Chaos theory has been widely used in the design of image encryption schemes. Some low-dimensional chaotic maps have been proved to be easily predictable because of their small chaotic space. On the other hand, high-dimensional chaotic maps have a larger chaotic space. However, their structures are too complicated, and consequently, they are not suitable for real-time image encryption. Motivated by this, we propose a new fractional one-dimensional chaotic map with a large chaotic space. The proposed map has a simple structure and a high chaotic behavior in an extensive range of its control parameters values. Several chaos theoretical tools and tests have been carried out to analyze and prove the proposed map’s high chaotic behavior. Moreover, we use the proposed map in the design of a novel real-time image encryption scheme. In this new scheme, we combine the substitution and permutation stages to simultaneously modify both of the pixels’ positions and values. The merge of these two stages and the use of the new simple one-dimensional chaotic map significantly increase the proposed scheme’s security and speed. Besides, the simulation and experimental analysis prove that the proposed scheme has high performances.