Suppose we represent the letters 'a' though 'z' by 1 through 26and a space by 0.

a) To encrypt messages I propose the formulaC =(18P + 7)mod 27, where P is the "plain text" (the original letter value) and C is the "cipher text" (the encrypted letter value).For example, if P = 2(the letter 'b' ),C would be 16 (the letter 'p') since (18(2) + 7) mod 27 = 16.There is a problem though: When I send the message 'b' to my friend, encrypted as 'p', they don't know whether the original message was 'b' or another letter that also encrypts to 'p'.What other letter(s) would also encrypt to 'p' besides 'b' in this system?Hint: Consider the congruence 18x + 7 16(mod 27).(Recall, a congruence ax b (mod m), may be simplified by dividing all three numbers by their gcd. After that, solve in the usual manner by multiplying each side of the congruence by a modular inverse of the coefficient for x.)

b) We decide to use a different formula for encryption:C =(16P + 7) mod 27.This time everything works perfectly. When I send an encrypted message to my friend, they always know exactly what the original letters were. Carefully explain why the formula C =(16P + 7) mod 27 works better than the formula C = (18P + 7)mod 27.

c)You intercept my encryption keyC = (16P + 7) mod 27 and you want to figure out the decryption key that will be used to decode messages.That means something of the form P = ________________, where anexpression involvingC goes in the blank. What is the decryption key?Show how you figured it out.

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