A new public key cryptography using $M_{q}$ matrix

We consider a new class of square Fibonacci $(q+1)\times(q+1)$-matrices in public key cryptography. This extends previous cryptography using generalized Fibonacci matrices. For a given integer $q$, a $(q+1)\times(q+1)$ binary matrix $M_{q}$ is a matrix which nonzero entries are located either on the super diagonal or on the last row of the matrix. In this article, we have proposed a modified public key cryptography using such matrices as key in Hill cipher and key agreement for encryption-decryption of terms of $M_{q}$-matrix. In this scheme, instead of exchanging the whole key matrix, only a pair of numbers needed to be exchanged, which reduces the time complexity as well as the space complexity of the transmission and has a large key space.