Image encryption using chaotic maps has recently become a crucial research focus because chaotic maps offer high randomness, ergodicity, and sensitivity to controlling parameters and initial states. However, higher-dimensional chaotic sequences are more random at the expense of increased complexity, posing a challenge. Therefore, our goal is to achieve a similar level of randomness using low-dimensional chaotic maps. In this paper, we introduce a new low-dimensional chaotic map. It has a better chaotic range, complex behavior, unpredictability, ergodicity, and sensitivity to initial values and controlling parameters. In our proposed image encryption scheme, we permute the pixels using a swap-based operation between the old location and the new location of the pixel, where the new location is found using the indices of the proposed chaotic map and the previous swapped pixel. Then, we change the permuted pixel values using a substitution technique where modular addition and XOR are used. The proposed framework is more robust against various attacks and offers enhanced security for image encryption applications. The experimental findings show our method’s efficacy in terms of security. It outperforms current techniques in terms of correlation coefficients, NPCR, UACI, and entropy.