Background. One-dimensional chaotic maps present attractive properties for cryptographic applications but are limited by restricted chaotic parameter ranges, insufficient entropy generation, and numerical instability in finite-precision implementations. Methods. We introduce the Logistic-Rational Map (LRM), a novel chaotic system that integrates a logistic component with a bounded rational term under modular arithmetic. The dynamical characteristics are rigorously evaluated through bifurcation analysis, Lyapunov exponent computation, sensitivity to initial conditions assessment, Shannon entropy measurement, and correlation analysis. Results. The proposed LRM exhibits significantly extended chaotic behavior across parameter space, near-optimal statistical distribution (Shannon entropy approaching 10), minimal correlation coefficients, and improved numerical robustness achieved through elimination of singularities within the operational domain (0,1). Conclusions. The demonstrated properties of strong mixing, minimal predictability, and enhanced numerical stability establish the LRM as a promising foundation for cryptographically secure pseudorandom number generation systems.