# ENG 204 Data Security Homework

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Homework 2 Rule: Finish all of the following on your own. 1. Compute the following with detailed steps. (Hints: Use Fermat Theorem, Euler Theorem, properties of totient functions, etc, or write program code as assistance) (1) 123416 mod 17 (2) 5451 mod 17 (3) ø(51) (4) gcd(33, 121) (5) 2-1 mod 17 (i.e., multiplicative inverse of 2 mod 17) (6) ind2,5(4) (7) ø(8000) (8) ø(98803519) 2. Write a computer program to implement the extended Euclid’s algorithm, use your code to compute the following. Submit your code and results. a. GCD(10012012,2314213) b. GCD(28176412,29108188) c. GCD(38172,23812188) d. The multiplicative inverse of 12091 mod 24123123. e. The multiplicative inverse of 28173928 mod 129182771. 3. 4. 5. 6. Prove that a=n-1 is always a solution to a2=1 mod n. What are the differences between symmetric and asymmetric key crypto. Explain the feasibility and security of RSA. Alice designs a double-RSA cipher. She first generates two secret primes p and q, and compute n=p*q, then choose two public encryption exponents e1 and e2 that are relatively prime to ø(n). So becomes the public key. She tells people to encrypt message M by computing C1=M e1 mod n and then C= C1e2 mod n, finally sending just C to her. a. Show the decryption process (i.e., how Alice can obtain the plaintext M from the final ciphertext C). b. Is the double-RSA cipher more secure, less secure, or just as secure as the regular RSA cipher with the same modulus n but only one encryption exponent? Why? c. Charlie got Alice’s instructions confused, and encrypt message M for Alice using e1 and e2 in the reverse order (i.e., Charlie uses C1=M e2 mod n then C= C1e1 mod n). What would happen when Alice, unaware of Charlie’s error, tries to decipher the ciphertext using her usual procedure?

Prof_Befly
School: Rice University

Attached.

Homework 2
Rule: Finish all of the following on your own.
1. Compute the following with detailed steps. (Hints: Use Fermat Theorem, Euler Theorem,
properties of totient functions, etc, or write program code as assistance)
(1) 123416 mod 17
Based on the Fermat theorem
17 is a prime, gcd (1234,17) = 1,
Therefore, 123416 mod 17 = 1
(2) 5451 mod 17
17 is a prime, gcd (54,17) = 1 , ø(17) = 16
Then,
(54)51 mod 17 = (54 mod 17)(51 mod 16) mod 17
= (3)3 mod 17 = (27) mod 17 = 10
(3) ø(51)

Thus, ø(51) = 32
(4) gcd(33, 121)
gcd (33, 121) = gcd (121, 33)= gcd (33, 22)
= gcd (22, 11) = gcd (11, 0) = 11

(5) 2-1 mod 17 (i.e., multiplicative inverse of 2 mod 17)
2−1 mod 17 = −8 mod 17 = 9
Through Extended Euclid algorithm,
y3 = 1 = gcd (2, 17)
y2 = −8 = 2−1 mod 17
(6) ind2,5(4)
2x mod 5 = 4 then, x=2
(7) ø(8000)

Therefore, ø(8000) = 64
(8) ø(98803519)

Hence, ø(98803519) = 98783640

2. Write a computer program to implement the extended Euclid’s algorithm, use
a. GCD(10012012,2314213)
b. GCD(28176412,29108188)
c. GCD(38172,23812188)
d. The multiplicative inverse of 12091 mod 24123123.
Mat Lab Code:

e. The multiplicative inverse of 28173928 mod 129182771.
Mat Lab Code:

3. Prove that ...

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Anonymous
Tutor went the extra mile to help me with this essay. Citations were a bit shaky but I appreciated how well he handled APA styles and how ok he was to change them even though I didnt specify. Got a B+ which is believable and acceptable.

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