## 7 Conclusions  
We improved the diffusion models so that they can, for the first time, unconditionally generate high-resolution volumetric (\(256\times 256\times 256\)) medical images. Moreover, on the basis of this new diffusion model, we proposed another formulation of image anonymization with which the processed images can satisfy provable Gaussian local differential privacy and we can generate images semantically different from the original image but belonging to the same class. Furthermore, we validated the formulation with high-resolution volumetric medical image anonymization. This method assumes no specific class of images, making it potentially applicable to any type of natural image. Low-resolution medical images are not well-suited for practical medical systems, and traditional standard diffusion models have struggled to model high-resolution volumetric medical images. In this study, we overcame this challenge and paved the way for applying deep generative models to practical medical systems. We believe that this improvement of the DDPM and the formulation of LDP algorithms combined with the DDPM can contribute to the secure sharing of practical images upstream of data processing.  
Conceptualization, H.S.; methodology, H.S.; software, H.S.; validation, H.S.; formal analysis, H.S.; investigation, T.N., T.K. and Y.N. ([PERSON]); resources, T.Y.; data curation, Y.N. ([PERSON]); writing--original draft preparation, H.S.; writing--review and editing, H.S., S.H., T.N., T.K., Y.N. ([PERSON]), Y.N. ([PERSON]), T.Y. and O.A.; visualization, H.S.; supervision, T.Y. and O.A.; project administration, H.S.; funding acquisition, H.S. and S.H. All authors have read and agreed to the published version of the manuscript.  
This work was supported by Japan Science and Technology Agency (JST), CREST Grant Number JPMJCR21M2, including the FY2023 AIP challenge program (Establishment of the diffusion models for high-resolution volumetric images; PI: [PERSON]), Japan.  
The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Review Board (or Ethics Committee) of the University of Tokyo Hospital (protocol code: 1461-(9) and date of approval: 16 September 2020).  
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.  
The code presented in this study is available on reasonable request from the corresponding author. The data are not publicly available due to privacy protection.  
The Department of Computational Diagnostic Radiology and Preventive Medicine, the University of Tokyo Hospital, is sponsored by HIMEDIC Inc., and Siemens Healthcare K.K. This research was conducted using the FUJITSU Supercomputer PRIMEHPC FX1000 and FUJITSU Server PRIMERGY GX25/0 (Wisteria/BDEC-01) at the Information Technology Center, the University of Tokyo.  
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.  
The following abbreviations are used in this manuscript:  
CIT Computed Tomography DDPM Denoising Diffusion Probabilistic Models GAN Generative Adversarial Networks LDP Local Differential Privacy MR Magnetic Resonance SGD Stochastic Gradient Descent