First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.5

Solution

Use the mapping $A=1,\dots,Z=26$ .

Word: CODE

C=3,\ O=15,\ D=4,\ E=5

Encryption key:

k=2-i

Encryption (multiply by $k$ )

\begin{aligned} C:&\ 3(2-i)=6-3i\\ O:&\ 15(2-i)=30-15i\\ D:&\ 4(2-i)=8-4i\\ E:&\ 5(2-i)=10-5i \end{aligned}

Encrypted values:

(6-3i,\ 30-15i,\ 8-4i,\ 10-5i)

Decryption (multiply by $k^{-1}$ )

\begin{aligned} \frac{1}{2-i}&=\frac{2+i}{(2-i)(2+i)}=\frac{2+i}{5} \end{aligned}

Now:

(6-3i)\cdot\frac{2+i}{5}=3,\quad (30-15i)\cdot\frac{2+i}{5}=15,\quad (8-4i)\cdot\frac{2+i}{5}=4,\quad (10-5i)\cdot\frac{2+i}{5}=5

So the decrypted word is CODE.

\boxed{\text{Encrypted: }(6-3i,30-15i,8-4i,10-5i)\ \to\ \text{Decrypted: CODE}}

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