Do any hardened Linux distributions exist?

Awesome! Here are a few things that come to mind:

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Make sure you have some aliases/functions for common operations:

• audit2allow -a to view audit violations (or -d for dmesg audits)
  • also -r to add a requires statement for module construction
• restorecon -Rv to recursively apply file contexts from policy (or -FRv to also apply user context)
• rm -f /var/log/audit/audit.log.*; >/var/log/audit/audit.log to clear audit logs
  • note: sometimes lots of logfiles (audit.log.1, etc.) collect, slowing down audit2allow
• chown -R user:user PATH; chcon -R -u user_u PATH to recursively change labels to user
  • could be generalized for arbitrary Linux/SELinux users
• semanage fcontext -a -t TYPE PATH -s $SEUSER to add a custom file context to the policy
  • e.g. semanage fcontext -a -t "user_secrets_t" "/home/[^/]+/.secrets(/.*)?" -s user_u
  • I've had better luck with this approach than the standard method of creating a .fc file, but in any case a custom policy is needed to create custom types
• semanage fcontext -d PATH to remove a custom file context
• semanage fcontext -lC to list custom file contexts
• semodule -DB to rebuild policy with all dontaudit rules disabled
  • often, something will not work, but audit2allow doesn't show anything
• semodule -B to rebuild policy (with dontaudit rules)
• semodule -i MODULE.pp to install a module
• semodule -r MODULE to remove a module

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Also a few scripts for policy creation and management are essential. There are two basic approaches to policy creation: modules and policy modules.

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Modules: can be used to modify AVC rules and are pretty simple

# a violation has occurred that you want to allow or dontaudit
echo "module my_allow 1.0;" > my_allow.te
audit2allow -ar >> my_allow.te

# verify that my_allow.te has what you expect
cat my_allow.te

# build and install the module (replace mcs with whatever policy you are using)
make -f /usr/share/selinux/mcs/include/Makefile my_allow.pp
semodule -i my_allow.pp

# clear audit logs
rm -f /var/log/audit/audit.log.*; >/var/log/audit/audit.log

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Policy modules: can do anything, but are complicated, and the tools for creating them are mostly based on Red Hat.

Creating a new type:

# generate foo.fc, foo.if, and foo.te
sepolicy generate --newtype -t foo_var_lib_t -n foo

# note: see sepolicy-generate(8); sepolicy generate only supports the following
#       type suffixes, but its output files can be adapted to your use case
# _tmp_t
# _unit_file_t
# _var_cache_t
# _var_lib_t
# _var_log_t
# _var_run_t
# _var_spool_t
# _port_t

# modify the .fc file with the desired file contexts, for example (with s0 for mcs)
# /path/to/context/target	--	gen_context(system_u:object_r:type_t,s0)
#
# note: the "--" matches regular files, -d for directories, -c for character
#       devices, -l for symbolic links, -b for block devices, or can be omitted
#       to match anything. also, as mentioned before, I often have better luck
#       with `semanage fcontext`, especially for user directories
vi foo.fc

# build and install the policy module
make -f /usr/share/selinux/mcs/include/Makefile foo.pp
semodule -i foo.pp

# use restorecon to adjust the file contexts of any paths you have 

# by default, all operations involving this type will be denied
# (and are sometimes not audited)
semodule -DB # --disable_dontaudit
# ... use the type, collect violations ...
audit2allow -ar >> foo.te
# if dontaudit is disabled, you'll likely have a lot things to remove from here
vi foo.te

# ... repeat until rules regarding type are fully defined

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Creating a new application type:

# sepolicy-generate is made for Red Hat,
# but you can use --application to get started

# creates a bunch of files that define bar_t and bar_exec_t
sepolicy generate --application -n bar [-u USER] CMD

# remove the line making the app permissive (up to you, but
# I prefer using audit violations to define the permissions)
perl -i -00 -pe 's/^permissive bar_t;\n\n//g' bar.te

# ensure that the file bar_exec_t file context points to the right bin:
vi bar.fc

# build and install the policy module
make -f /usr/share/selinux/mcs/include/Makefile bar.pp
semodule -i bar.pp

# ... use the application, update AVC rules, repeat ...

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If your target application is interpreted, you'll need to write a custom C program that launches the interpreter in a specific context, then write your policy around that application. For example, you should execv something like this: /usr/bin/runcon -u user_u -t my_script_t /bin/bash PROG.