Japan's government aims to promote the linkage of medical records, including medical genomic testing data and personal health records (PHRs), via cloud computing (the cloud). However, linking national medical records and using them for health care research can be controversial. Additionally, many ethical issues with using cloud networks with health care and genome data have been noted. However, no research has yet explored the Japanese public's opinions about their PHRs, including genome data, being shared for health care research or the use of the cloud for storing and analyzing such data. Therefore, we conducted a survey in March 2021 to clarify the public's attitudes toward sharing their PHRs, including genome data and using the cloud for health care research. We analyzed data to experimentally create digital health basic literacy scores (BLSs). Our results showed that the Japanese public had concerns about data sharing that overlapped with structural cloud computing issues. The effect of incentives on changes in participants' willingness to share data (WTSD) was limited. Instead, there could be a correlation between WTSD and BLSs. Finally, we argue that it is vital to consider not only researchers but also research participants as value cocreators in health care research conducted through the cloud to overcome both parties' vulnerability.