The goal is to study a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring balancing and Lucas-balancing numbers. These functions are indeed the extension of Binet formulas for both balancing and Lucas-balancing numbers in continuous domain. Some identities concerning hyperbolic balancing and Lucas-balancing functions are also established. Further, a new class of square matrices, a generalization of balancing QB-matrices for continuous domain, is considered. These matrices indeed enable us to develop a cryptography method for secrecy purpose.