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Mathematics LibreTexts

18.16: Cryptography

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    41810
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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.

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