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1 Exercises 1. Caesar wants to arrange a secret meeting with Marc Antony, either at the Tiber (the river) or at the Coliseum (the arena). He sends

I wanted to know how to work questions 1 thru 12 on the attachment

1 Exercises 1. Caesar wants to arrange a secret meeting with Marc Antony, either at the Tiber (the river) or at the Coliseum (the arena). He sends the ci- phertext EVIRE . However, Antony does not know the key, so he tries all possibilities. Where will he meet Caesar? ( Hint: This is a trick question.) 2. The ciphertext UCR was encrypted using the afne Function 9 x + 2 mod 26. ±ind the plaintext. 3. Encrypt howareyou using the afne Function 5 x + 7 (mod 26). What is the decryption Function? Check that it works. 4. Consider an afne cipher (mod 26). You do a chosen plaintext attack using hahaha . The ciphertext is NONONO . Determine the encryption Function. 5. The Following ciphertext was encrypted by an afne cipher mod 26: CRWWZ. The plaintext starts ha . Decrypt the message. 6. Suppose you encrypt using an afne cipher, then encrypt the encryption using another afne cipher (both are working mod 26). Is there any ad- vantage to doing this, rather than using a single afne cipher? Why or why not? 7. Suppose we work mod 27 instead oF mod 26 For afne ciphers. How many keys are possible? What iF we work mod 29? 8. Suppose that you want to encrypt a message using an afne cipher. You let a = 0 , b = 1 , ..., z = 25, but you also include ? = 26 , ; = 27 , ” = 28 , ! = 29. ThereFore, you use x m→ αx + β (mod 30) For your encryption Function, For some integers α and β . (a) Show that there are exactly eight possible choices For the integer α (that is, there are only eight choices oF α (with 0 < α < 30) that allow you to decrypt). (b) Suppose you try to use α = 10 , β = 0. ±ind two plaintext letters that encrypt to the same ciphertext letter. 9. You want to carry out an afne encryption using the Function αx + β , but you have gcd( α, 26) = d > 1. Show that iF x 1 = x 2 + (26 /d ), then αx 1 + β αx 2 + β (mod 26). This shows that you will not be able to decrypt uniquely in this case. 10. Suppose there is a language that has only the letters a and b . The Fre- quency oF the letter a is .1 and the Frequency oF b is .9. A message is encrypted using a Vigen` ere cipher (working mod 2 instead oF mod 26). The ciphertext is BABABAAABA. 1
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(a) Show that the key length is probably 2. (b) Using the information on the frequencies of the letters, determine the key and decrypt the message. 11. Suppose you have a language with only the 3 letters a, b, c , and they occur with frequencies .7, .2, .1, respectively. The following ciphertext was encrypted by the Vigen` ere method (shifts are mod 3 instead of mod 26, of course): ABCBABBBAC. Suppose you are told that the key length is 1, 2, or 3. Show that the key length is probably 2, and determine the most probable key. 12. If v and w are two vectors in n -dimensional space, v · w = | v || w | cos θ , where θ is the angle between the two vectors (measured in the two- dimensional plane spanned by the two vectors), and | v | denotes the length of v . Use this fact to show that, in the notation of Section 2.3, the dot product A 0 · A i is largest when i = 0. 13. The ciphertext YIFZMA was encrypted by a Hill cipher with matrix p 9 13 2 3 P . Find the plaintext. 14. The ciphertext text GEZXDS was encrypted by a Hill cipher with a 2 × 2 matrix. The plaintext is solved . Find the encryption matrix M . 15. Eve captures Bob’s Hill cipher machine, which uses a 2-by-2 matrix M mod 26. She tries a chosen plaintext attack. She ±nds that the plaintext ba encrypts to HC and the plaintext zz encrypts to GT . What is the matrix M . 16. (a) The ciphertext text ELNI was encrypted by a Hill cipher with a 2 × 2 matrix. The plaintext is dont . Find the encryption matrix. (b) Suppose the ciphertext is ELNK and the plaintext is still dont . Find the encryption matrix. Note that the second column of the matrix is changed. This shows that the entire second column of the encryption matrix is involved in obtaining the last character of the ciphertext (see the end of Section 2.7). 17. Suppose the matrix p 1 2 3 4 P is used for an encryption matrix in a Hill cipher. Find two plaintexts that encrypt to the same ciphertext. 18. Let a,b,c,d,e,f be integers mod 26. Consider the following combination of the Hill and a²ne ciphers: Represent a block of plaintext as a pair ( x,y ) mod 26. The corresponding ciphertext ( u,v ) is ( x y ) p a b c d P + ( e f ) ( u v ) (mod 26) . 2
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