Compute x as far as possible without a calculator. Where appropriate, make use o

Question

Compute x as far as possible without a calculator. Where appropriate, make use of a smart decomposition of the exponent as shown in the example in Sect. 1.4.1: x = 3^2 mod 13 x = 7^2 mod 13 x = 3^10 mod 13 x = 7^100 mod 13 7^x = 11 mod 13 Suppose we want to encrypt using affine cipher with a = 5 and b = 8. What is the encryption function? What is the decryption function? Decrypt the ciphertext: HPCCXAQ using this affine cipher. Suppose we want to encrypt using affine cipher with a = 5 and b = 8. What is the encryption function? What is the decryption function? Encrypt the plaintext: ITS COOL using this affine cipher. We consider the ring Z_4. Construct a table which describes the addition of all elements in the ring with each other: Construct the multiplication table for Z_4. Construct the addition and multiplication tables for Z_5. Construct the addition and multiplication tables for Z_6. There are elements in Z_4 and Z_6 without a multiplicative inverse. Which elements are these? Why does a multiplicative inverse exist for all nonzero elements in Z_5? Using the basic form of Euclid's algorithm, compute the greatest common divisor of 7469 and 2464 2689 and 4001 For this problem use only a pocket calculator. Show every iteration step of Euclid's algorithm, ie, don't write just the answer, which is only a number. Also, for every god, provide the chain of gcd computations, i.e., gcd(r_0, r_1) = gcd(r_1, r_2) =

Explanation / Answer

Question 4)
a) x=3^2 mod 13
3*3 mod 13
9 mod 13
x=9

b) 7^2 mod 13
7*7 mod 13
49 mod 13
x=10 (13*4=52 which is grater that 49 hence 13*3=39 and 49-39=10)

c) 3^10 mod 13
exponent rule is
3^10=3^5*3^5
3^5 mod 13=243 mod 13
9
x=((3^5 mod 13)*(3^5 mod 13)) mod 13
x=(9*9)mod 13
x=3

d) 7^100 mod 13
exponent rule
7^100=7^10*7^10*7^10*7^10*7^10*7^10*7^10*7^10*7^10*7^10
7^10 mod 13=4
x=((7^10 mod13)(7^10 mod 13) (7^10 mod13)(7^10 mod 13) (7^10 mod13)(7^10 mod 13) (7^10 mod13)(7^10 mod 13) (7^10 mod13)(7^10 mod 13)) mod 13
x=(4*4*4*4*4*4*4*4*4*4)mod 13
x=9

Question 5)

a) encryption function
E=(ax+b)mod 26
(a and b are the specified values and x is an alphabet number, E=encryption)

b) decryption function

D=a^-1(x-b) mod 26
a^-1=21(defined)
hence,
D=21(x-b)mod 26
(a and b are the specified values and x is an alphabet number, d=decryption)

C) HPCCXAQ=FREEDOM

Question 6)

a) encryption function
E=(ax+b)mod 26
(a and b are the specified values and x is an alphabet number, E=encryption)

b) decryption function

D=a^-1(x-b) mod 26
a^-1=21(defined)
hence,
D=21(x-b)mod 26
(a and b are the specified values and x is an alphabet number, d=decryption)

c) ITS COOL=WZU SAA

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