Cryptology is the study of making and breaking secret codes. Secret codes are often used to send messages over the Internet. By devising a code that is difficult to break, the sender hopes to prevent the messages from being read by an unauthorized person.
In practice, complicated one-to-one functions and their inverses are used to encode and decode messages. The following procedure uses the simple function $f(x)=2 x-1$ to illustrate the basic concepts that are involved.
Assign to each letter of the alphabet, and a blank space, a two-digit numerical value, as shown below.
$$
\begin{array}{llllllll}
\text { A } & 10 & \text { H } & 17 & \text { O } & 24 & \text { V } & 31 \\
\text { B } & 11 & \text { I } & 18 & \text { P } & 25 & \text { W } & 32 \\
\text { C } & 12 & \text { J } & 19 & \text { Q } & 26 & \text { X } & 33 \\
\text { D } & 13 & \text { K } & 20 & \text { R } & 27 & \text { Y } & 34 \\
\text { E } & 14 & \text { L } & 21 & \text { S } & 28 & \text { Z } & 35 \\
\text { F } & 15 & \text { M } & 22 & \text { T } & 29 & & 36 \\
\text { G } & 16 & \text { N } & 23 & \text { U } & 30
\end{array}
$$
Using these numerical values, the message MEET YOU AT NOON would be represented by
$$
\begin{array}{llllllllllllllll}
22 & 14 & 14 & 29 & 34 & 24 & 30 & 36 & 29 & 23 & 24 & 23
\end{array}
$$
Let $f(x)=2 x-1$ define a coding function. The above message can be encoded by finding $f(22), f(14), f(14), f(29)$, $f(36), f(34), f(24), \ldots, f(23)$, which yields
$$
43272757716747597119577145474745
$$
The inverse of $f$, which is
$$
f^{-1}(x)=\frac{x+1}{2}
$$
is used by the receiver of the message to decode the message. For instance,
$$
f^{-1}(43)=\frac{43+1}{2}=22
$$
which represents $M$, and
$$
f^{-1}(27)=\frac{27+1}{2}=14
$$
which represents E.
a. Use the above coding procedure to encode the message DO YOUR HOMEWORK.
b. Use $f^{-1}(x)$ to decode the message
$$
49334745277133474327
$$
c. Explain why it is important to use a one-to-one function to encode a message.