HW Wk 6: Number Theory

Use Fermat's little theorem to compute the following expression. Arrange the steps in the correct order to compute 3302 mod 11. (You must provide an answer before moving to the next part.)

1. 3^10 ≡ 1 (mod 11) 2. 3^300 = (3^10)^30 ≡ 1 (mod 11) 3. 3^302 = 3^2 · 3^300 ≡ 9 (mod 11) 4. 3^302 mod 11 = 9

Use Fermat's little theorem to compute the following expression. Arrange the steps in the correct order to compute 3302 mod 5. (You must provide an answer before moving to the next part.)

1. 3^4 ≡ 1 (mod 5) 2. 3^300 = (3^4)^75 ≡ 1^75 ≡ 1 (mod 5) 3. 3^302 = 32 · 3300 ≡ 9 · 1 = 9 (mod 5) 4. 3^302 mod 5 = 4

Use Fermat's little theorem to compute the following expression. Arrange the steps in the correct order to compute 3302 mod 7. (You must provide an answer before moving to the next part.)

1. 3^6 ≡ 1 (mod 7) 2. 3^300 = (3^6)^50 ≡ 1 (mod 7) 3. 3^302 = 3^2 · 3^300 ≡ 9 (mod 7) 4. 3^302 mod 7 = 2

Arrange the steps in correct order to solve the congruence 2x ≡ 7 (mod 17) using the inverse of 2 modulo 17, which is 9.

1. 9 is an inverse of 2 modulo 17. The given equation is 2x ≡ 7 (mod 17). 2. Multiplying both sides of the equation by 9, we get x ≡ 9 · 7 (mod 17). 3. Since 63 mod 17 = 12, the solutions are all integers congruent to 12 modulo 17, such as 12, 29, and −5.

Arrange the steps to find an inverse of a modulo m for each of the following pairs of relatively prime integers using the Euclidean algorithm in the order. a = 200, m = 1001

1. By using the Euclidean algorithm, 1001 = 5 · 200 + 1. 2. Let 200s + 1001t = 1, where s is an inverse of 200 modulo 1001. 3. The gcd in terms of 1001 and 200 is written as 1 = 1001 − 5 · 200. 4. s = −5, so an inverse of 200 modulo 1001 is −5.

Arrange the steps to find an inverse of a modulo m for each of the following pairs of relatively prime integers using the Euclidean algorithm in the order. a = 144, m = 233

1. Let 144s + 233t = 1, where s is an inverse of 144 modulo 233. 2. The steps to find gcd(144, 233) = 1 using the Euclidean algorithm is as follows. 233 = 144 + 89144 = 89 + 5589 = 55 + 3455 = 34 + 2134 = 21 + 1321 = 13 + 813 = 8 + 58 = 5 + 35 = 3 + 23 = 2 + 1 3. The gcd in terms of 144 and 233 is written as 1 = 3 − 2 = 3 − (5 − 3) = 2 · 3 − 5 = 2 · (8 − 5) − 5 = 2 · 8 − 3 · 5 = 2 · 8 − 3 · (13 − 8) = 5 · 8 − 3 · 13 = 5 · (21 − 13) − 3 · 13 = 5 · 21 − 8 · 13 = 5 · 21 − 8 · (34 − 21) = 13 · 21 − 8 · 34 = 13 · (55 − 34) − 8 · 34 = 13 · 55 − 21 · 34 = 13 · 55 − 21 · (89 − 55) = 34 · 55 − 21 · 89 = 34 · (144 − 89) − 21 · 89 = 34 · 144 − 55 · 89 = 34 · 144 − 55 · (233 − 144) = 89 · 144 − 55 · 233 4. s = 89, so an inverse of 144 modulo 233 is 89.

Decrypt these messages that were encrypted using the Caesar cipher. EOXH MHDQV The decrypted message of EOXH MHDQV is

BLUE JEANS

Decrypt these messages that were encrypted using the Caesar cipher. HDW GLP VXP The decrypted message of HDW GLP VXP is

EAT DIM SUM

Consider the message "DO NOT PASS GO." Translate the encrypted numbers to letters for the function f(p) = f(p + 3) mod 26. (You must provide an answer before moving to the next part.)

GR QRW SDVV JR

Decrypt these messages that were encrypted using the Caesar cipher. WHVW WRGDB The decrypted message of WHVW WRGDB is

TEST TODAY

Use Fermat's little theorem to compute the following expression. Identify the value of 3302 mod 385 using the results of previous part (a), (b), and (c) of this question and the Chinese remainder theorem.

9

Arrange the steps in the correct order to solve the system of congruences x ≡ 2 (mod 3), x ≡ 1 (mod 4), and x ≡ 3 (mod 5) using the method of back substitution.

1. The first congruence can be written as x = 3t + 2, where tis an integer. Substituting this expression for x into the second congruence gives 3t + 2 ≡ 1 (mod 4). 2. This implies t ≡ 1 (mod 4). Therefore, t = 4u + 1 for some integer u. 3. Thus, x = 3t + 2 = 3(4u + 1) + 2 = 12u + 5. We substitute this into the third congruence to obtain 12u + 5 ≡ 3 (mod 5), which implies u ≡ 4 (mod 5). 4. Hence, u = 5v + 4 and so x = 12u + 5 = 12(5v + 4) + 5 = 60v + 53, where v is an integer. 5. Translating x = 60v + 53 back into a congruence, we find the solution to the simultaneous congruences x ≡ 53 (mod 60).

Arrange the steps in correct order to encrypt the message ATTACK using the RSA system with n = 43 · 59 and e = 13, translating each letter into integers and grouping together pairs of integers.

1. Translate the letters into numbers. 0019 1900 0210 2. Compute C = P13 mod 2537 for P = 19, P = 1900, and P = 210. 3. The results are 2299, 1317, and 2117, respectively. 4. The message is 2299 1317 2117.

Which memory locations are assigned by the hashing function h(k) = kmod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 501338753 is

3

What sequence of pseudorandom numbers is generated using the linear congruential generator xn + 1 = (4xn + 1) mod 7 with seed x0 = 3? Enter the sequence below, starting with x0. The sequence is

3,6,4,3

Consider the message "DO NOT PASS GO." Translate the letters in the above message to numbers by using their position in the alphabet. (You must provide an answer before moving to the next part.)

3-14 13-14-19 15-0-18-18 6-14

Which memory locations are assigned by the hashing function h(k) = kmod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 372201919 is

52

Which memory locations are assigned by the hashing function h(k) = kmod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 104578690 is

58

Consider the message "DO NOT PASS GO." Identify the number obtained after applying the encryption function f(p) = (p + 3) mod26 to the the number translated from the letters of the above message. (You must provide an answer before moving to the next part.)

6-17 16-17-22 18-3-21-21 9-17

Which memory locations are assigned by the hashing function h(k) = kmod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 432222187 is

60