It is very difficult to find the inverse of a matrix in Galois field using standard matrix inversion algorithms. Hence, any block-based encryption process involving matrix as a key will take considerable amount of time for decryption. The inverse of a self-invertible matrix is the matrix itself. So, if these matrices are used for encryption, the computational time of the decryption algorithm reduces significantly. In this paper, a new method of generating self-invertible matrix is presented. In addition to this, a new method of generating sparse matrices based on a polynomial function and the process of inversion of this matrix without using standard matrix inversion algorithms is also presented.

The product of these two types of matrices constitute the public key matrix whereas the matrices individually act as the private keys. This matrix will have a large domain and can also be used to design an asymmetric encryption technique. The inverse of the key matrix can be calculated easily by the receiver provided the components of the key i.e. the self-invertible and the sparse matrices are known. This public key is used to encrypt images using standard image encryption algorithm and it is tested with various gray-scale images. After encryption, the images are found to be completely scrambled. The image encryption process has very low computational complexity which is evident from comparison with AES(128). Moreover, since the number of key matrices are huge, brute force attack becomes very difficult.