1. Beutelspacher, page 18, exercise 1. In addition to the plaintext, state the key (the circumference of the scytale) and provide a brief description of how you solved the problem and/or show your work as to how you derived the solution.

2. Beutelspacher, page 19, exercise 9. Find at least two pairs of words.

3. Decipher the following text which has been enciphered using an additive cipher. What is the key? Also explain how you solved the problem and/or show your work.

   UFIME MPMDW MZPEF ADYKZ USTF
   

4. Decipher the following text which has been enciphered using a multiplicative cipher. What is the key? Also explain how you solved the problem and/or show your work.

   JDKCI TUKHD IVIBJ ILJCD AQFBN IIGCS JABIU KHDIV
   

5. Solve the following equations for x. In each case, find the canonical representation for x, that is, the smallest positive x which makes the equation true.
   • (x+12) mod 5 = 3
   • (x-6) mod 26 = 23
   • (7x-2) mod 26 = 21

   For example, the equation (x+18) mod 10 = 4 is true for x = 6 (6+18 mod 10 = 24 mod 10 = 4), x = 16 (16+18 mod 10 = 34 mod 10 = 4), x = 26, etc. x = 6 is the canonical form because it is the smallest - all of the others are just multiples of 10 added to 6.

6. The message below has been enciphered using an affine cipher with key [8,7]. What is the inverse key? (i.e. the key [s,t] so that if you apply the affine cipher with key [s,t] to the ciphertext, you'll get the original plaintext back) Explain how you found the inverse key - for full credit, you should work out a method for determining the inverse key which can be applied to any affine key, instead of using trial-and-error to guess the inverse key. Use the inverse key and the affine cipher tool to decipher the message (or you can apply the inverse key by hand).

   KXFZM IHEFA MGOGN HMKUN HEZET NUZOX ANHMM XOAQK 
   

   Hint: Try to write an equation which, when applied to a plaintext letter x, describes the result of applying the original key [8,7] and then the inverse key [s,t]. What should be the result of these two operations?

Reminder: The collaboration policy outlines the allowed collaboration on homeworks: you may work in a group, but (a) you must list the names of the people you worked with on your handin, (b) you must fully understand the solution, and (c) the solution must be written up on your own, apart from the group. You are not allowed to simply look up solutions on the Internet.