I \\ I I for modulus 28: (p (28) = 12 ; so there are 12 different multiplicative keys availabie I for modulus 29: go (29) = 28 ; so there are 28 different multiplicative keys available I for modulus 30: go (30) = 8 ; so there are 8 different multiplicative keys available Generally, the number of different possible key values is less when the modulus has more factors. That's why increasing to mod 29 makes a big difference: because 29 is a prime number meaning it will be relatively prime to all positive integers less that 29, providing a total of 28 different key options. An issue with increasing the modulus value is that the modulus value is connected with the number of characters in the ciphertext: if you use modulus 29, then your ciphertext needs 29 characters. Since the English alphabet only has 26 characters, using a modulus value of 29 requires having three additional characters in the ciphertext which might look something like A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,@,#,$. in fact, those extra characters can also be added to the plaintext character set. Unfortunately, having to nd additional characters for larger modulus values gets messy and complicated. Just think of how many characters would need to added to use the modulus value of 517 '. For this reason, the ciphertext created by modern electronic cipher systems use numbers instead of characters. For example, the plaintext "HELLO" in a substitution cipher system that uses a multiplicative key value of 3 would become the ciphertext "24 15 36 36 45\" which allows the modulus value to be increased without having to worry about guring out additional characters. In fact, most ciphers in use today E+> use modulus values that are the result of multiplying two very large (100+ digit) prime numbers. Questions to Answer instructions: Answer the questions by hand-writing on a piece of paper the work you did to come up with your answer. Then, upload pictures of the papers with your hand-written work. Your grade will be based primarily on the work showing how you got to your answer; providing just the correct answer will receive a grade of at most 50% 1. How many different multiplicative key values are available when using a modulus 32? Verify your answer by computing (,0 (32) 2. The modulus value of 37 is a good choice because 37 is prime. But using 37 as the modulus requires an additional 11 characters. List the 11 additional charactIJrs that you think would be MOST USEFUL to have for writing plaintext. Briey explain how and why you chose those specic 11 characters