Solve your IGNOU Doubts

Question:

b) A 64 bit key for the DES is given below

$\begin{array}{cc}11000111&10000101\\11110111&11000001\\11111011&10101011\\10011101&10010001\end{array}$

i) Check whether the key is error free using the parity bits.

ii) Find the keys for the second round.

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Question:

a) Decrypt the ciphertext 101000111001 which was encrypted with the Toy block cipher once using the key 101010010. Show all the steps

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Question:

a) 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 7_26-

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:

$AL\left(\bar{0,\bar{11}}\right)$

$LI\left(\bar{11,\bar{8}}\right)$

$SW\left(\bar{18,\bar{22}}\right)$

$EL\left(\bar{4,\bar{11}}\right)$

$LX\left(\bar{11,\bar{23}}\right)$

Next, we choose an inveritble 2×2 matrix with coefficients in Z_26, for example. $A=\begin{bmatrix}\overline{3}&\overline{1}\\\overline{7}&\overline{4}\end{bmatrix}.$ This matrix has determinant $\overline{3}\cdot\overline{4}-\overline{7}\cdot\overline{1}=\overline{12}-\overline{7}=\overline{5}$ 5 is a unit in Z₂₆with inverse $\bar{21}.$ We write each pair of elements in Z26 as a column vector and multiply it by A:

$A\begin{bmatrix}\overline{0}\\\overline{11}\end{bmatrix}=\begin{bmatrix}\overline{11}\\\overline{18}\end{bmatrix},A\begin{bmatrix}\overline{11}\\\overline{8}\end{bmatrix}=\begin{bmatrix}\overline{15}\\\overline{5}\end{bmatrix},\dots$

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

$A^{-1}=\overline{5}^{-1}\begin{bmatrix}\overline{4}&\overline{-1}\\\overline{-7}&\overline{3}\end{bmatrix}=\overline{21}\begin{bmatrix}\overline{4}&\overline{-1}\\\overline{-7}&\overline{3}\end{bmatrix}=\begin{bmatrix}\overline{6}&\overline{5}\\\overline{9}&\overline{11}\end{bmatrix}\text{and we have}\begin{bmatrix}\overline{6}&\overline{5}\\\overline{9}&\overline{11}\end{bmatrix}\begin{bmatrix}\overline{11}\\\overline{18}\end{bmatrix}=\begin{bmatrix}\overline{0}\\\overline{18}\end{bmatrix},\dots$

Decrypt the text "TWDXHUJLUENN" which was encrypted using the Hill’s cipher with the $\text{matrix}\begin{bmatrix}\overline{3}&\overline{1}\\\overline{0}&\overline{9}\end{bmatrix}\text{as the encryption matrix.}$

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Question:

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

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Question:

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

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Question:

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

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Question:

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:

a) 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".

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Question:

ii) Compute $\frac{(\gamma^4+\gamma^2)+(\gamma^3+\gamma+1)}{(1+\gamma^2+\gamma^4)(1+\gamma^3)}\text{and}\frac{\gamma^2(\gamma^2+\gamma+1)}{(\gamma^3+\gamma^2)(1+\gamma^5)}$ using the logarithm an antilogarithm tables

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Question:

i) Prepare logarithm and antilogarithm tables as given in page 23 of block 1

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Question:

b) $\text{Let}f(x)=x^4+x+1\in\mathbb{F}_2[x].\text{We represent the field}\mathbb{F}_{2^4}\text{by}\mathbb{F}_2[x]/(f(x)).\text{Let us write}$

$\gamma=x+(f(x)).$ ). The table of values is given below:

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

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Question:

a) $\text{Let}f(x)=x^3-x-1\in\mathbb{Z}_5[x].$ Find the product of $x^2+2x+1+(f(x))\text{and}x^2+3x-1+(f(x))$ using the algorithm in page 23, block 1. You should show all the steps as in example 11, pag 22, block 1.