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# 18.16: Cryptography

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1. $$\mathrm{ZLU ~ KZB ~ WWS ~ PLZ}$$ 3. $$\mathrm{SHRED ~ EVIDENCE}$$

5. $$\mathrm{O2H ~ DO5 ~ HDV}$$ 7. $$\mathrm{MERGER ~ ON}$$

9. $$\mathrm{MNI ~ YNE ~ TBA ~ AEH ~ RTA ~ TEA ~ TAI ~ LRE ~ A}$$

11. $$\mathrm{THE ~ STASH ~ IS ~ HIDDEN ~ AT ~ MARVINS ~ QNS}$$

13. $$\mathrm{UEM ~ IYN ~ IOB ~ WYL ~ TTL ~ N}$$

15. $$\mathrm{HIRE ~ THIRTY ~ NEW ~ EMPLOYEES ~ MONDAY}$$

17. $$\mathrm{ZMW ~ NDG ~ CDA ~ YVK}$$

19. a) $$3$$ b) $$0$$ c) $$4$$

21. We test out all $$n$$ from 1 to 10

$$\begin{array}{|r|r|r|} \hline \mathrm{n} & 4^{\mathrm{n}} & 4^{\mathrm{n}} \bmod 11 \\ \hline 1 & 4 & 4 \\ \hline 2 & 16 & 5 \\ \hline 3 & 64 & 9 \\ \hline 4 & 256 & 3 \\ \hline 5 & 1024 & 1 \\ \hline 6 & 4096 & 4 \\ \hline 7 & 16384 & 5 \\ \hline 8 & 65536 & 9 \\ \hline 9 & 262144 & 3 \\ \hline 10 & 1048576 & 1 \\ \hline \end{array}$$

Since we have repeats, and not all values from 1 to 10 are produced (for example, there is no $$\left.n \text { is } 4^{n} \bmod 11=7\right)$$, 4 is not a generator $$\bmod 11$$.

23. $$157^{10} \bmod 5=(157 \bmod 5)^{10} \bmod 5=2^{10} \bmod 5=1024 \bmod 5=4$$

25. $$3^{7} \bmod 23=2$$

27. Bob would send $$5^{7}$$ mod $$33=14$$. Alice would decrypt it as $$14^{3} \bmod 33=5$$

31.

a. $$67^{8} \bmod 83=\left(67^{4} \bmod 83\right)^{2} \bmod 83=49^{2} \bmod 83=2401 \bmod 83=77$$

$$67^{16} \bmod 83=\left(67^{8} \bmod 83\right)^{2} \bmod 83=77^{2} \bmod 83=5929 \bmod 83=36$$

b. $$17000 \bmod 83=(100 \bmod 83)^{*}(170 \bmod 83) \bmod 83=(17)(4) \bmod 83=68$$

c. $$67^{5} \bmod 83=\left(67^{4} \bmod 83\right)(67 \bmod 83) \bmod 83=(49)(67) \bmod 83=3283 \bmod 83=46$$

d. $$67^{7} \bmod 83=\left(67^{4} \bmod 83\right)\left(67^{2} \bmod 83\right)(67 \bmod 83) \bmod 83=(49)(7)(67) \bmod 83=22981 \bmod 83=73$$

e. $$67^{24}=67^{16} 67^{8}$$ so $$67^{24} \bmod 83=\left(67^{16} \bmod 83\right)\left(67^{8} \bmod 83\right) \bmod 83=(77)(36) \bmod 83=2272 \bmod 83 = 33$$

18.16: Cryptography is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.