## 这是一篇来自英国的关于学习对称密码系统的数学基础作业代写

**Assignment guidance**

Provide answers to the three exercises below. Answer all three exercises.

**Assessment tasks**

**Exercise 1: **Alice and Bob use the Hill cipher over Z3 to communicate.

(a) Let *G *= (Z2*×*23*, **·*) be the multiplicative group of 2×2 matrices with elements in Z3, i.e. matrices of the form

*g *=[*a b *

*c d*]*,*

where *a, b, c, d **∈ *Z3 and det(*g*) = *ad **− **bc **∈ *Z*∗ *3. What is the order of the group *|**G**|*? Explain how you can count the number of elements in the group. **[4 marks]**

(b) Show that the key *K *=[1 2

1 1] a valid encryption key for the Hill cipher in *G*. **[1 mark]**

(c) Find the inverse key *K**−*1 for the key *K *=[1 2

1 1] in *G*. **Hint: **

The inverse matrix *A**−*1 of a 2 *× *2 matrix

*A *=[*a b*

*c d]*

can be expressed as *A**−*1 = det(*A*) *−*1*A**adj *where the adjoint matrix is defifined as

*A**adj *=[*d **−**b*

*−**c a]*

and all the computations are performed in Z3. **[3 marks] **

**Exercise 2: **Alice and Bob communicate using the autokey cipher and the plaintext is encoded in the 26-character alphabet.

(a) Bob receives the ciphertext ‘PVMXWPIHUUNEOEKGMXHQMFIOG’ and knows that the encryption key is ‘TRADE’. Explain the steps of how Bob performs decryption in the autokey cipher and fifind the plaintext. **[3 marks] **

(b) Explain why the autokey cipher is not vulnerable to a frequency attack even if the key length is known. Mention two other ciphertext only attacks that can be used against the autokey cipher when the key length is short. **[3 marks] **

**Exercise 3: **Alice and Bob use the DES block cipher for secure communication.

(a) The 3DES cipher involves three rounds of DES applied successively with a total key length of 168 bits. Explain why 4DES using four rounds of DES does not increase the security compared to 3DES when the attacker has access to a known plaintextciphertext pair and explain the method of attack. **[4 marks] **

(b) Alice and Bob use DES in the cipher feedback (CFB) mode. Suppose that during transmission, one bit in the *i*’th ciphertext block *C**i *is flflipped. How many plaintext blocks will be affffected during decryption and how? Justify your answer by appealing to the CFB decryption algorithm diagram below. **[2 marks] **

**General guidance and study support**

The MS Teams group for COMP3223 Cryptography will be used for general support for this assignment. If your question would reveal parts of the answer to any problem, please send instead a private message to the module leader on MS Teams.

**Assessment criteria and marking process**

Assessment marks and feedback will be available on Minerva within three weeks of the submission deadline. Late submissions are allowed,standard late penalties apply.

**Presentation and referencing**

When writing mathematical formulas, use similar notation and symbols as during the lectures and tutorials. Hand-written sections for mathematical notation are acceptable but need to be clearly readable.

You may assume theorems and other results that have been presented during lectures and tutorials as known. Any other theorems need to be cited using standard citation practice.

**Submission requirements**

This is an individual piece of work. Submit your answers through Turnitin as one PDF document (generated either in Word or with LaTeX).

You may use hand-written and scanned pages for mathematical formulas, but these need to be clearly legible and the document must contain at least some typeset text or Turnitin will reject it. All submissions will be checked for academic integrity.

**Academic misconduct and plagiarism**

Academic integrity means engaging in good academic practice. This involves essential academic skills, such as keeping track of where you fifind ideas and information and referencing these accurately in your work.

By submitting this assignment you are confifirming that the work is a true expression of your own work and ideas and that you have given credit to others where their work has contributed to yours.

**Assessment/marking criteria grid**

Total number of marks is 20, divided as follows:

Exercise 1 (Hill cipher): 8 marks

Exercise 2 (Autokey cipher): 6 marks

Exercise 3 (DES and block cipher operation modes): 6 marks