Asymmetric Image Encryption based on Cipher Matrices

Abstract

In most of the cryptological methods, the encrypted data or the cipher texts
maintain same statistics of the plain texts, whereas matrix encryption method
does not keep the statistics of individual cipher texts. However, it maintains the
statistics of block of characters of size m where m is the size of the key matrix. One
of the important features of the cipher matrix in Residue Number System (RNS)
is that it is highly dicult and time consuming to obtain its inverse by standard
inverse algorithms. Matrix in RNS does not have all the eigen values as dened
in complex eld. The eigen factors of a matrix is dened as the irreducible factors
of the characteristic equation(eigen function). All the above properties are valid
for cipher matrix in Galois Field. The public key is generated by using two types
of matrices. One of these matrices is a self-invertible matrix or an orthonormal
matrix in Galois eld whereas the other matrix is a diagonally dominant matrix.
Matrix inversion is very dicult and time consuming when size of matrix and
modulo number are large. The computational overhead in generalized Hill cipher
can be reduced substantially by using self-invertible matrices. Self-invertible ma-
trices uses less space compared to invertible matrices. In order to overcome this
problem, p(modulo) is made very large so that there would be at least pn=2 possible
matrices making it extremely dicult for the intruder to nd the key matrix. In
this thesis several methods of generating self-invertible matrix are proposed.
Orthogonal Transform is used in signal processing. Modular Orthogonal Trans-
form such as Walsh, Hadamard, Discrete Cosine Transform, Discrete Sine Trans-
form, Discrete Fourier Transform have been used for encryption of image. The
orthogonal matrices can be used as asymmetric key for encryption. In this work
various methods of generating orthogonal matrices have been proposed. Matrix
having primitive polynomial as eigen factors is used resulting in robust encryp-
tion.
A novel operation called exponentiation and its inverse has been dened in this
thesis. All the properties of this new operation have been analyzed in Zp. This
operation is used for encryption of image. The original image can be obtained by using the same exponentiation operation.
Chaotic sequence and chaotic signal generation is widely used in communica-
tion. Two stages of image encryption scheme using chaotic sequence is proposed
in this work. First stage of encryption by chaotic sequence generated in GF(p)
and the second stge of encryption is carried out by one of the encryption methods
discussed in the previous chapters.
Standard images have been used for encryption during simulation.
Keywords: Encryption, Decryption, Cipher matrix, Public key, Private key,
Residue number system, Eigen function, self-invertible matrix, Orthogonal, Ga-
lois Field, Exponentiation, Chaotic sequence.