In this paper, we introduce a new one-dimensional modular chaotic system based on the composition of the cubic map and the exponential function (1-DCE). The evaluation results demonstrate that chaos performances of the 1-DCE chaotic map can be enhanced significantly, such as a better Lyapunov exponent, a high initial state sensitivity, and infinite chaotic ranges when compared with the classical cubic map and other existing chaotic maps. Furthermore, we investigate its applications in color image encryption, where a new efficient image encryption scheme is proposed using the confusion-diffusion pattern. We notice that an efficient diffusion method is performed based on a constructed mask and a ribbon before applying bitxor operator, where we change the pixel values completely and randomly. This technique gives a unique encryption results for each encryption time and makes it unpredictable and sensitive to any infinitesimal changes. Moreover, the computer results and security analysis prove the encryption method efficiency when compared with existing works in terms of the required high security level against statistical and differential attacks, and also its significant encryption rapidity.