A process may get an entirely private mount namespace in case it---or one of its ancestors---was started by an invocation of the clone(2) system call that had the CLONE_NEWNS flag set. This handles the '/' part of the pathname.
If the pathname does not start with the '/' character, the starting lookup directory of the resolution process is the current working directory of the process --- or in the case of openat(2)-style system calls, the dfd argument (or the current working directory if AT_FDCWD is passed as the dfd argument). The current working directory is inherited from the parent, and can be changed by use of the chdir(2) system call.)
Pathnames starting with a '/' character are called absolute pathnames. Pathnames not starting with a '/' are called relative pathnames.
If the process does not have search permission on the current lookup directory, an EACCES error is returned ("Permission denied").
If the component is not found, an ENOENT error is returned ("No such file or directory").
If the component is found, but is neither a directory nor a symbolic link, an ENOTDIR error is returned ("Not a directory").
If the component is found and is a directory, we set the current lookup directory to that directory, and go to the next component.
If the component is found and is a symbolic link (symlink), we first resolve this symbolic link (with the current lookup directory as starting lookup directory). Upon error, that error is returned. If the result is not a directory, an ENOTDIR error is returned. If the resolution of the symbolic link is successful and returns a directory, we set the current lookup directory to that directory, and go to the next component. Note that the resolution process here can involve recursion if the prefix ('dirname') component of a pathname contains a filename that is a symbolic link that resolves to a directory (where the prefix component of that directory may contain a symbolic link, and so on). In order to protect the kernel against stack overflow, and also to protect against denial of service, there are limits on the maximum recursion depth, and on the maximum number of symbolic links followed. An ELOOP error is returned when the maximum is exceeded ("Too many levels of symbolic links").
As currently implemented on Linux, the maximum number of symbolic links that will be followed while resolving a pathname is 40. In kernels before 2.6.18, the limit on the recursion depth was 5. Starting with Linux 2.6.18, this limit was raised to 8. In Linux 4.2, the kernel's pathname-resolution code was reworked to eliminate the use of recursion, so that the only limit that remains is the maximum of 40 resolutions for the entire pathname.
The resolution of symbolic links during this stage can be blocked by using openat2(2), with the RESOLVE_NO_SYMLINKS flag set.
The path resolution process will assume that these entries have their conventional meanings, regardless of whether they are actually present in the physical filesystem.
One cannot walk up past the root: "/.." is the same as "/".
One can walk out of a mounted filesystem: "path/.." refers to the parent directory of "path", outside of the filesystem hierarchy on "dev".
Traversal of mount points can be blocked by using openat2(2), with the RESOLVE_NO_XDEV flag set (though note that this also restricts bind mount traversal).
Of the three bits used, the first bit determines read permission, the second write permission, and the last execute permission in case of ordinary files, or search permission in case of directories.
Linux uses the fsuid instead of the effective user ID in permission checks. Ordinarily the fsuid will equal the effective user ID, but the fsuid can be changed by the system call setfsuid(2).
(Here "fsuid" stands for something like "filesystem user ID". The concept was required for the implementation of a user space NFS server at a time when processes could send a signal to a process with the same effective user ID. It is obsolete now. Nobody should use setfsuid(2).)
Similarly, Linux uses the fsgid ("filesystem group ID") instead of the effective group ID. See setfsgid(2).
On Linux, superuser privileges are divided into capabilities (see capabilities(7)). Two capabilities are relevant for file permissions checks: CAP_DAC_OVERRIDE and CAP_DAC_READ_SEARCH. (A process has these capabilities if its fsuid is 0.)
The CAP_DAC_OVERRIDE capability overrides all permission checking, but grants execute permission only when at least one of the file's three execute permission bits is set.
The CAP_DAC_READ_SEARCH capability grants read and search permission on directories, and read permission on ordinary files.