Description

Encryption , Image compression (20 Marks) Guidelines:
a) Code has to be developed in PYTHON (Open Source compiler IDEs)
b) Assignment will have to be carried out in teams of size TWO. Form your team and register in the Google form ( will be shared in the Group)
Drive shared folder
d) Approx 4 – 5 Weeks of time will be available before submission. Actual Demo dates will be broadcast. Hence look out !! ( Most likely to be a Google Meet)
Problem Definition, Data Sets, Testing and Logging Stats
Problem:
a) Implement 1-D & 2-D Fourier Transform & RSA Encryption on a M x N matrix to achieve Fast Polynomial Multiplication, Secure transmission and Lossy Image compression.
b) Step (a) to be done in the following sequence:
i) Implement 1-D DFT ,on coefficient vectors of two polynomials A(x), B(x) by multiplication of Vandermonde matrix . ( O(n2 ) – Complexity) ii) Implement 1-D FFT on the same vectors, of A(x) and B(x). Ensure above two steps produce same results. ( O(n logn) – Complexity) iii) Pointwise multiply results of Step (ii) to produce C(x) in P-V form iv) RSA encrypt (128-bit , 256-bit and 512-bit ) , with public key , the C(x) in PV form, for transmission security and decrypt with a private key and verify .
v) Implement 1-D Inverse FFT (I-FFT) on C(x), in PV form (Interpolation) to get C(x) in Coefficient form (CR) Polynomial. vi) Verify correctness of C(x) , by comparing with the coefficients generated by a Elementary “Convolution For Loop” on the Coefficients of A(x) and B(x) vii) Implement a 2-D FFT and 2-D I-FFT module using your 1-D version (This just means , applying FFT on the Rows First and Columns Next on
M x N matrix of numbers !!)
viii) Verify your of Step (vii) correctness on a Grayscale matrix ( which has random integer values in the range 0-255; 255 → White & 0 → Black)) ix) Apply your 2D-FFT on TIFF/JPG (lossless) Grayscale image and drop Fourier coefficients below some specified magnitude and save the 2D- image to a new file.
( relates to % compression – permanent Lossy compression)
( by sorting and retaining only coefficients greater than some(quantization) value. Rest are made 0.)
x) Apply 2D I-FFT, on the Quantized Grayscale image and render it to observe Image Quality.

Data Set Generation and Preparatory Reading Links :
a) For 1-D, DFT, FFT and Inverse DFT, Inverse FFT use randomly generated polynomial coefficient vectors for A(x) and B(x) of varying degree-bound sizes namely, n = 4, 8, 16, 32, 64, … 1024 and 2048
b) For 2-D Grayscale image, use randomly generated pixel values in the range 0-255 for testing 2-D FFT and 2-D Inverse FFT.
c) For testing on actual image, use a Grayscale TIFF or lossless JPEG image and use the Python OpenCV image manipulation API e.g imread( ) , imshape( ) etc., for accessing the raw pixel data of the 2-D image matrix
d) For comparing the Efficiency/Asymptotic complexity of your DFT, FFT, Inverse DFT and Inverse FFT, for increasing values of n, compare with Python Numpy built-in FFT/IFFT functions .
e) Below are set of of links for your review, before getting into the Assignment :

L1) Fast Fourier Transform. How to implement the Fast Fourier… | by Cory Maklin | Towards Data Science
L2) Understanding the FFT Algorithm | Pythonic Perambulations
L3) Image Transforms – Fourier Transform ( 2-D FFT)
L4) (2) Image Compression and the FFT – YouTube ( Steve Brunton )
L5) (2) Image Compression and the FFT (Examples in Python) – YouTube ( Steve Brunton) L6) OpenCV: Basic Operations on Imag
L7) Unraveling The JPEG
L8) The RSA Homonym
L9) RSA Encryption/Decryption Example – YouTube

Demo:
– All of the team members should be present ( Most likely to be a Google Meet)
– Should produce pdf of the report with Performance Metrics / Plots and Screen snap shots
– Should be able to demo the1-D & 2-D FFT, Inverse FFT, Encryption, Decryption for variable size of Polynomial degree and RSA key size
Report:
– About 10-15 pages in size
– Should contain your design and implementation details, snap shots results, critical code section developed by your team
– Timings for various degree polynomial multiplication and Timings for Encryption and Decryption, and FFT/IFFT with comparative numbers from Python Numpy
built-in functions
Last para of your report should contain your observations on the Learning Outcomes of this project.