Debian yes, but don't install from flatpaks or docker. Neither is secure.
AppImage can be secure if the release is signed.
Docker can pull images securely, but it's disabled by default and many developers don't sign their releases, so even if you enable it client-side there's a risk you'll download something malicious.
Flatpak is never secure because it doesn't support signing of releases at all.
Apt is always secure because all packages must be cryptographically signed (by default).
You should switch to something that's actually secure. Flatpak devs haven't addressed this since 2015, and I doubt they ever will. They don't seem to care about security.
Your earlier comment complains about pulling images securely, presumably meaning signature verification, which I believe Flatpak does.
The report you linked is about tying downloaded sources to their author using public key infrastructure, which is a different issue. APT and dpkg don't do that, either. (I know this because I build and publish with those, too.)
Can you name a packaging system that does? I can't. I would like to see it (along with reproducible builds) integrated into the software ecosystem, and I think we're moving in that direction, but it will take time to become common.
I have my own criticisms of Flatpak, mostly regarding the backwards permissions model (packages grant themselves permissions by default) and sloppy sandboxing policies on Flathub, so I caution against blindly assuming it's safe. But claiming that it doesn’t support signing of releases is just plain false.
If the flatpak (flathub?) repo was compromised and started serving malicious packages, the client would happily download & install them because it doesn't have any cryptographic authenticity checks.
APT and dpkg don’t do that
Apt does verify the authenticity of everything it downloads (by default) using PGP signatures on SHA256SUMS manifest files. This provides cryptographic authenticity of everything it downloads. Flatpak doesn't do this.
I'm talking about the end-user securely downloading packages from the repo, not how the package maintainer obtains the software upstream.
How a package maintainer obtains the software from the source is dynamic and depends on the package. Ideally those releases are signed by the developer. In any case, if the package is poisoned when grabbing the source, it's much easier for the community to detect than a targeted MITM attack on a client obtaining it from the repo.
I can say that I do maintain a software project that's in the repo, and we do sign it with our PGP release key. Our Debian package maintainer does verify its authenticity by checking the release's signature. So the authenticity is checked both at the source and when downloading the package.
This is not correct. APT always verifies cryptographic signature unless you explicitly disable it.
Yet it's very important to understand who is signing packages. What kind of review process did the software go through? What kind of vetting did the package maintainer themselves go through?
If software is signed only by the upstream developer and no 3rd party review is done by a distribution this means trusting a stranger's account on a software forge.
Update: the Debian infrastructure supports checking gpg signatures from upstream developers i.e. on the tarballs published on software forges.