Question:

Discuss these tough questions and give honest results.

1. Determine if given the that 38 is the ciphertext, the plain text code for {1 6 8 15 24} which is the Knapsack.

2. Between a jumbled knapsack and a knapsack problem that is super increasing, which problem is easy to solve?

3. Given that you may apply the RSA algorithm where PT message equals 88, determine the value of CT given that p equals 11, q equals 17 and that you are supposed to choose e=7.

4. Calculate for the private key (d, p, q) which is given by the public key denoted by (e=23, n=233 241=56,153).

5. Determine the private key of a user of an RSA system where the public key is given as e = 31, n = 3599.

6. Determine the ciphertext for the message 'WHY' given that the plaintext message consists of mainly single letters that are of 5-bit numerical values in the range (00000)2 to (11001)2. Also, the super-increasing 5- tuple (2, 3, 7, 15, 31), m = 61 and a = 17 is the secret deciphering key.

7. Can we say that the public key in Merkle-Hellman Cryptosystem is used to decrypt messages but cannot be used to encrypt messages. Is it true that the private key on the other hand is used to encrypt messages?

8. Determine another name that is used to refer to Merkle-Hellman Cryptosystem.

9. Determine the value of n given a knapsack that has been formed from the weights of the super-increasing series {1, 2, 4, 9, 20, and 38} whose weight is 23 units.

10. Given a set {1, 2, 3, 9, 10, and 24}, is it true that it is super-increasing?

(Keith Chugg) Draw connection lines in Fig. 1 that link the assumptions and mode of stochastic convergence (on the left) associated with each of the convergence theorems (on the right). independent, 3'3\"" identically distributed 1M\") uncorrelated Almost Sure Convergence Sure Convergence Convergence in Distribution Convergence in Pmbabilily Mean Square Same Convergence Figure 1: Connect the related concepts by drawing lines. 511mg Law of Large Numbers Weak Law of Large Numbers Central Limit Theorem 10 points Let {1:th . . } be independent identiicz'allj,r distrEuted random 1variables from a distribution with finite mean a and variance oz. Denote by Kn the sample mean of the first 11 random variables. Using the Central Limit Theorem, the Weak Law of Large Numbers, and Slutslqr's Theorem, or otherwise, show that an - a} +9 NE}, a) where the convergence is in distribution to a Normal random 1variable with mean 0 and variance H22 and identify the constant E2. Hint. Factor E: - ,u". Let X1, X2, . .. bei.i.d. U|0, 1] random variables. Define X,= n Xi. (a) Show, by the Weak Law of Large Numbers, that X converges in distribution to 1/2. (b) Show, by the Weak Law of Large Numbers, that n- E, X? converges in distribution to 1/3. (c) Show, by the Central Limit Theorem, that Vn (X, - 1/2) con- verges in distribution to a /(0, 1/12) random variable. (d) The Central Limit Theorem asserts that n-1/2 C" (X? - 1/3) converges in distribution to a random variable Z. Specify the distribution of Z.\f