Question

Encrypt the message WATCH YOUR STEP by translating the letters into numbers, applying the given encryption function, and then translating the numbers back into letters.

a) f(p)=(p+14) mod 26

b) f(p)=(14 p+21) mod 26

c) f(p)=(-7 p+1) mod 26

Short answer

$$\begin{gathered} \mathit{a}\mathbf{)}\mathbf{}\mathit{f}\mathbf{(}\mathit{p}\mathbf{)}\mathbf{=}\mathbf{(}\mathit{p}\mathbf{+}\mathbf{14}\mathbf{)}\mathbf{mod}\mathbf{26}\mathbf{=}\mathbf{}\mathbf{KOHQVMCIFGHSD}\mathbf{} \\ \mathit{b}\mathbf{)}\mathbf{}\mathit{f}\mathbf{\left(}\mathit{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{14}\mathit{p}\mathbf{+}\mathbf{21}\mathbf{)}\mathbf{mod}\mathbf{26}\mathbf{=}\mathbf{}\mathbf{RVBXPTJPZNBZX}\mathbf{} \\ \mathit{c}\mathbf{)}\mathbf{}\mathit{f}\mathbf{(}\mathit{p}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{7}\mathit{p}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{mod}\mathbf{26}\mathbf{=}\mathbf{}\mathbf{DBYNEPHRMFYZA} \end{gathered}$$

Step-by-step solution

Step 1

Given message: WATCH YOUR STEP

$$\begin{gathered} \mathbf{Let}\mathbf{}\mathit{A}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{B}\mathbf{=}\mathbf{1}\mathbf{,}\mathit{C}\mathbf{=}\mathbf{2}\mathbf{,}\mathit{D}\mathbf{=}\mathbf{3}\mathbf{,}\mathit{E}\mathbf{=}\mathbf{4}\mathbf{,}\mathit{F}\mathbf{=}\mathbf{5}\mathbf{,}\mathit{G}\mathbf{=}\mathbf{6}\mathbf{,}\mathit{H}\mathbf{=}\mathbf{7}\mathbf{,}\mathit{I}\mathbf{=}\mathbf{8}\mathbf{,}\mathit{J}\mathbf{=}\mathbf{9}\mathbf{,}\mathit{K}\mathbf{=}\mathbf{10}\mathbf{,}\mathit{L}\mathbf{=}\mathbf{11}\mathbf{,}\mathit{M}\mathbf{=}\mathbf{12}\mathbf{,}\mathit{N}\mathbf{=}\mathbf{13}\mathbf{,} \\ \mathit{O}\mathbf{=}\mathbf{14}\mathbf{,}\mathit{P}\mathbf{=}\mathbf{15}\mathbf{,}\mathit{Q}\mathbf{=}\mathbf{16}\mathbf{,}\mathit{R}\mathbf{=}\mathbf{17}\mathbf{,}\mathit{S}\mathbf{=}\mathbf{18}\mathbf{,}\mathit{T}\mathbf{=}\mathbf{19}\mathbf{,}\mathit{U}\mathbf{=}\mathbf{20}\mathbf{,}\mathit{V}\mathbf{=}\mathbf{21}\mathbf{,}\mathit{W}\mathbf{=}\mathbf{22}\mathbf{,}\mathit{X}\mathbf{=}\mathbf{23}\mathbf{,}\mathit{Y}\mathbf{=}\mathbf{24}\mathbf{,}\mathit{Z}\mathbf{=}\mathbf{25} \\ \mathbf{2201927241420171819415} \end{gathered}$$

Step 2

(a)

f(p)=(p+14) mod 26

Let us apply the function to$\mathbf{2201927241420171819415}$

$\mathbf{101471621}\mathbf{}\mathbf{12285}\mathbf{}\mathbf{67183}$

$$\begin{gathered} \mathbf{Let}\mathbf{}\mathit{A}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{B}\mathbf{=}\mathbf{1}\mathbf{,}\mathit{C}\mathbf{=}\mathbf{2}\mathbf{,}\mathit{D}\mathbf{=}\mathbf{3}\mathbf{,}\mathit{E}\mathbf{=}\mathbf{4}\mathbf{,}\mathit{F}\mathbf{=}\mathbf{5}\mathbf{,}\mathit{G}\mathbf{=}\mathbf{6}\mathbf{,}\mathit{H}\mathbf{=}\mathbf{7}\mathbf{,}\mathit{I}\mathbf{=}\mathbf{8}\mathbf{,}\mathit{J}\mathbf{=}\mathbf{9}\mathbf{,}\mathit{K}\mathbf{=}\mathbf{10}\mathbf{,}\mathit{L}\mathbf{=}\mathbf{11}\mathbf{,}\mathit{M}\mathbf{=}\mathbf{12}\mathbf{,}\mathit{N}\mathbf{=}\mathbf{13}\mathbf{,} \\ \mathit{O}\mathbf{=}\mathbf{14}\mathbf{,}\mathit{P}\mathbf{=}\mathbf{15}\mathbf{,}\mathit{Q}\mathbf{=}\mathbf{16}\mathbf{,}\mathit{R}\mathbf{=}\mathbf{17}\mathbf{,}\mathit{S}\mathbf{=}\mathbf{18}\mathbf{,}\mathit{T}\mathbf{=}\mathbf{19}\mathbf{,}\mathit{U}\mathbf{=}\mathbf{20}\mathbf{,}\mathit{V}\mathbf{=}\mathbf{21}\mathbf{,}\mathit{W}\mathbf{=}\mathbf{22}\mathbf{,}\mathit{X}\mathbf{=}\mathbf{23}\mathbf{,}\mathit{Y}\mathbf{=}\mathbf{24}\mathbf{,}\mathit{Z}\mathbf{=}\mathbf{25} \end{gathered}$$

KOHQVMCIFGHSD

Step 3

(b)

f(p)=(14p+21) mod 26

Let us apply the function to$\mathbf{2201927241420171819415}$

$$\begin{gathered} \mathbf{Let}\mathbf{}\mathit{A}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{B}\mathbf{=}\mathbf{1}\mathbf{,}\mathit{C}\mathbf{=}\mathbf{2}\mathbf{,}\mathit{D}\mathbf{=}\mathbf{3}\mathbf{,}\mathit{E}\mathbf{=}\mathbf{4}\mathbf{,}\mathit{F}\mathbf{=}\mathbf{5}\mathbf{,}\mathit{G}\mathbf{=}\mathbf{6}\mathbf{,}\mathit{H}\mathbf{=}\mathbf{7}\mathbf{,}\mathit{I}\mathbf{=}\mathbf{8}\mathbf{,}\mathit{J}\mathbf{=}\mathbf{9}\mathbf{,}\mathit{K}\mathbf{=}\mathbf{10}\mathbf{,}\mathit{L}\mathbf{=}\mathbf{11}\mathbf{,}\mathit{M}\mathbf{=}\mathbf{12}\mathbf{,}\mathit{N}\mathbf{=}\mathbf{13}\mathbf{,} \\ \mathit{O}\mathbf{=}\mathbf{14}\mathbf{,}\mathit{P}\mathbf{=}\mathbf{15}\mathbf{,}\mathit{Q}\mathbf{=}\mathbf{16}\mathbf{,}\mathit{R}\mathbf{=}\mathbf{17}\mathbf{,}\mathit{S}\mathbf{=}\mathbf{18}\mathbf{,}\mathit{T}\mathbf{=}\mathbf{19}\mathbf{,}\mathit{U}\mathbf{=}\mathbf{20}\mathbf{,}\mathit{V}\mathbf{=}\mathbf{21}\mathbf{,}\mathit{W}\mathbf{=}\mathbf{22}\mathbf{,}\mathit{X}\mathbf{=}\mathbf{23}\mathbf{,}\mathit{Y}\mathbf{=}\mathbf{24}\mathbf{,}\mathit{Z}\mathbf{=}\mathbf{25} \end{gathered}$$

RVBXP TJPZ NBZX

Step 4

(c)

f(p)=(-7 p+1) mod 26

Let us apply the function to $\mathbf{2201927241420171819415}$

3124134 1571712 524250

$$\begin{gathered} \mathbf{Let}\mathbf{}\mathit{A}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{B}\mathbf{=}\mathbf{1}\mathbf{,}\mathit{C}\mathbf{=}\mathbf{2}\mathbf{,}\mathit{D}\mathbf{=}\mathbf{3}\mathbf{,}\mathit{E}\mathbf{=}\mathbf{4}\mathbf{,}\mathit{F}\mathbf{=}\mathbf{5}\mathbf{,}\mathit{G}\mathbf{=}\mathbf{6}\mathbf{,}\mathit{H}\mathbf{=}\mathbf{7}\mathbf{,}\mathit{I}\mathbf{=}\mathbf{8}\mathbf{,}\mathit{J}\mathbf{=}\mathbf{9}\mathbf{,}\mathit{K}\mathbf{=}\mathbf{10}\mathbf{,}\mathit{L}\mathbf{=}\mathbf{11}\mathbf{,}\mathit{M}\mathbf{=}\mathbf{12}\mathbf{,}\mathit{N}\mathbf{=}\mathbf{13}\mathbf{,} \\ \mathit{O}\mathbf{=}\mathbf{14}\mathbf{,}\mathit{P}\mathbf{=}\mathbf{15}\mathbf{,}\mathit{Q}\mathbf{=}\mathbf{16}\mathbf{,}\mathit{R}\mathbf{=}\mathbf{17}\mathbf{,}\mathit{S}\mathbf{=}\mathbf{18}\mathbf{,}\mathit{T}\mathbf{=}\mathbf{19}\mathbf{,}\mathit{U}\mathbf{=}\mathbf{20}\mathbf{,}\mathit{V}\mathbf{=}\mathbf{21}\mathbf{,}\mathit{W}\mathbf{=}\mathbf{22}\mathbf{,}\mathit{X}\mathbf{=}\mathbf{23}\mathbf{,}\mathit{Y}\mathbf{=}\mathbf{24}\mathbf{,}\mathit{Z}\mathbf{=}\mathbf{25} \end{gathered}$$

DBYNEPHRMFYZA