SIT 281: Hashing - Discrete Logarithm - ElGamal Cryptosystem - IT Assignment Help

Assignment Task:

Task:

Part A: Secret Sharing
 

1. In a (3, 7) Shamir secret sharing scheme with modulus 59, three of the shares are (5, 11), (10, 43) and (20, 34). Another share is (7, k), but the value of k is unreadable. Find the correct value of k (3 marks). Verify your result with Maple (2 marks).

2. The general formula for a plane in the 3D space is z = ax + by + c, where a, b, c are the parameters. Alice encodes three English words to three numbers by using the Maple command “text2num”. She then set up a (3,4) secret sharing scheme with the idea of the Blakley method. The four shares are four planes given by:
z = 4x + 19y + 2515211725275120 (mod 2515211819051461)
z = 52x + 27y + 2515210613496048 (mod 2515211819051461)
z = 36x + 65y + 2515210981587340 (mod 2515211819051461)
z = 6x + 60y + 2515211676449260 (mod 2515211819051461).
Find the secret English words of Alice with Maple command (2 marks).

Part B: Hashing
3. Describe how to use a block cipher, say for example AES, to design a secure hash function, and draw the diagram of this function (3 marks).

4. Say there are 365 days in a year, and the probability that each day is taken as someone’s birthday is the same, i.e., 1/365. There are 40 students in the classroom, work out the probability that someone in the classroom has the same birthday as others (3 marks).

Part C: Discrete Logarithm
5. Use the baby step, giant step method to solve 2 k = 7 mod 13 (3 marks).

6. For the ElGamal cryptosystem, let the public key of Alice be (p, g, h), where p is a big prime, and g is a generator for mod p, and h=g a . The secret key for Alice is a. Bob wants to communicate with Alice, he will encrypt the messages m 1 m 2 m 3 m 4 … in order as specified by ElGamal and send the ciphertexts c 1 c 2 c 3 c 4 … to Alice in ordere. Eve wants to attack this encrypted communication between Alice and Bob. Assume that Eve can have one plaintext m 1 , show an attack that how Eve can recover all the encrypted c 2 c 3 c 4 … if Bob uses a broken random number generator that only outputs a constant value, say b = 2 (2 marks). One step further, if the random number generator is not broken, but it is bad in the sense that it outputs b = 2 with probability 90%, then what is the probability that Eve can recover c 2 , can recover c 3 , and can recover c 3 c 4 (2 marks)?

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