Reference no: EM133013272
Task: Explore how Public Key Cryptography works.
Use the Wiener attack to break your private key and is this way steal your identity by using your private key to encode another number and masquerade as you.
In your answers to each part properly number each question and your answer to it. Please refer to the ‘How to write a technical report" in the resources directory.
Making your own Keys:
Explore how Public Key Cryptography works and summarise this in your own words with particular attention to how the method generates these keys.
From your exploration above, generate Private and Public Keys using an example.
Using your student ID, determine how bits the number part can be represented with.
For a particular number of binary bits determine the number of prime numbers available in that bit range.
(a) Using the Wolframalpha "nextprime" function and your student ID create two prime numbers which are of the form 4x+1. Such that P1=4x 1+1 and P2=4x 2+1.
(b) Check that they have the same number of binary bits.
(a) Using the Wolframalpha "nextprime" function and your student ID create two prime numbers which are of the form 4x+3. Such that P3=4x 3+3 and P4=4x 4+3.
(b) Check that they have the same number of binary bits.
(a) Use P1 and P2 to generate your public keys and encrypt your student ID.
Use P3 and P4 to generate your public keys and encrypt your student ID.
Create a private key and use this to decrypt and recover your student ID.
Breaking the Keys:
Using Overmars triangles: Let P1=C1 and P2=C2 and let the smallest value of the two squares be n express the sides of each triangle as:
c=n^2+(n+2m-1)^2 b=2n(n+2m-1) a=(2m-1)(2m+2n-1)
N:N_1= P1 P2. Find N_1 as two sums of two squares?
Using Euler's factorization method, show how the original prime numbers can be recovered using the sum of squares method.
Can N_2= P3 P4 be expressed as the sum of two squares? Can Euler's Factorization be used? Why?
Represent P3 and P4 as the difference of two squares
N:N_2= P3 P4 as two differences of two squares
Factorise with Fermat's Method.
If φ(n)=(P_1-1)(P_2-1) show that if φ(n) and N are known, the primes P1 and P2 can be recovered.
Using your public key, show how your private keys can be recovered using the Wiener attack.
As the villain, armed with this information and masquerading as you, send a different encrypted Student ID (not yours) encrypted with the recovered key. Use the same public key as earlier to recover the masquerading Student ID.
Final Report:
Your final report is a professional representation of parts (1) and (2) above, drawing on the relevant literature (including journals) where relevant and properly referencing these. You should also look at linkages between parts (1) and (2) and explore the structures of primes with particular attention to types 4x+1 and 4x+3.
Use the references provided in the Resources directory as well as any others that you find useful. Do not pad out the references. If references are used ensure that there is a point to mentioning the reference and that it is relevant to the points you are making.
The book Chapter by Overmars provides useful methods for "Making" and "Breaking" keys.
Marks will be awarded for your ability to connect the questions in "Making" and "Breaking" keys together in a mathematical context.
Please refer to the ‘How to write a technical report" in the Resources/Assignment" directory. Also the marking rubric will provide a guide to how the assignment report will be marked. Some useful references can also be found there. This is not a definitive list and students should use their initiative to find additional sources of information.
Attachment:- Cryptography.rar