Chaotic maps such as the quadratic and tent maps have demonstrated great potential in constructing secure and lightweight cryptosystems due to their sensitive dependence on initial conditions and ergodic behavior. However, quadratic maps suffer from limited parameter ranges and quickly settle into periodic behavior, reducing unpredictability and weakening their cryptographic strength over time while tent maps display symmetry and linearity that can be exploited analytically, making them vulnerable to attacks if not properly hybridized or perturbed.  In this paper, we present a detailed investigation of these maps as core elements of a secure chaotic cryptosystem. We analyze their dynamical behavior using bifurcation diagrams, Lyapunov exponent plots, time series analysis and entropies in order to study the sequential behaviors of the maps. Furthermore, we hybridized the two maps to form a new chaotic map named Modified Hybrid Map (MHM). Our results show that both maps offer unique advantages in security design, and the integration of the two maps improves randomness, confusion, and diffusion properties which are very essential in cryptography.