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.
We are given the bytes 10001001 and 10101010, which represent elements in the finite field F₂[X]/⟨g(X)⟩, where: \[ g(X) = X^8 + X^4 + X^3 + X + 1 \] ### Step 1: Convert Binary Numbers to Polynomials 1. 10001001 (from left to right): - \(1 \cdot X^7 + 0 \cdot X^6 + 0 \cdot X^5 + 0 \cdot X^4 + 1 \cdot X^3 + 0 \cdot X^2 + 0 \cdot X^1 + 1\) - Polynomial Representation: \[ X^7 + X^3 + 1 \] 2. 10101010 (from left to right): - \(1 \cdot X^7 + 0 \cdot X^6 + 1 \cdot X^5 + 0 \cdot X^4 + 1 \cdot X^3 + 0 \cdot X^2 + 1 \cdot X^1 + 0\) _______ _______ _____ ________ _________ __________ _____ _________ _______.
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b) Use Miller-Rabin test to check whether 75521 is a strong pseuodprime to the base 2.
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.
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 the frequency test, serial test and autocorrelation test to the following sequence at level of significance α = 0.05:
011001110000110010011100.
Apply runs test to the following sequence:
1001101000010000101111011
0111010010110110010011010
0110011100001100100111000
1100001101010111101001110
0010001111000001101010010
1000110100000110100101101
1110001001
a) Apply the frequency test, serial test and autocorrelation test to the following sequence at level of significance α = 0.05: 011001110000110010011100
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.