## Exercises6.2Exercises Set 13

###### 1.

Finish all investigations from in-class activities.

###### 2.

Suppose Alice's RSA modulus is $n=17512109\text{,}$ her public encryption exponent is $e=52627\text{,}$ and her secret decryption exponent is $d=1663\text{.}$

1. Alice wants to sign the message $M=3276872\text{.}$ Calculate her signature.

2. Bob receives the unencrypted message signature pair $(M,\sigma)=(2813786, 8184081)\text{.}$ Does he regard the pair as likely to be an authentic message from Alice?

3. What about the unencrypted message signature pair $(M,\sigma)=(2813787,8184083)?$

###### 3.

Use the keyword WARBLERS to construct a playfair square.

1. Encrypt the message FUN HOMEWORK

2. Decrypt the message LW PS ID PX GS AC

###### 4.

Use the keyword DEATH to construct a playfair square.

1. Encrypt the message GOOD MOVIE

2. Decrypt the message ME IK QO TX CQ TE ZX CO MW QC TE HN FB IK ME HA KR QC UN GI KM AV

3. Bonus: In what movie, does this message appear?

###### 5.

Decode UT VM HT MP QA HK KU GV OW which was encrypted with a Playfair square. The keyword is the answer to the ‘riddle’: What does RSA stand for in the RSA cryptosystem?

###### 6.

Decrypt the message GN YO DO NM BI that was encrypted with the Playfair square below.

 I H G F E J U T S D K V Y R C L W X Q B M N O P A

###### 7.

Warm-up for next block cipher. Recall how to multiply 2x2 matrices: $\left( \begin{matrix} a \amp b \\ c \amp d\end{matrix} \right) \left( \begin{matrix} r \amp s\\ t \amp u \end{matrix} \right) = \left( \begin{matrix}ar+bt \amp as+bu \\ cr+dt \amp cs+du \end{matrix} \right)$

Multiply the following matrices.

1. $\displaystyle \left( \begin{matrix} 1 \amp 3 \\ 4 \amp 5\end{matrix} \right) \left( \begin{matrix} 2 \amp 9\\ 2 \amp 8 \end{matrix} \right)$

2. $\displaystyle \left( \begin{matrix} 3 \amp 9 \\ 11 \amp 12\end{matrix} \right) \left( \begin{matrix} 5 \amp 6\\ 7 \amp 8 \end{matrix} \right)$

3. $\left( \begin{matrix} 1 \amp 3 \\ 4 \amp 5\end{matrix} \right) \left( \begin{matrix} 2 \amp 9\\ 2 \amp 8 \end{matrix} \right)$ (mod 26)

4. $\left( \begin{matrix} 3 \amp 9 \\ 11 \amp 12\end{matrix} \right) \left( \begin{matrix} 5 \amp 6\\ 7 \amp 8 \end{matrix} \right)$ (mod 26)

###### 8.

A Playfair square was used to encrypt a message, you know the following relationships between plain text and cipher text (plain text is on the left and cipher text is on the right). Find the Playfair square.

 plaintext ciphertext MA NG YV RW CA YO FT KY EN FM DA FO UT ZC IW FX PA YH MT SR QU LX TA YB XT ZP PH HI
###### 9.

The ciphertext below has been encrypted with an unknown Playfair square. You do know that it contains the phrase BEWARE THE SPIDER WEB. Find where it is located. Can you decrypt it? AS KD MR IE SV RK CO GS HW GR BD VS IQ PY LM GK SB QK TV VT QZ

###### 10.

The messages below were encrypted with a Playfair square with unknown keyword. (The same square for all four of the messages below.)

1. Each cipher text contains the phrase THE WEATHER in the plaintext. Find where it is in the ciphertext. Note: it must occur as TH EW EA TH ER or _T HE WE AT HE R_. Remember that a letter cannot get encrypted as itself in the Playfair cipher.

of oa le nh qm bo ud yr bo af cf om nd hq qf or bo kd dq kn hn vk od rq ry mr cl fd un iz yr hf fc rs yo gf oa gi ca bd fd bn cf yv ib kn sb oc ek cm yf

hr sz km qr ht mf yn hq du nm hq fm ef mf lc qr km ef mf sy yt cf id ft my cf ay ms yx rc ov

xg fm nu fm td wb fd et yr mo yz ct ho hq du nm hq fm cv vy rp im cf pg ex tf vd kz fl ib dn ca hq fd rg ny dc nk th bo hn qv kn bd ck em he eq

xg om bo mr bo ud yr bo mi ne cf du bo hq fm hq du nm hq fm km df yx bo hq fm hq du nm hq fm km hn pn xg om bo mr bo ud yr bo mi ob th ud qv pv nm hq fm hq du nm hq fm xg yr nu fm hq du bo hq fm xg om bo yz nk bs fb ht af th

2. Each of the ciphertexts above contains one of the phrases below in its plaintext. Find which phrase belongs in which ciphertext.

THE WEATHER NINETENTHS

WHEREVER

WHETHER THE WEATHER

3. Find the playfair square and decrypt the messages.

###### 11.

Prove by induction that $1^3 +2^3 + \cdots n^3 = \left( \frac{n(n+1)}{2} \right)^2\text{.}$

###### 12.

Prove by induction that 3 divides $4^n-1$ for all positive integers $n\text{.}$