Partial Differential Equations for Data Compression and Encryption

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Partial Differential Equations for Data Compression and Encryption Ramaz Botchorishvili Faculty of Exact and

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

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

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

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

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

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

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 8 GGSWBS 12 11/27/2020

Lossy compression �JPEG 83261 bytes 15138 bytes 9 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 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

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 =

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 =

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 =

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 =

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 =

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

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 18 GGSWBS 12 11/27/2020

Cosine transform �Different interpretation �Least square approach �Interpolation For accurate representation of smooth functions

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

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

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

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

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

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

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 26 GGSWBS 12 11/27/2020

Smoothing example 27 GGSWBS 12 11/27/2020

Smoothing example 27 GGSWBS 12 11/27/2020

Smoothing example 28 GGSWBS 12 11/27/2020

Smoothing example 28 GGSWBS 12 11/27/2020

Other models Parameters 29 GGSWBS 12 11/27/2020

Other models Parameters 29 GGSWBS 12 11/27/2020

Estimates 30 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 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 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

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

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 Source, W. H. Press 35 GGSWBS 12 11/27/2020

Laplace interpolation 90% of pixels deleted restored Source, W. H. Press 36 GGSWBS 12

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

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

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

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

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

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

PDE based cryptosystem 42 GGSWBS 12 11/27/2020

Blakley-Rundell Cryptosystem main idea �Based on solution of hard problem �Initial function – data

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 44 GGSWBS 12 11/27/2020

Connecting text to function 45 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

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

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

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

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

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

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

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

Importance of different meshes 53 GGSWBS 12 11/27/2020

Sensitivity to numerical methods 54 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

Different time = different encrypted file 55 GGSWBS 12 11/27/2020

Niche – where could be used Seems not to be suitable for standard communication

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

Thank you for your attention 57 GGSWBS 12 11/27/2020