With the wide/rapid spread of distributed systems for information processing, such as cloud computing and social networking, not only transmission but also processing is done on the internet. Therefore, a lot of studies on secure, efficient and flexible communications have been reported. Moreover, huge training data sets are required for machine learning and deep learning algorithms to obtain high performance. However, it requires large cost to collect enough training data while maintaining people's privacy. Nobody wants to include their personal data into datasets because providers can directly check the data. Full encryption with a state-of-the-art cipher (like RSA, or AES) is the most secure option for securing multimedia data. However, in cloud environments, data have to be computed/manipulated somewhere on the internet. Thus, many multimedia applications have been seeking a trade-off in security to enable other requirements, e.g., low processing demands, and processing and learning in the encrypted domain, Accordingly, we first focus on compressible image encryption schemes, which have been proposed for encryption-then-compression (EtC) systems, although the traditional way for secure image transmission is to use a compression-then encryption (CtE) system. EtC systems allow us to close unencrypted images to network providers, because encrypted images can be directly compressed even when the images are multiply recompressed by providers. Next, we address the issue of learnable encryption. Cloud computing and machine learning are widely used in many fields. However, they have some serious issues for end users, such as unauthorized access, data leaks, and privacy compromise, due to unreliability of providers and some accidents.