Introduction to Cryptography with Coding Theory, 3rd edition Wade Trappe Solution Manual

Type:

Solution Manual

Resource:

Introduction to Cryptography with Coding Theory

Edition:

3rd Edition

Author(s):

Wade Trappe Lawrence Washington

INSTRUCTOR’S SOLUTIONS MANUAL INTRODUCTION TO CRYPTOGRAPHY WITH CODING THEORY THIRD EDITION

Wade Trappe Rutgers University

Lawrence C. Washington University of Maryland

Chapter 10 - Exercises 1. (a) Perhaps the easiest way to do this is to list the powers of 2 mod 13 until we get 3: 2, 22 ≡ 4, 23 ≡ 8, 24 ≡ 16 ≡ 3 (mod 13). Therefore L2 (3) = 4. (b) 27 = 128 ≡ 11 (mod 13), which implies that L2 (11) = 7.

3. (a) 62 ≡ 3 (mod 11), 64 ≡ 32 ≡ 9, 65 ≡ 6 · 64 ≡ 6 · 9 ≡ 10. (b) Since 2 is a primitive root, 25 = 2(11−1)/2 ≡ −1 (mod 11). Therefore, −1 ≡ 65 ≡ (2x )5 ≡ (−1)x , so x is odd.

5. Since 5 is a primitive root, 5611 ≡ −1 (mod 1223). Therefore, 1 ≡ 3611 ≡ 611x 5 ≡ (−1)x , so x is even.

7. (a) L2 (24) ≡ 3L2 (2) + L2 (3) ≡ 3 + 69 ≡ 72 (mod 100). Therefore, L2 (24) = 72. (b) L2 (24) ≡ 3L2 (5) ≡ 3 · 24 ≡ 72 (mod 100). Therefore, L2 (24) = 72.

9. Since 11 = 44/22 , and since 3136 ≡ 1 (mod 137), we have 3x ≡ 36−2·10 ≡ −14 3 ≡ 3−14+136 ≡ 3122 . Therefore, x = 122.

11. (a) If someone knows 2x (mod p), it is difficult to find x since this is the discrete logarithm problem. (b) If p has only 5 digits, it is easy to compute 2k (mod p) for k = 1, 2, . . . p − 1 until the number 2x (mod p) is found. Then k = x is the password.

13. Alice computes 33 ≡ 10 (mod 17) and sends 10 to Bob. Bob computes 5 3 ≡ 5 (mod 17) and sends 5 to Alice. Alice computes 53 ≡ 6 (mod 17) and Bob computes 105 ≡ 6 (mod 17). The shared secret is 6.

15. Eve computes b1 with bb1 ≡ 1 (mod p − 1). Then xb21 ≡ αbb1 ≡ α1 ≡ α (mod p).

17. (a) Write d in base N , so d = a0 + a1 N with 0 ≤ ai < N . Then m ≡ cd ≡ ca0 +a1 N implies that mc−a1 N ≡ ca0 . Therefore, there is a match for j = a0 and k = a1 . This gives a0 + a1 N as a candidate for d. (b) c = m = 1 gives a match for every j, k, so it is unlikely that the first match yields the correct d. √ (c) The lists are of length approximately N . This is approximately the time √ required to factor n by dividing by all the primes up to N .

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