Question

Encrypt the message DO NOT PASS GO by translating the letters into numbers, applying the given encryption function, and then translating the numbers back into letters.

a) f(p)=(p+3) mod 26(the Caesar cipher)

b) f(p)=(p+13) mod 26

c) f(p)=(3 p+7) mod 26

Short answer

$$\begin{gathered} \mathit{a}\mathbf{)}\mathbf{}\mathit{f}\mathbf{(}\mathit{p}\mathbf{)}\mathbf{=}\mathbf{(}\mathit{p}\mathbf{+}\mathbf{3}\mathbf{)}\mathbf{mod}\mathbf{26}\mathbf{=}\mathbf{}\mathit{G}\mathit{R}\mathit{Q}\mathit{R}\mathit{W}\mathit{S}\mathit{D}\mathit{V}\mathit{V}\mathit{J}\mathit{R}\mathbf{} \\ \mathit{b}\mathbf{)}\mathbf{}\mathit{f}\mathbf{\left(}\mathit{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathit{p}\mathbf{+}\mathbf{13}\mathbf{)}\mathbf{mod}\mathbf{26}\mathbf{=}\mathbf{}\mathit{Q}\mathit{B}\mathit{A}\mathit{B}\mathit{G}\mathit{C}\mathit{N}\mathit{F}\mathit{F}\mathit{T}\mathit{B}\mathbf{} \\ \mathit{c}\mathbf{)}\mathbf{}\mathit{f}\mathbf{(}\mathit{p}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{3}\mathit{p}\mathbf{+}\mathbf{7}\mathbf{)}\mathbf{mod}\mathbf{26}\mathbf{=}\mathbf{}\mathit{Q}\mathit{X}\mathit{U}\mathit{X}\mathit{M}\mathit{A}\mathit{H}\mathit{J}\mathit{J}\mathit{Z}\mathit{X}\mathbf{} \end{gathered}$$

Step-by-step solution

Step 1

DO NOT PASS GO$\mathbf{3}\mathbf{,}\mathbf{14}\mathbf{,}\mathbf{13}\mathbf{,}\mathbf{14}\mathbf{,}\mathbf{19}\mathbf{,}\mathbf{15}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{18}\mathbf{,}\mathbf{18}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{14}$

The Caesar Cipher

$$\begin{gathered} \mathit{A}\mathbf{=}\mathbf{0}\mathit{B}\mathbf{=}\mathbf{1}\mathit{C}\mathbf{=}\mathbf{2}\mathit{D}\mathbf{=}\mathbf{3} \\ \mathit{E}\mathbf{=}\mathbf{4}\mathit{F}\mathbf{=}\mathbf{5}\mathit{G}\mathbf{=}\mathbf{6}\mathit{H}\mathbf{=}\mathbf{71}\mathbf{=}\mathbf{8}\mathbf{J}\mathbf{=}\mathbf{9}\mathbf{K}\mathbf{=}\mathbf{10}\mathbf{L}\mathbf{=}\mathbf{11} \\ \mathit{M}\mathbf{=}\mathbf{12}\mathbf{N}\mathbf{=}\mathbf{13}\mathbf{H}\mathbf{=}\mathbf{14}\mathbf{P}\mathbf{=}\mathbf{15}\mathbf{Q}\mathbf{=}\mathbf{16}\mathbf{R}\mathbf{=}\mathbf{17}\mathbf{S}\mathbf{=}\mathbf{18}\mathbf{T}\mathbf{=}\mathbf{19}\mathbf{U}\mathbf{=}\mathbf{20}\mathbf{V} \\ \mathbf{=}\mathbf{21}\mathbf{W}\mathbf{=}\mathbf{22}\mathbf{X}\mathbf{=}\mathbf{23} \\ \mathbf{Y}\mathbf{=}\mathbf{242}\mathbf{=}\mathbf{25} \end{gathered}$$

Step 2

A) f(p)=(p+3)mod26

DO NOTPASS GO $\mathbf{3}\mathbf{,}\mathbf{14}\mathbf{,}\mathbf{13}\mathbf{,}\mathbf{14}\mathbf{,}\mathbf{19}\mathbf{,}\mathbf{15}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{18}\mathbf{,}\mathbf{18}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{14}$

$$\begin{gathered} \vee \vee \vee \vee \vee \vee \vee \vee \vee \vee \vee \vee \\ \mathbf{1}\mathbf{,}\mathbf{17}\mathbf{,}\mathbf{16}\mathbf{,}\mathbf{17}\mathbf{,}\mathbf{22}\mathbf{,}\mathbf{18}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{21}\mathbf{,}\mathbf{21}\mathbf{,}\mathbf{9}\mathbf{,}\mathbf{17}\mathbf{GRQRWSDVJJR} \end{gathered}$$

Step 3

B) f(p)=(p+13) mod 26

DO NOT PASS GO$\mathbf{3}\mathbf{,}\mathbf{14}\mathbf{,}\mathbf{13}\mathbf{,}\mathbf{14}\mathbf{,}\mathbf{19}\mathbf{,}\mathbf{15}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{18}\mathbf{,}\mathbf{18}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{14}$

$$\begin{gathered} \vee \vee \vee \vee \vee \vee \vee \vee \vee \vee \vee \vee \\ \mathbf{16}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{13}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{19}\mathbf{,}\mathbf{1}\mathbf{QB}\mathbf{}\mathbf{ABGCNFFTB} \end{gathered}$$

Step 4

C) f(p)=(3 p+7) mod 26

DO NOT PASS GO $\mathbf{3}\mathbf{,}\mathbf{14}\mathbf{,}\mathbf{13}\mathbf{,}\mathbf{14}\mathbf{,}\mathbf{19}\mathbf{,}\mathbf{15}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{18}\mathbf{,}\mathbf{18}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{14}$

role="math" localid="1668600557951" $$\begin{gathered} \vee \vee \vee \vee \vee \vee \vee \vee \vee \vee \vee \\ \mathbf{16}\mathbf{,}\mathbf{23}\mathbf{,}\mathbf{20}\mathbf{,}\mathbf{23}\mathbf{,}\mathbf{12}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{7}\mathbf{,}\mathbf{9}\mathbf{,}\mathbf{9}\mathbf{,}\mathbf{25}\mathbf{,}\mathbf{23}\mathbf{QXUXM}\mathbf{}\mathbf{AHJJZX} \end{gathered}$$