Apply runs test to the following sequence:
1001101000010000101111011
0111010010110110010011010
0110011100001100100111000
1100001101010111101001110
0010001111000001101010010
1000110100000110100101101
1110001001
The runs test is used to determine if a sequence of binary data (0s and 1s) is randomly distributed by analyzing the "runs" — consecutive sequences of the same number. In this case, we'll be testing the given sequence for randomness using the runs test. ### Sequence Breakdown: Here’s the binary sequence: ``` 1001101000010000101111011 0111010010110110010011010 0110011100001100100111000 1100001101010111101001110 0010001111000001101010010 1000110100000110100101101 1110001001 ``` We can start by counting the runs in each sequence. A "run" is a series of consecutive identical ________ ________ ___ _____ ____ _______.
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i) Prepare logarithm and antilogarithm tables as given in page 23 of block 1
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 |
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
b)
). 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) |
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.
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"
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.