Find a subgroup of a permutation group isomorphic

Assignment Help Mathematics

Reference no: EM132691310

Question 1: Write the following permutations as a product of cycles in a compact notation.
a. [1 2 3 4 6 5]
b. [5 4 6 1 2 3]
c. [1 2 3 4 5 6 7]
Find the order of each permutation in Exercise 3.

Question 2: Consider the permutation group (S5, ?), and let π = [2 1 4 5 3].
a. Find π-1
b. Find π-2
c. Find π2
d. Find (π2)-1 and verify whether it equals to π-2 in (b) or not.
e. Find ord(π)
f. Find < π >
g. Is S5 cyclic? Briefly justify your answer.
h. Show that the group S5 is non-abelian by a counterexample.

Question 3: Consider the symmetric group S12.

c. Find a permutation in S12 of order 20.
e. Find a permutation in S12 of a maximum order.

Question 4: Let π be a permutation of order p in some symmetric group Sn, where p is a large prime. Find the order of the permutation (π4)

Find a subgroup of a permutation group isomorphic to Z*9, and show the isomorphism mapping

Question 5: Apply the Shift-Xor algorithm to multiply P1=(00001101) by P2=(11001101) in GF(256) using the irreducible polynomial
x8 + x4 + x3 + x + 1 as a modulus.
Show all the steps in a table as explained in the class.

Question 6: Consider the generator for GF(23) in Table 5.5. This table is used to efficiently perform operations in this field.
(a) Reconstruct Table 5.5 for GF(23) using the modulus x3 + x2 + 1 (No need for the Hex column).
(b) Use your table to compute

Question 7: Answer the following questions about AES:

(a) What are the main advantages and disadvantages of 3DES? How does AES keep the advantages and eliminate the disadvantages?

(b) What are the main requirements of AES stated by NIST?

(d) The 4 stages in the AES round together provide high security. Explain the weakness of each stage alone.

(e) What are the main strength points of the design of the S-box in AES?

Question 10: Compare DES, 3DES and the three AES versions in terms of: block size, number of rounds, the main key length, and the round-key length.
Organize your answer in a 6x5 table (including the headers)

Attachment:- Permutation Exercise.rar

Reference no: EM132691310

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