The rapid development of the Internet leads to a surge in the amount of information transmission and brings many security problems. For multimedia information transmission, especially digital images, it is necessary to compress and encrypt at the same time. The emergence of compressive sensing solves this problem. Compressive sensing can compress and encrypt at the same time, which can not only reduce the transmission bandwidth of the network but also improve the security of the system. However, when using compressive sensing encryption, the whole measurement matrix needs to be stored, and the compressive sensing can be combined with a chaotic system, so only the generation parameters of the matrix need to be stored, and the security of the system can be further improved by using the sensitivity of the chaotic system. This paper introduces a secure and efficient image compression-encryption scheme using a new chaotic structure and compressive sensing. The chaotic map used in the scheme is generated by our new and universal chaotic structure, which not only expands the chaotic range of the chaotic system but also improves the performance of the chaotic system. After analyzing the performance comparison of traditional one-dimensional chaotic maps and some existing methods, the image compression-encryption scheme based on a new chaotic structure and compressive sensing has a good encryption effect and large keyspace, which can resist brute force attack and statistical attack.