Partial Differential Equations for Data Compression and Encryption
- Slides: 57
Partial Differential Equations for Data Compression and Encryption Ramaz Botchorishvili Faculty of Exact and Natural Sciences Tbilisi State University 1 GGSWBS 12 11/27/2020
Data Compression Original file Compressed file Compression Size = big 2 GGSWBS 12 Size = small 11/27/2020
Data Compression Original file Compressed file Compression Size = big Size = small Size = big Decompression Recovered file 3 GGSWBS 12 11/27/2020
Data Compression Original file Compressed file Compression Size = big Size = small Size = big Decompression Recovered file 4 GGSWBS 12 11/27/2020
Data Compression Lossy or lossless ? Original file Compressed file Compression Size = big Size = small Size = big Decompression Recovered file 5 GGSWBS 12 11/27/2020
Data Compression Lossless compression Original file = Recovered file Compression Size = big Decompression Size = big Compressed file 6 GGSWBS 12 11/27/2020
Data Compression Lossless compression Original file = Recovered file Compression Size = big Decompression Size = big Original file is not Recovered file Lossy compression 7 GGSWBS 12 11/27/2020
Lossy compression �JPEG 83261 bytes 8 GGSWBS 12 11/27/2020
Lossy compression �JPEG 83261 bytes 15138 bytes 9 GGSWBS 12 11/27/2020
Lossy compression �JPEG 4725 bytes 83261 bytes 15138 bytes 10 GGSWBS 12 11/27/2020
Lossy compression �JPEG, wikipedia 4725 bytes 83261 bytes 1523 bytes 15138 bytes 11 GGSWBS 12 11/27/2020
Jpeg – cosine transform � Approach for data compression �Data = function �Algorithm = � Expand function e. g. Fourier series � Approximate function � Set threshold � Throw away Fourier coefficients smaller then threshold 12 GGSWBS 12 � Store big coefficients only! 11/27/2020
Jpeg – cosine transform � Approach for data compression �Data = function �Algorithm = � Expand function e. g. Fourier series � Approximate function � Set threshold � Throw away Fourier coefficients smaller then threshold 13 GGSWBS 12 � Store big coefficients only! 11/27/2020
Jpeg – cosine transform � Approach for data compression �Data = function �Algorithm = � Expand function e. g. Fourier series � Approximate function � Set threshold � Throw away Fourier coefficients smaller then threshold 14 GGSWBS 12 � Store big coefficients only! 11/27/2020
Jpeg – cosine transform � Approach for data compression �Data = function �Algorithm = � Expand function e. g. Fourier series � Approximate function � Set threshold � Throw away Fourier coefficients smaller then threshold 15 GGSWBS 12 � Store big coefficients only! 11/27/2020
Jpeg – cosine transform � Approach for data compression �Data = function �Algorithm = � Expand function e. g. Fourier series � Approximate function � Set threshold � Throw away Fourier coefficients smaller then threshold 16 GGSWBS 12 � Store big coefficients only! 11/27/2020
JPEG – Discrete cosine transform �Cosine basis functions 17 GGSWBS 12 11/27/2020
Cosine transform �Different interpretation �Least square approach �Interpolation 18 GGSWBS 12 11/27/2020
Cosine transform �Different interpretation �Least square approach �Interpolation For accurate representation of smooth functions with small variation less coefficients are needed compared to highly variable functions 19 GGSWBS 12 11/27/2020
New approach �preprocessing before compression �Smooth out data �Compress smoothed data with existing methods 20 GGSWBS 12 11/27/2020
New approach �preprocessing before compression �Smooth out data �Compress smoothed data with existing methods How to smooth data? 21 GGSWBS 12 11/27/2020
New approach �preprocessing before compression �Smooth out data �Compress smoothed data with existing methods How to smooth data? Apply Diffusion ? 22 GGSWBS 12 11/27/2020
New approach �preprocessing before compression �Smooth out data �Compress smoothed data with existing methods How to smooth data? Apply Diffusion ? No, time can not be inverted 23 GGSWBS 12 11/27/2020
Other models �Time can be inverted �Smoothing property by analogy of diffusion 24 GGSWBS 12 11/27/2020
Other models �Time can be inverted �Smoothing property by analogy of diffusion 25 GGSWBS 12 11/27/2020
Smoothing example 26 GGSWBS 12 11/27/2020
Smoothing example 27 GGSWBS 12 11/27/2020
Smoothing example 28 GGSWBS 12 11/27/2020
Other models Parameters 29 GGSWBS 12 11/27/2020
Estimates 30 GGSWBS 12 11/27/2020
Laplace interpolation Source, W. H. Press 31 GGSWBS 12 11/27/2020
Laplace interpolation original Source, W. H. Press 32 GGSWBS 12 11/27/2020
Laplace interpolation 50% of pixels deleted original Source, W. H. Press 33 GGSWBS 12 11/27/2020
Laplace interpolation 50% of pixels deleted original restored Source, W. H. Press 34 GGSWBS 12 11/27/2020
Laplace interpolation 90% of pixels deleted Source, W. H. Press 35 GGSWBS 12 11/27/2020
Laplace interpolation 90% of pixels deleted restored Source, W. H. Press 36 GGSWBS 12 11/27/2020
PDE inpainting and interpolation Image Confidence function Unknown function Differential operator, e. g. Laplacian Source, J. Weickert et a 37 GGSWBS 12 11/27/2020
PDE interpolation and compression �Diffusion equations �Homogenous diffusion �Nonlinear diffusion Source, J. Weickert et a 38 GGSWBS 12 11/27/2020
PDE compression � Select points/boundary conditions and let them diffuse Some approaches can already beat JPEG, source Weickert et al 39 GGSWBS 12 11/27/2020
PDE compression � Select points/boundary conditions and let them diffuse Some approaches can already beat JPEG, source Weickert et al � Smooth data, select points/boundary conditions and let them diffuse Future projects for students 40 GGSWBS 12 11/27/2020
PDE compression � Select points/boundary conditions and let them diffuse Some approaches can already beat JPEG, source Weic � Smooth data, select points/boundary conditions and let them diffuse Future projects for students Which equations? Why not to start from refinement? 41 GGSWBS 12 Bitsadze-Samarski problem 11/27/2020
PDE based cryptosystem 42 GGSWBS 12 11/27/2020
Blakley-Rundell Cryptosystem main idea �Based on solution of hard problem �Initial function – data �Solution – encryption �Inverse problem – decryption �Key – coefficients of PDEs 43 GGSWBS 12 11/27/2020
Connecting text to function 44 GGSWBS 12 11/27/2020
Connecting text to function 45 GGSWBS 12 11/27/2020
Connecting text to function Piecewise constant Encription block size 46 GGSWBS 12 11/27/2020
Encription Key Information to be encrypted u(T, x) – encrypted message Problem: decryption is not possible – heat equation can not be inverted in tim Solution: using pseudo parabolic equations 47 GGSWBS 12 11/27/2020
PDE based cryptosystem - principles �Based on really hard problems – what computational power is needed for few seconds of computations? 48 GGSWBS 12 11/27/2020
PDE based cryptosystem - principles �Based on really hard problems – what computational power is needed for few seconds of computations? �File for encryption – convert to real valued function on complex computational domain �No small block sizes for encryption but entire data 49 GGSWBS 12 11/27/2020
PDE based cryptosystem - principles �Based on really hard problems – what computational power is needed for few seconds of computations? �File for encryption – convert to real valued function on complex computational domain �No small block sizes for encryption but entire data �Computational domain – could be any figure in N dimensional space, N=2, 3, 4, … �Different meshes could be used for the same data in the same domain 50 GGSWBS 12 11/27/2020
PDE based cryptosystem - principles � Based on really hard problems – what computational power is needed for few seconds of computations? � File for encryption – convert to real valued function on complex computational domain � No small block sizes for encryption but entire data � Computational domain – could be any figure in N dimensional space, N=2, 3, 4, … � Different meshes could be used for the same data in the same domain � Different encryption for different time moment � Inventing different boundary conditions � Inventing different equations and problems, e. g. nonlocal in time � sensitive to numerical methods 51 GGSWBS 12 11/27/2020
PDE based cryptosystem - principles � Based on really hard problems – what computational power is needed for few seconds of computations? � File for encryption – convert to real valued function on complex computational domain � No small block sizes for encryption but entire data � Computational domain – could be any figure in N dimensional space, N=2, 3, 4, … � Different meshes could be used for the same data in the same domain � Different encryption for different time moment � Inventing different boundary conditions � Inventing different equations and problems, e. g. nonlocal in time � sensitive to numerical methods � Key – combination of all the above 52 GGSWBS 12 11/27/2020
Importance of different meshes 53 GGSWBS 12 11/27/2020
Sensitivity to numerical methods 54 GGSWBS 12 11/27/2020
Different time = different encrypted file 55 GGSWBS 12 11/27/2020
Niche – where could be used Seems not to be suitable for standard communication between tw Could be used for encrypting databases inside organization 56 GGSWBS 12 11/27/2020
Thank you for your attention 57 GGSWBS 12 11/27/2020
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