First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.5

Solution

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

Word: QUIZ

Q=17,\ U=21,\ I=9,\ Z=26

Encryption key:

k=3-3i

Encryption (multiply by $k$ )

\begin{aligned} Q:&\ 17(3-3i)=51-51i\\ U:&\ 21(3-3i)=63-63i\\ I:&\ 9(3-3i)=27-27i\\ Z:&\ 26(3-3i)=78-78i \end{aligned}

Encrypted values:

(51-51i,\ 63-63i,\ 27-27i,\ 78-78i)

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

\begin{aligned} \frac{1}{3-3i}&=\frac{3+3i}{(3-3i)(3+3i)}=\frac{3+3i}{18}=\frac{1+i}{6} \end{aligned}

Now:

(51-51i)\cdot\frac{1+i}{6}=17,\quad (63-63i)\cdot\frac{1+i}{6}=21,\quad (27-27i)\cdot\frac{1+i}{6}=9,\quad (78-78i)\cdot\frac{1+i}{6}=26

So the decrypted word is QUIZ.

\boxed{\text{Encrypted: }(51-51i,63-63i,27-27i,78-78i)\ \to\ \text{Decrypted: QUIZ}}

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