The consortium decide to implement the final approach described in question 1, using Elgamal public key encryption with the following parameters: (p, g) = (103, 5). Aldebaran’s public key is pkA = 51, Borealis’ public key is pkB = 55 and Chandra’s public key is pkC = 38. For each of the following questions, you may use an online tool or python to compute modular exponentiation and reduction, but show your working. (Hint: use the pow(x,y,z) function in python to compute x y (mod z), or using WolframAlpha, query “xˆy mod z ’’.) (a) Aldebaran wishes to send a message m = 33 to Borealis. i. (2 marks) Confirm whether or not Aldebaran’s public key corresponds to the secret key skA = 7. ii. (5 marks) In the first step, Aldebaran must compute the Elgamal encryption (c1, c2) = Enc(pkB, m). Suppose during encryption, Aldebaran randomly samples a = 33, as in where c1 = g a . What is (c1, c2)? Note: Aldebaran will perform the rest of the steps to convey this message by themselves.
Question
The consortium decide to implement the final approach described in question 1, using Elgamal public key encryption with the following parameters: (p, g) = (103, 5). Aldebaran’s public key is pkA = 51, Borealis’ public key is pkB = 55 and Chandra’s public key is pkC = 38. For each of the following questions, you may use an online tool or python to compute modular exponentiation and reduction, but show your working. (Hint: use the pow(x,y,z) function in python to compute x y (mod z), or using WolframAlpha, query “xˆy mod z ’’.) (a) Aldebaran wishes to send a message m = 33 to Borealis. i. (2 marks) Confirm whether or not Aldebaran’s public key corresponds to the secret key skA = 7. ii. (5 marks) In the first step, Aldebaran must compute the Elgamal encryption (c1, c2) = Enc(pkB, m). Suppose during encryption, Aldebaran randomly samples a = 33, as in where c1 = g a . What is (c1, c2)? Note: Aldebaran will perform the rest of the steps to convey this message by themselves.
Solution
Sure, let's solve this step by step.
i. To confirm whether Aldebaran’s public key corresponds to the secret key skA = 7, we need to check if the public key is indeed g^skA mod p. In this case, g = 5, skA = 7, and p = 103.
So, we calculate 5^7 mod 103. Using Python's pow function, we get:
print(pow(5, 7, 103))
This gives us 51, which is indeed Aldebaran’s public key. So, the public key corresponds to the secret key skA = 7.
ii. To compute the Elgamal encryption (c1, c2) = Enc(pkB, m), we need to calculate c1 and c2.
c1 is calculated as g^a mod p, where g = 5, a = 33, and p = 103.
So, we calculate 5^33 mod 103. Using Python's pow function, we get:
c1 = pow(5, 33, 103)
print(c1)
c2 is calculated as m * (pkB^a) mod p, where m = 33, pkB = 55, a = 33, and p = 103.
So, we calculate 33 * (55^33 mod 103). Using Python's pow function, we get:
c2 = (33 * pow(55, 33, 103)) % 103
print(c2)
This will give us the values of c1 and c2, which are the Elgamal encryption of the message m.
Similar Questions
) The consortium decide to implement the final approach described in question 1, using Elgamal public key encryption with the following parameters: (p, g) = (103, 5). Aldebaran’s public key is pkA = 51, Borealis’ public key is pkB = 55 and Chandra’s public key is pkC = 38. Some time later, Chandra receives a different broadcast (38, cmsg, cdest) where cdest = (55, 10) and cmsg = (c1, c2) = ((101, 28),(90, 94)). i. (2 marks) Confirm whether or not Chandra’s public key corresponds to her secret key skC = 22. ii. (5 marks) Who is the final intended recipient of the message? (Hint: compute the Elgamal decryption Dec(skC, cdest) and compare with the known public keys.) iii. (6 marks) Hence, what does Chandra broadcast? (Hint: compute the Elgamal decryptions Dec(skC, c1) and Dec(skC, c2))
(a) Aldebaran wishes to send a message m = 33 to Borealis. i. (2 marks) Confirm whether or not Aldebaran’s public key corresponds to the secret key skA = 7. ii. (5 marks) In the first step, Aldebaran must compute the Elgamal encryption (c1, c2) = Enc(pkB, m). Suppose during encryption, Aldebaran randomly samples a = 33, as in where c1 = g a . What is (c1, c2)? Note: Aldebaran will perform the rest of the steps to convey this message by themselves.
Aldebaran computes cmsg = Enc(pkC , m), cdest = Enc(pkC , pkB ) and broadcasts (pkC , cmsg, cdest). Chandra observes the broadcast containing her public key. She then computes m = Dec(skC , cmsg) and pkdest = Dec(skC , cdest). Lastly, she re-encrypts c′ = Enc(pkdest, m) and broadcasts (pkdest, c′). Borealis identifies their public key in the broadcast and obtains the message m = Dec(skB , c′). state Secure or Insecure, and explain why that approach does or does not achieve the two desired notions of confidentiality described above.
Some time later, Chandra receives a different broadcast (38, cmsg, cdest) where cdest = (55, 10) and cmsg = (c1, c2) = ((101, 28),(90, 94)). i. (2 marks) Confirm whether or not Chandra’s public key corresponds to her secret key skC = 22. ii. (5 marks) Who is the final intended recipient of the message? (Hint: compute the Elgamal decryption Dec(skC, cdest) and compare with the known public keys.) iii. (6 marks) Hence, what does Chandra broadcast? (Hint: compute the Elgamal decryptions Dec(skC, c1) and Dec(skC, c2))
Choose the Elliptic curve with prime p = 11 , a = 1, b = 3 Generator G = (2, 7) and Message=(3,5), receiver private key is 2 and sender random integer k = 3 Compute the encryption anddecryption processes completely
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