Data security has become one of the most critical issues in information technology. Among the approaches developed to ensure secure data transmission and storage, chaotic maps play a central role due to their inherent unpredictability and sensitivity to initial conditions. In particular,1D chaotic maps have attracted significant attention because of their low computational complexity and ease of implementation in practical systems. In this study, a novel 1D hybrid chaotic map is proposed through the hybridization of two widely used optimization benchmark functions, namely Problem 14 and Problem 20. The construction of the proposed map is based on multiplying and combining the initial and latter segments of these benchmark functions, followed by the incorporation of a control parameter to enhance its dynamic behavior. The newly designed chaotic system was rigorously evaluated through a series of standard security analyses, including statistical randomness tests, Lyapunov exponent, and sensitivity evaluations. Furthermore, the proposed map was integrated into an image encryption framework to validate its cryptographic applicability. A set of 10 natural images obtained from public datasets, along with the Lena color image, was employed to assess the practical performance of the encryption scheme. Extensive experiments covering information entropy, correlation analysis, chi-square and variance tests, the number of pixels change rate, and the unified average changing intensity metrics, as well as robustness evaluations under cropping and noise attacks, confirm that the proposed encryption scheme achieves high security and reliability. Comparative results demonstrate that the method performs competitively with, and in some metrics better than, state-of-the-art techniques.