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Chaotic systems, presenting complex and nonreproducible dynamics, may be found in nature, from the interaction between planets to the evolution of weather, but can also be tailored using current technologies for advanced signal processing. However, the realization of chaotic signal generators remains challenging due to the involved dynamics of the underlying physics. In this paper, we experimentally and numerically present a disruptive approach to generate a chaotic signal from a micromechanical resonator. This technique overcomes the long-established complexity of controlling the buckling in micro/nanomechanical structures by modulating either the amplitude or the frequency of the driving force applied to the resonator in the nonlinear regime. The experimental characteristic parameters of the chaotic regime, namely, the Poincaré sections and Lyapunov exponents, are directly comparable to simulations for different configurations. These results confirm that this dynamical approach is transposable to any kind of micro/nanomechanical resonator, from accelerometers to microphones. We demonstrate a direct application exploiting the mixing properties of the chaotic regime by transforming an off-the-shelf microdiaphragm into a true random number generator conforming to the National Institute of Standards and Technology specifications. The versatility of this original method opens new paths to combine the unique properties of chaos with the exceptional sensitivity of microstructures, leading to emergent microsystems.