Design of an Image

Cryptosystem using Singular

Value Decomposition

Aseem Kumar Patel and Arup Kumar Pal

Department of Computer Science and Engineering

Bhubaneswar Institute of Technology

Bhubaneswar-752054, Orissa, India

Coverage of the Presentation

Introduction

Fundamentals of Image Compression,SVD and Secure Communication.

The proposed image cryptosystem (basedon SVD)

Experimental Results

Security Analysis

Conclusions

Introduction

Image Cryptosystem is useful to transmit importantimage through public channel.

It consists of mainly two parts Compression – Reducing network traffic

Encryption – Making data secret or confidential.

SVD is an efficient tool for image compression andsame has been used in our work.

Secure transmission of SVD compressed image usingcryptography techniques.

Image Compression Image Compression exploits various types of redundancies

present in the real images.

Two key factors are: Compression ratio

Tolerance in the quality degradation

Lossless compression Perfect reconstruction from compressed data (text documents,

medical images etc.)

Lossy compression Perfect reconstruction is not possible (is not essential) and we can

afford the partial loss in the image data as long as it is withintolerance.

4

Singular Value Decomposition (SVD)

Given any mn matrix M, algorithm to findmatrices U, D and V such that

Where U and V are column-orthogonal matrix and D is a diagonalmatrix.

Image Compression: Discard the trailinginsignificant singular values and keep the firstr (where r < k) singular values

5

Tm n m k k k n kM U D V

Tm n m r r r n rM U D V

Secure Communication

6

Encryption Key(K) Decryption Key(K)

Plain textCipher text

(Open Channel)

Enemy or

Adversary

Alice Encryption Decryption Bob

Plain text

Eve

7

The Proposed

Technique(Encoding)

Encoding

8

SVD

XOR

Key Formation

Proposed Algorithmic Steps for EncodingBeginStep 1: Decompose input secret image into three matrices U, D and

V respectively.Step 2: Select first r number of column vectors from each

decomposed matrix and discard rest of the column vectors.Step 3: Use truncated matrix, Vr obtained from Step 2 to generate a

key matrix.Step 4: Perform XOR operation between the produced key matrix

and the truncated matrix, Ur obtained from step 2.End

Decoding

10

XOR

Key Formation

T

r r rU D V

Algorithmic Steps (Decoding)

BeginStep 1: Generate a key matrix from the matrix Vr

using a secret PNS.Step 2: Perform XOR operation between the

produced key matrix and the matrix, EUr.Step 3: Reconstruct the image using the following

equationEnd

T

r r rU D V

12

Experimental Results

(b) Pepper Image(a) Lena Image

Original Test Images

Contd..(Experimental Results on Test Image Lena)

Decrypted ImageEncrypted image

Contd..(Experimental Results on Test Image Pepper)

Decrypted ImageEncrypted image

Security Analysis

The security of any encryption algorithm is measured bythe size of the key space.

In the proposed cryptosystem the secrecy depends on theselection of secret PNS.

In our experiment we have taken a PNS of length 65536.

The PNS guessing in our scheme is practically infeasiblebecause there will be one optimal PNS out of totalpossible 65536! PNSs.

Conclusions

In the proposed scheme, SVD is applied onthe secret image for compression andsubsequently the compressed files are partiallyencrypted for ensuring confidentiality.The proposed scheme reduces overallcomputation cost for encryption of the secretimage.The proposed scheme has been simulatedusing MATLAB and the satisfactory results havebeen found.

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