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
We are given that Alice wants to use the ElGamal Digital Signature Scheme with the following values: * Prime modulus: $p = 47$ * Primitive root: $\alpha = 2$ * Alice's private key: $a = 7$ * Alice's public key: $\beta = \alpha^a \mod p = 2^7 \mod 47 = 128 \mod 47 = 34$ ✅ * Message: $M = 20$ * Ephemeral key: $k = _______ ________ _____ ______ _______ ________.
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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.
Apply poker test to the following sequence with level of significane α = 0.05. (4) 1001101000010000101111011 01110100101101100100110.
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.
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.
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
i) Prepare logarithm and antilogarithm tables as given in page 23 of block 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.