Introduction to Cryptography, CS, Assignment, 7 questions.pdf - Q1[3 Compute(124x134x23x49x13 mod 3 using modular arithmetic Solution(124 13)3

# Introduction to Cryptography, CS, Assignment, 7 questions.pdf

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Q1 [3] Compute (124x134x23x49x13) mod 3 using modular arithmetic. Solution : (124? 134? 23? 49? 13) 3 = (243441016? 4 ) 3 = 243441016 3 (? 4 ) 3 = 243441016 3 ? 12 Q2 [4] What is the last digit of 17^17 ? Solution: 7 1 = 7 7 2 = 49 7 3 = 343 7 4 = 2401 7 5 = 16807 7 6 = 117649 Now, let's take the power and divide by 4: 4? + 1 = 17 𝑖? ? = 4 17 17 = 17 (4.4+1) Knowing that unit place will repeat after 4 circles. Therefore: 17 1 ℎ𝑎? 7 ?𝑎?? ?𝑖?𝑖?, ?𝑎?? ?𝑖?ℎ 17 17 ?𝑖?? ℎ𝑎?? ?𝑎?? ?𝑖?𝑖? ?? 7 Hence, the answer is 7 as the last digit for 17 17 .
Q3 [8] Find a) multiplicative b) additive inverses of each non zero elements in Z9 (Note: Z9 = {0,1,2,3,4,5,6,7,8}). Show your steps. Solutions: Z 9 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 2 2 4 6 8 1 3 5 7 3 3 6 0 3 6 0 3 6 (a) 4 4 8 3 7 2 6 1 5 5 5 1 6 2 7 3 8 4 6 6 3 0 6 3 0 6 3 7 7 5 3 1 8 6 4 2 8 8 7 6 5 4 3 2 1 (b) a 1 2 3 4 5 6 7 8 a 0 1 5 X 7 2 X 4 8 Q4 [8] In a public-key system using RSA. You intercept the cipher text C = 10 sent to a user whose public key is e = 5, n =35. Find plaintext M and private key d. Solution : Here, the two prime numbers p and q.

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