Bob uses ElGamal cryptosystem with parameters ,  and the secret value . What values will he make public? 

 Decrypt the message  that was encrypted using RSA algorithm with  and . 

 Apply runs test to the following sequence:

 1001101000010000101111011
 0111010010110110010011010
 0110011100001100100111000
 1100001101010111101001110
 0010001111000001101010010
 1000110100000110100101101
 1110001001 

Apply poker test to the following sequence with level of significance .
 1001101000010000101111011
 01110100101101100100110. 

 Apply the frequency test, serial test and autocorrelation test to the following sequence at level of significance :

 011001110000110010011100. 

 Find a recurrence that generates the sequence 110110110110110. 

Find the keys for the second round.
6) a) Considering the bytes 10001001 and 10101010 as elements of the field , where g(X) is the polynomial X⁸ + X⁴ + X³ + X + 1, find their product and quotient. 

 Decrypt the ciphertext 101000111001 which was encrypted with the Toy block cipher once using the key 101010010. Show all the steps. 
b) A 64 bit key for the DES is given below
 11000111  10000101
 11110111  11000001
 11111011  10101011
 10011101  10010001
 i) Check whether the key is error free using the parity bits.

In this exercise, we introduce you to Hill cipher. In this cipher, we convert our message to numbers, just as in affine cipher. However, instead of encrypting character by character, we encrypt pairs of characters by multiplying them with an invertible matrix with co-efficients in Z₂₆.

Here is an example: Suppose we want to ENCRYPT "ALLISWELL". Since we require the plaintext to have even number of characters, we pad the message with the character ‘X’. We break up the message into pairs of characters AL, LI, SW, EL and LX. We convert each pair of characters into a pair elements in Z₂₆ as follows:

">

Next, we choose an invertible  matrix with coefficients in , for example, .
This matrix has determinant  and  is a unit in  with inverse . We write each pair of elements in  as a column vector and multiply it by A:

We then convert each pair of numbers to a pair of characters and write them down. In this example, we get the cipher text "LSPFYGXUEN" corresponding to the plain text "ALLWELL". To decrypt, we convert pairs of characters to pairs of numbers and multiply by  and we have

Decrypt the text "TWDXHUJLUENN" which was encrypted using the Hill’s cipher with the matrix  as the encryption matrix.

Use Miller-Rabin test to check whether 75521 is a strong pseuodprime to the base 2.

Find the inverse of 13 (mod 51) using extended euclidean algorithm.

Decrypt each of the following cipher texts:
    i) Text: "CBBGYAEBBFZCFEPXYAEBB", encrypted with affine cipher with key (7,2). 
    ii) Text:"KSTYZKESLNZUV", encrypted with Vigenère cipher with key "RESULT". 
   b) Another version of the columnar transposition cipher is the cipher using a key word. In this cipher, we encrypt as follows: Given a key word, we remove all the duplicate characters in the key word. For example, if the key word is ‘SECRET’, we remove the second ‘E’ and use ‘SECRT’ as the key word. To encrypt, we form a table as follows: In the first row, we write down the key word. In the following rows, we write the plaintext. Suppose we want to encrypt the text ‘ATTACKATDAWN’. We make a table as follows:

Then we read off the columns in alphabetical order. We first read the column under ‘C’, followed by the columns under ‘E’, ‘R’, ‘S’ and ‘T’. We get the cipher text TTX TAN ADX AKW CAX. To decrypt, we reverse the process. Note that, since we know the length of the keyword, we can find the length of the columns by dividing the length of the message by the length of the keyword.

Given the ciphertext ‘HNDWUEOESSRORUTXLARFASUXTINOOGFNEGASTORX’ and the key word ‘LANCE’, find the plaintext.

) Let . We represent the field  by . Let us write . The table of values is given below:
 

 i) Prepare logarithm and antilogarithm tables as given in page 23 of block 1. 
    ii) Compute  and  using the logarithm and antilogarithm tables

Let . Find the product of x² + 2x + 1 + (f(x)) and x² + 3x - 1 + (f(x)) using the algorithm in page 23, block 1. You should show all the steps as in example 11, page 22, block 1. 

 For married couples living in a certain locality, the probability that the husband will vote in a school board election is 0.21, the probability that they both will vote is 0.15. What is the probability that

i) at least one of them will vote?

ii) neither of them will vote?

) Let X be a gamma variable with parameters αand λ, having E(X ) = 6 and Var(X ) = .3 Find αand λ. Also, find the m.g.f. of a gamma variable, and hence verify that mean of X is 6 and variance of X is 3 using m.g.f.

The mean and standard deviation of 20 items is found to be 10 and 2 respectively. At the time of checking it was found that one item having value 8 was incorrect. Calculate the mean and standard deviation if the wrong item is omitted.

) Let X₁ be an observation from an exponential distribution with the p.d.f.

Test the null hypothesis that the mean of the distribution is  against the alternative hypothesis that is . The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. Find the probabilities of type-I and type-II errors.

The probability that a card will have a flat tyre while crossing a certain bridge is 0.00005. Find the probability that, among 10,000 cars crossing the bridge,
 i) exactly two cars will have a flat tyre.
 ii) at most two cards will have a flat tyre.

For a mesokurtic distribution with standard deviation 5, find fourth central moment m₄.