E3-RevQs-Sol - Exam #3 Review Question Solutions 1) What is...

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Exam #3 Review Question Solutions 1) What is the prime factorization of 175? 175 = 5 2 x 7 2) Prove that 2 is a generator mod 13. 2 1 ≡ 2 mod 13 2 7 ≡ 11 mod 13 2 2 ≡ 4 mod 13 2 8 ≡ 9 mod 13 2 3 ≡ 8 mod 13 2 9 ≡ 5 mod 13 2 4 ≡ 3 mod 13 2 10 ≡ 10 mod 13 2 5 ≡ 6 mod 13 2 11 ≡ 7 mod 13 2 6 ≡ 12 mod 13 2 12 ≡ 1 mod 13 This shows that the smallest positive integer for which 2 k ≡ 1 mod 13 is 12, this 2 is a generator. 3) What is the remainder when 37 129 is divided by 80? φ(80) = φ((5)(16)) = φ(5) φ(16) = (5 – 1)(16 – 8) = 32. (4 pts) Thus, 37 32 ≡ 1 mod 80. 37 129 ≡(37 32 ) 4 (37) mod 80 (3 pts) ≡(1) 4 (37) mod 80 ≡ 37 mod 80 (1 pt) 4) In an RSA scheme, p = 7, q = 13 and e = 5. What is d? n = 7x13 = 91 φ(n) = (7 – 1)(13 – 1) = 72. (3 pts) ed ≡ 1 mod 72 5d ≡ 1 mod 72 Thus, we must find 5 -1 mod 72. Use the Extended Euclidean Algorithm: 72 = 14x5 + 2 5 = 2x2 + 1 (3 pts) 5 – 2x2 = 1 5 – 2(72 – 14x5) = 1 5 – 2x72 + 28x5 = 1 29x5 – 2x72 = 1 (3 pts)
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E3-RevQs-Sol - Exam #3 Review Question Solutions 1) What is...

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