IGNOU MMTE 6 SOLVED ASSIGNMENT 2025

Ques 1.

Let f (x) = x³ − x − 1 ∈ Z₅[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. 

Ques 2.

Let f (x) = x⁴ + x + 1 ∈ F₂[x]. We represent the field F₂4 by F₂[x]/( f (x)). Let us write γ = x + ( f (x)). The table of values is given below:

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

ii) Compute (γ 4+γ 2 )+ γ 3+γ+1/(1+γ 2+γ 4)(1+γ 3) and γ 2 (γ 2+γ+1) / (γ 3+γ 2)(1+γ 5) using the logarithm an 

antilogarithm tables.

Ques 3.

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"

Ques 4.

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.

Ques 5.

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

Ques 6.

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

Ques 7.

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 inveritble 2×2 matrix with coefficients in Z26, for example, A = 

This matrix has determinant  is a unit in Z₂₆ with inverse  We write each pair of elements in Z₂₆ 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 

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

Ques 8.

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

Ques 9.

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.

ii) Find the keys for the second round

Ques 10.

Considering the bytes 10001001 and 10101010 as elements of the field F₂[X]/hg(X)i, where g(X) is the polynomial X ⁸ +X ⁴ +X ³ +X +1, find their product and quotient.

Ques 11.

Find a recurrence that generates the sequence 110110110110110.

Ques 12.

Apply the frequency test, serial test and autocorrelation test to the following sequence at level of significance α = 0.05:

011001110000110010011100.

Ques 13.

Apply poker test to the following sequence with level of significane α = 0.05. (4) 1001101000010000101111011 01110100101101100100110.

Ques 14.

Apply runs test to the following sequence:

1001101000010000101111011

0111010010110110010011010

0110011100001100100111000

1100001101010111101001110

0010001111000001101010010

1000110100000110100101101

1110001001

Ques 15.

Decrypt the message c = 23 that was encrypted using RSA algorithm with e = 43 and n = 77.

Ques 16.

i) Bob uses ElGamal cyrptosystem with parameters p = 47, g = 5 and the secret value x = 3. What values will he make public? 

ii) Alice wants to send Bob the message M = 15. She chooses k = 5. How will she compute the cipher text? What information does she send to Bob?

iii) Explain how Bob will decrypt the message. 

Ques 17.

Solve the discrete logarithm problem 5^x ≡ 22 (mod 47) using Baby-Step, Giant-Step algorithm.

Ques 18.

Alice wants to use the ElGamal digital signature scheme with public parameters p = 47, α = 2, secret value a = 7 and β = 34. She wants to sign the message M = 20 and send it to Bob. She chooses k = 5 as the secret value. Explain the procedure that Alice will use for computing the signature of the message. What information will she send Bob?

Ques 19.

Alice wants to use the Digital Signature algorithm for signing messages. She chooses p = 83, q = 41, g = 2 and a = 3. Alice wants to sign the message M = 20. She chooses the secret value k = 8. Explain the procedure that Alice will use for computing the signature. What information will she send Bob?

Ques 20.

a)  Find the product of using the algorithm in page 23, block 1. You should show all the steps as in example 11, pag 22, block 1.

Ques 21.

b) 

). The table of values is given below:

Ques 22.

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

Ques 23.

ii) Compute  using the logarithm an antilogarithm tables

Ques 24.

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".

Ques 25.

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.

Ques 26.

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

Ques 27.

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

Ques 28.

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

Ques 29.

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:

Next, we choose an inveritble 2×2 matrix with coefficients in Z_26, for example.This matrix has determinant5 is a unit in Z₂₆ with inverse We write each pair of elements in Z26 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

Decrypt the text "TWDXHUJLUENN" which was encrypted using the Hill’s cipher with the

Ques 30.

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

Ques 31.

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

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

ii) Find the keys for the second round.

Ques 32.

a) Considering the bytes 10001001 and 10101010 as elements of the  g(X)) is the polynomia  find their product and quotient.

Ques 33.

Find a recurrence that generates the sequence 110110110110110.

Ques 34.

a) Apply the frequency test, serial test and autocorrelation test to the following sequence at level of significance α = 0.05: 011001110000110010011100

Ques 35.

Apply poker test to the following sequence with level of significane α = 0.05. (4) 1001101000010000101111011 01110100101101100100110.

c) Apply runs test to the following sequence

Ques 36.

a) Decrypt the message c = 23 that was encrypted using RSA algorithm with e = 43 and n = 77.

Ques 37.

b) i) Bob uses ElGamal cyrptosystem with parametersand the secret value x = 3. What values will he make public?

ii) Alice wants to send Bob the messageShe chooses k = 5. How will she compute the cipher text? What information does she send to Bob?

iii) Explain how Bob will decrypt the message

Ques 38.

Solve the discrete logarithm problem 5^x ≡ 22 (mod 47) using Baby-Step, Giant-Step algorithm.

Ques 39.

b) Alice wants to use the ElGamal digital signature scheme with public parameters  secret value a = 7 and β = 34. She wants to sign the message M = 20 and send it to Bob. She chooses k = 5 as the secret value. Explain the procedure that Alice will use for computing the signature of the message. What information will she send Bob

Ques 40.

c) Alice wants to use the Digital Signature algorithm for signing messages. She chooses p = 83, q = 41, g = 2 and a = 3. Alice wants to sign the message M = 20. She chooses the secret value k = 8. Explain the procedure that Alice will use for computing the signature. What information will she send Bob?

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