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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?

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?

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

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. 

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

Apply runs test to the following sequence:

1001101000010000101111011

0111010010110110010011010

0110011100001100100111000

1100001101010111101001110

0010001111000001101010010

1000110100000110100101101

1110001001

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

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

011001110000110010011100.

Find a recurrence that generates the sequence 110110110110110.

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.

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

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

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.

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.

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.

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"

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.