## Exercises2.11Exercises Set 4

###### 1.

Finish all investigations from in-class activities.

###### 2.

The following humorous excuses for failing to complete a mathematics homework have been encrypted with a keyword columnar cipher and given keyword. Decrypt them.

1. Keyword: FLAMES

Ciphertext: TFJJT RYUHF DDQRT WTRYR BQXYV MFURN QGFGY VBPVE HTUHM LDFNY E

2. Keyword: SMALL

Ciphertext: CMSEP WMPDN AAUBL WWMPN PCRZW NAAVW AINCW PCWCZ WMCRV SNYCZ

3. Keyword: BASEBALL

Ciphertext: JHBMH BRIDJ KARDU HQGSO MUGJU MBKOA QRRJU OXWRG TJKAR QWGQL URDBR JRIQK LUGAU O

###### 3.

The keyword CRAZY was used to construct a keyword columnar cipher

PT: ABCDEFGHIJKLMNOPQRSTUVWXYZ

CT: CBHMSXRDINTAEJOUZFKPVYGLQW

The alphabetic string HIJKL is not in the keyword. In the ciphertext arrangement, How far apart are the letters H to I, I to J, J to K, and K to L? How is this related to the length of the keyword?

###### 4.

The keyword STRING was used to construct a keyword columnar cipher

PT: ABCDEFGHIJKLMNOPQRSTUVWXYZ

CT: SAHPYTBJQZRCKUIDLVNEMWGFOX

The alphabetic string ABCDEF is not in the keyword. In the ciphertext arrangement, How far apart are the letters A to B, B to C, C to D, D to E, and E to F? How is this related to the length of the keyword?

###### 5.

An unknown keyword was used to create the following in a keyword columnar cipher.

PT: ABCDEFGHIJKLMNOPQRSTUVWXYZ

CT: ABKVPDLWREMXIFNYCGQZOHSTJU

Using alphabetical strings what can you say about the length of the keyword. Bonus: Find the keyword.

###### 6.

The following message was encrypted with a keyword columnar monoalphabetic cipher. Decrypt it and find the keyword. Hint: the text contains the word INTERESTING.

ZD, CW CS I FQOT CZWQOQSWCZY ZAVBQO; CW CS WNQ SVIRRQSW ZAVBQO QPEOQSSCBRQ IS I SAV DU WKD GABQS CZ WKD LCUUQOQZW KITS, WNQ WKD KITS BQCZY DZQ GABQL ERAS WKQRFQ GABQL IZL ZCZQ GABQL ERAS WQZ GABQL. --SOCZCFISI OIVIZAHIZ

###### 7.

The following exercises are taken from Colonel Tiltman's Signal's Intelligence course written in the early 1900's. John Tiltman, who eventually became Brigadier John Tiltman was a great British cryptanalyst working for Government Communications Headquarters (GCHQ, Great Britain’s counterpart to NSA). They are monoalphabetic ciphers, but each letter is replaced by a number of unspecified number of digits. Solve them!

1. Exercise 5. Solve the following cipher text which has been enciphered by the method of “simple alphabet substitution.” The clear text starts with the word “possession.” (N.B.-The fact that the text is divided into 5 figure groups has nothing to do with the “unit of substitution.” Messages sent by cable or W /T are most frequently transmitted in 5 letter or 5 figure groups.)

28827 73213 21167 32132 12112 77266 00021 13210 00343 32134 31232 44244 38700 02662 11266 16700 02882 77211 26633 23210 00277 17800 03322 00167 00024 41233 65000

2. Exercise 6. The following cipher text represents the same clear text as in Exercise 5. Find the method of encipherment.

18172 12172 12111 17160 11210 23212 33141 42701 61116 70181 71116 22210 17802 21070 14325

###### 8.

The ciphertext below was encrypted with an affine cipher. Decrypt it. You do not need to write the entire plain text, just the first few words is enough. You should however show all calculations you made and explain how you deciphered the message.

OFZGT PGYBE PGYBA EGVPM CMPMV YBBGT EWNGT GNGWO EYBLY PGVPM CYLGO EXGGK NMBGV OSMBN YEWOS PMMBG MRPNM EGVYB BGTEN GPMFV NYEKN OARRG ATWNM FMMUG VGHOK PFSFY UGNYQ PNOPN GWOEV GOVPY TGVMR CYLYB CPNGE OQGEX GGKNV YBBGT ORPGT VYBBG TWGFF EOYVP NGKNO RRGAT YLGCM POCMM VYVGO WNSVM BPYCY LGPNG EXGGK NEYBK GYLGN GOTVY PEMQO BSPYQ GEEMO FZGTP EKNOA RRGAT COLGP NGEXG GKNXG TRGKP FSOBV GLGBO BEWGT GVORG WIAGE PYMBE PNGBO XTMRG EEMTE PMMVA XOBVO EUGVN YQOTG OFFSP MACNI AGEPY MBOZM APOBP YQOPP GTWNY KNPNG KNOAR RGATK MAFVB POBEW GTEYT PNGOB EWGTP MSMAT IAGEP YMBYE EMGOE SPNOP YFFFG PQSKN OARRG ATOBE WGTYP H

###### 9.

The ciphertext below was encrypted with an affine cipher. Decrypt it. You do not need to write the entire plain text, just the first few words is enough. You should however show all calculations you made and explain how you deciphered the message.

UXQDN LMKDN LMVHB VBMKD TJQCS BWDBM ZLNDV MVBNJ QDLWB JQWVY ZMKLY HKDBB NJQDJ OLZLN WCDMK LYUJR DQLYS CLBMB CJYZD QMKLY NJYJU JCPVM BOQDD VMHLY WDUCL PDSLY PTKDQ DLQHK VNDSD BSVSV MVYLW LMKMX WQVHK LQSGM QXSDL X

###### 10.

Find the greatest common divisors for each of the following:

1. gcd(12,16)

2. gcd(6,21)

3. gcd(5,4)

4. gcd(10,25)

5. gcd(100,80)

###### 11.

The function $\phi(n)$ computes the number of integers, $a\text{,}$ such that $1 \leq a \leq n$ and $\gcd(a,n)=1\text{.}$ Compute $\phi(n)$ for the numbers below.

1. $\displaystyle \phi(14)$

2. $\displaystyle \phi(20)$

3. $\displaystyle \phi(21)$

4. $\displaystyle \phi(22)$

5. $\displaystyle \phi(24)$

###### 12.

Suppose we have a language with only 14 letters. How many affine ciphers would there be?

###### 13.

Suppose we have a language with only 20 letters. How many affine ciphers would there be?

###### 14.

The function $\phi(n)$ computes the number of integers, $a\text{,}$ such that $1 \leq a \leq n$ and $\gcd(a,n)=1\text{.}$ Use $\phi(n)$ to construct a formula for the number of affine ciphers in a language with $n$ letters.

###### 15.

Compute $\phi(n)$ for the numbers below.

1. $\displaystyle \phi(5)$

2. $\displaystyle \phi(7)$

3. $\displaystyle \phi(13)$

4. $\displaystyle \phi(17)$

5. $\displaystyle \phi(19)$

6. Make a conjecture and pick at least one more value with which to test your conjecture.

7. Explain why your conjecture is true.

###### 16.

Compute $\phi(n)$ for the numbers below.

1. $\displaystyle \phi(2)$

2. $\displaystyle \phi(4)$

3. $\displaystyle \phi(8)$

4. $\displaystyle \phi(16)$

5. $\displaystyle \phi(32)$

6. Make a conjecture and pick at least one more value with which to test your conjecture.

7. Explain why your conjecture is true.

###### 17.

Compute $\phi(n)$ for the numbers below.

1. $\displaystyle \phi(3)$

2. $\displaystyle \phi(9)$

3. $\displaystyle \phi(27)$

4. $\displaystyle \phi(81)$

5. Make a conjecture and pick at least one more value with which to test your conjecture.

6. Explain why your conjecture is true.

###### 18.

Converse and Contrapositive. Consider the mathematical statement: If $a$ is even then $6a$ is even.

1. Verify the statement above is true using the definition of even. (multiple of 2)

2. State the converse of the statement.

3. Is the converse true? Either verify it using the definition of even or give a counterexample to show that it is false.

4. State the contrapositive of the mathematical statement. Explain why it is true.

###### 19.

Converse and Contrapositive. Consider the mathematical statement: If $a$ is even then $3a$ is even.

1. Verify the statement above is true using the definition of even. (multiple of 2)

2. State the converse of the statement.

3. State the contrapositive of the converse. Is it true? Either verify it or give a counterexample to show that it is false.

###### 20.

More Converse and Contrapositive. Consider the mathematical statement: If $a$ is odd then $6a$ is odd.

1. Determine if the statement above is true or false. If true, verify it using the definition of odd. If false, give a counterexample.

2. State the converse of the statement.

3. Is the converse true? Either verify it using the definition of odd or give a counterexample to show that it is false.

4. State the contrapositive of the mathematical statement. Explain whether it is true or false.

###### 21.

More Converse and Contrapositive. Consider the mathematical statement: If $a$ is odd then $3a$ is odd.

1. Determine if the statement above is true or false. If true, verify it using the definition of odd. If false, give a counterexample.

2. State the converse of the statement.

3. Is the converse true? Either verify it using the definition of odd or give a counterexample to show that it is false.

4. State the contrapositive of the mathematical statement. Explain whether it is true or false.

###### 22.

Watch the movie “The man who knew infinity” and write a paragraph about how the ideas of mathematical proof was important to the movie.

###### 23.

Read the memoirs of Elizebeth Friedman available at link to Marshall Foundation Write a paragraph about what you learned about Riverbank Laboratories and how Elizebeth got started working in the field of cryptology.