IGNOU MMTE 6 SOLVED ASSIGNMENT

Ques 1.

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. 

Ques 2.

) 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

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". 
   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 4.

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

Ques 5.

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

Ques 6.

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.

Ques 7.

 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.

Ques 8.

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. 

Ques 9.

 Find a recurrence that generates the sequence 110110110110110. 

Ques 10.

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

 011001110000110010011100. 

Ques 11.

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

Ques 12.

 Apply runs test to the following sequence:

 1001101000010000101111011
 0111010010110110010011010
 0110011100001100100111000
 1100001101010111101001110
 0010001111000001101010010
 1000110100000110100101101
 1110001001 

Ques 13.

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

Ques 14.

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

Ques 15.

 Alice wants to send Bob the message . She chooses . How will she compute the cipher text? What information does she send to Bob? 

Ques 16.

Explain how Bob will decrypt the message. 
9) a) Solve the discrete logarithm problem  using Baby-Step, Giant-Step algorithm. 

Ques 17.

Alice wants to use the ElGamal digital signature scheme with public parameters , , secret value  and . She wants to sign the message  and send it to Bob. She chooses  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 18.

 Alice wants to use the Digital Signature algorithm for signing messages. She chooses , ,  and . Alice wants to sign the message . She chooses the secret value . Explain the procedure that Alice will use for computing the signature. What information will she send Bob? 

Ques 19.

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. 

Ques 20.

) 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

Ques 21.

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.

Ques 22.

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

Ques 23.

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

Ques 24.

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.

Ques 25.

 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.

Ques 26.

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. 

Ques 27.

 Find a recurrence that generates the sequence 110110110110110. 

Ques 28.

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

 011001110000110010011100. 

Ques 29.

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

Ques 30.

 Apply runs test to the following sequence:

 1001101000010000101111011
 0111010010110110010011010
 0110011100001100100111000
 1100001101010111101001110
 0010001111000001101010010
 1000110100000110100101101
 1110001001 

Ques 31.

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

Ques 32.

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

Ques 33.

 Alice wants to send Bob the message . She chooses . How will she compute the cipher text? What information does she send to Bob? 

Ques 34.

Explain how Bob will decrypt the message. 
9) a) Solve the discrete logarithm problem  using Baby-Step, Giant-Step algorithm. 

Ques 35.

Alice wants to use the ElGamal digital signature scheme with public parameters , , secret value  and . She wants to sign the message  and send it to Bob. She chooses  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 36.

 Alice wants to use the Digital Signature algorithm for signing messages. She chooses , ,  and . Alice wants to sign the message . She chooses the secret value . Explain the procedure that Alice will use for computing the signature. What information will she send Bob? 

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