We are aware that a major cryptosystem element plays a crucial part in maintaining the security and robustness of cryptography. Various researchers are focusing on creating new forms of cryptography and improving those that already exist using the principles of number theory and linear algebra. In this article, we have proposed an Extended generalized Fibonacci matrix (recursive matrix of higher order) having a relation with Extended generalized Fibonacci sequences and established some properties in addition to that usual matrix algebra. Further, we proposed a modified public key cryptography using these matrices as keys in Affine-Hill Cipher and key agreement for encryption-decryption with the combination of terms of Extended generalized Fibonacci sequences under prime modulo. This system has a large key space and reduces the time complexity as well as space complexity of the key transmission by only requiring the exchange of pair of numbers(parameters) as opposed to the entire key matrix.

A Fibonacci coding method is introduced using Extended Generalized Fibonacci Polynomials in this paper. A new square matrix Qⁿm (a, b), the nᵗʰ power of Qm(a, b) of order m × m is defined whose elements are based on extended Generalized Fibonacci Polynomial. Matrix Qⁿ m (a, b) for integer x ≥ 1, a ≥ 1 and b ≥ 1 is considered as the encoding matrix and a matrix Q−ⁿm (a, b) is considered as decoding matrix. An error-detection and error-correction method is also defined in Extended Generalized Fibonacci polynomials.