Solve your IGNOU Doubts

Question:

Find the keys for the second round.
6) a) Considering the bytes 10001001 and 10101010 as elements of the field $\mathbb{F}_2[X]/\langle g(X)\rangle$ , where g(X) is the polynomial X⁸ + X⁴ + X³ + X + 1, find their product and quotient.

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Question:

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
$\quad$ 11000111 $\quad$ 10000101
$\quad$ 11110111 $\quad$ 11000001
$\quad$ 11111011 $\quad$ 10101011
$\quad$ 10011101 $\quad$ 10010001
$\quad$ i) Check whether the key is error free using the parity bits.

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Question:

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 Z₂₆as follows:

$<div style=$ $\underline{\overline{\begin{aligned}AL&\quad(0,11)\\LI&\quad(11,8)\\SW&\quad(18,22)\\EL&\quad(4,11)\\LX&\quad(11,23)\end{aligned}}}$ ">

Next, we choose an invertible $2 \times 2$ matrix with coefficients in $\mathbf{Z}_{26}$ , for example, $A = \begin{bmatrix} 3 & 1 \\ 7 & 4 \end{bmatrix}$ .
This matrix has determinant $\bar{3} \cdot \bar{4} - \bar{7} \cdot \bar{1} = \bar{5}$ and $\bar{5}$ is a unit in $\mathbf{Z}_{26}$ with inverse $\overline{21}$ . We write each pair of elements in $\mathbf{Z}_{26}$ as a column vector and multiply it by A:
$$A \begin{bmatrix} \bar{0} \\ \overline{11} \end{bmatrix} = \begin{bmatrix} \overline{11} \\ \overline{18} \end{bmatrix}, A \begin{bmatrix} \overline{11} \\ \bar{8} \end{bmatrix} = \begin{bmatrix} \overline{15} \\ \bar{5} \end{bmatrix}, \dots$
We then convert each pair of numbers to a pair of characters and write them down. In this example, we get the cipher text "LSPFYGXUEN" corresponding to the plain text "ALLWELL". To decrypt, we convert pairs of characters to pairs of numbers and multiply by $A^{-1} = \bar{5}^{-1} \begin{bmatrix} \bar{4} & \overline{-1} \\ \overline{-7} & \bar{3} \end{bmatrix} = \overline{21} \begin{bmatrix} \bar{4} & \overline{-1} \\ \overline{-7} & \bar{3} \end{bmatrix} = \begin{bmatrix} \bar{6} & \bar{5} \\ \bar{9} & \overline{11} \end{bmatrix}$ and we have
$$\begin{bmatrix} \bar{6} & \bar{5} \\ \bar{9} & \overline{11} \end{bmatrix} \begin{bmatrix} \overline{11} \\ \overline{18} \end{bmatrix} = \begin{bmatrix} \bar{0} \\ \overline{18} \end{bmatrix}, \dots$
Decrypt the text "TWDXHUJLUENN" which was encrypted using the Hill’s cipher with the matrix $\begin{bmatrix} \bar{3} & \bar{1} \\ \bar{0} & \bar{9} \end{bmatrix}$ as the encryption matrix.

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Question:

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

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Question:

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

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Question:

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:

S	E	C	R	T
A	T	T	A	C
K	A	T	D	A
W	N	X	X	X

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.

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Question:

) Let $f(x) = x^4 + x + 1 \in \mathbf{F}_2[x]$ . We represent the field $\mathbf{F}_{2^4}$ by $\mathbf{F}_2[x]/(f(x))$ . Let us write $\gamma = x + (f(x))$ . The table of values is given below:

`i`	`γⁱ`	Vector	`i`	`γⁱ`	Vector
0	`1`	`(0,0,0,1)`	8	`γ2+1`	`(0,1,0,1)`
1	`γ`	`(0,0,1,0)`	9	`γ3+γ`	`(1,0,1,0)`
2	`γ2`	`(0,1,0,0)`	10	`γ2+γ+1`	`(0,1,1,1)`
3	`γ3`	`(1,0,0,0)`	11	`γ3+γ2+γ`	`(1,1,1,0)`
4	`γ+1`	`(0,0,1,1)`	12	`γ3+γ2+γ+1`	`(1,1,1,1)`
5	`γ2+γ`	`(0,1,1,0)`	13	`γ3+γ2+1`	`(1,1,0,1)`
6	`γ3+γ2`	`(1,1,0,0)`	14	`γ3+1`	`(1,0,0,1)`
7	`γ3+γ+1`	`(1,0,1,1)`

i) Prepare logarithm and antilogarithm tables as given in page 23 of block 1.
ii) Compute $\dfrac{(\gamma^4 + \gamma^2) + (\gamma^3 + \gamma + 1)}{(1 + \gamma^2 + \gamma^4)(1 + \gamma^3)}$ and $\dfrac{\gamma^2(\gamma^2 + \gamma + 1)}{(\gamma^3 + \gamma^2)(1 + \gamma^5)}$ using the logarithm and antilogarithm tables

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Question:

Let $f(x) = x^3 - x - 1 \in \mathbf{Z}_5[x]$ . 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.

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Question:

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?

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Question:

) 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.

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Question:

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.

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Question:

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

$$f(x) = \frac{1}{\theta} e^{-x/\theta}; x > 0$
Test the null hypothesis that the mean of the distribution is $\theta = 2$ against the alternative hypothesis that is $\theta = 5$ . 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.

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Question:

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,
$\quad$ i) exactly two cars will have a flat tyre.
$\quad$ ii) at most two cards will have a flat tyre.

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Question:

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

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Question:

Let E₁, E₂, E₃ and E₄ be arbitrary events. Write the following events in set notations:
$\quad$ i) not more than one of E₁, E₂, E₃, E₄.
$\quad$ ii) one and only one of E₁, E₂, E₃, E₄.
$\quad$ iii) E₁ and at least one of E₂, E₃, E₄.
$\quad$ iv) none of E₂, E₃ and E₄ using E₁.
$\quad$ b) Let the probability density function of r.v. X be
$$f(x) = \begin{cases} 1 + x & ; \ -1 < x \le 0 \\ 1 - x & ; \ 0 < x < 1 \\ 0 & ; \ \text{otherwise} \end{cases}$
$\quad$ and if $u = X$ and $v = X^2$ , find Cov(u, v). Also check the independence of u and v.

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Question:

The mean and standard deviation of a variable x are m and $\sigma$ respectively. Obtain the mean and standard deviation of $\frac{(ax+b)}{c}$ , where a, b and c are constants.
$\quad$ b) If X is a random variable such that $E(X) = 3$ and $E(X^2) = 13$ , determine a lower bound for P(-2 < X < 8).

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Question:

. Consider the joint probability density function

$$f(x, y) = y^2 e^{-y(x+1)}; x \geq 0, y \geq 0.$

Are both x and y regressions linear? Give reasons for your answer.

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Question:

A die is thrown 60 times with the following results:

Face of die	1	2	3	4	5	6
Frequency	8	7	12	8	14	11

Test that the die is unbiased at $5\%$ level of significance. Given that at 5, 6 and 7 d.f. the value of $\chi^2$ are 11.070, 15.592 and 14.067 respectively.

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Question:

6 observations on (X, Y) yielded the following data:

$$\sum X_i = 30, \sum Y_i = 180, \sum X_i Y_i = 1000,$

$$\sum X_i^2 = 200, \sum Y_i^2 = 5642.$

$\quad$ i) Determine the correlation coefficient between X and Y.

ii) Given $X = 10$ , what will be the predicted value of Y?

iii) Given $Y = 15$ , what will be the predicted value of X?

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Question:

The mean I.Q. of a large number of children of age 14 was 100 and standard deviation 16. Assuming that the distribution was normal, find
$\quad$ i) the percentage of children having I.Q. under 80.

ii) the limits in which the I.Q. of the middle 40% of the children will lie.
$\quad$ You may like to use the following values:

$\quad P(Z > 1.25) = 0.1056$