a (3 .lettervalue + 5) + b = lettervalue(mod 26)
We solve
3a =-l (mod 26) and 5a + b =- O(mod 26)to get a = 9 and b = 7. The inverse is G(lettervalue) -9(lettervalue) + 7 (mod 26). We
now compose these two functions to decrypt the message as shown:
G o F(L) G o F(E) G o F(A) G o F(V) G o F(I) G o F(N)G o F(G) G o
F(T) G o F(O) G o F(D) G o F(A) G o F(Y)
= G(12) G(17) G(05) G(16) G(03) G(18) G(23) G(10) G(21) G(14)G(05) G(25)= 1104002108 1306 19 14030024
=LEAVINGTODAY
4.1.9 Increasing and Decreasing Functions
The reader has probably already encountered increasing and decreasing functions in a
mathematics course. It is common to speak of a function as being increasing or decreasing
on an interval. The function defined on IR,