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I have an external hard drive, which has a ntfs file system.

From output of sudo fdisk -l:

Disk /dev/sdb: 931.5 GiB, 1000170586112 bytes, 1953458176 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disklabel type: dos
Disk identifier: 0x00023f15

Device Boot Start End Sectors Size Id Type
/dev/sdb1 2048 1953458175 1953456128 931.5G 7 HPFS/NTFS/exFAT

From output of mount:

/dev/sdb1 on /media/t/My Passport type fuseblk (rw,nosuid,nodev,relatime,user_id=0,group_id=0,default_permissions,allow_other,blksize=4096,uhelper=udisks2)

If I set some permission bits on a directory in the file system , for example to be executable and readable and writable only by root, will the permission bits be respected when I plug the external hard drive to a Windows system?

I would like to find a way to restrict access to a directory on my external hard drive from any OS (in particular, Linux and Windows). Also see https://askubuntu.com/a/104544/1471

I just found that chmod doesn't work on the external hard drive, and I wonder why and how I can do about it:

$ ls -ld temp/
drwxrwxrwx 1 t t 144 May 27 18:31 temp/
$ sudo chmod o-rwx temp
$ ls -ld temp/
drwxrwxrwx 1 t t 144 May 27 18:31 temp/

Thanks.

If I set some permission bits on a directory in the file system , for example to be executable and readable and writable only by root, will the permission bits be respected when I plug the external hard drive to a Windows system?

No, UNIX file permissions don't work on Windows.

I would like to find a way to restrict access to a directory on my external hard drive from any OS.

In general, every OS has its different way to set permissions, so that's not possible. You'll need to set the appropriate permissions on the files depending on which OS the filesystem is currently mounted.

Quoting from this answer:

NTFS has Windows ACEs. Unix uses "mode bits" on each file.

On NTFS, each file can have an owner, and zero or more Windows access control entries (ACEs). An ACE consists of a principal (users and groups are principals), a set of operations (Read, Write, Execute, etc.) and whether those operations are allowed or denied. Files can have many ACEs. Other objects in Windows other than files can have ACEs as well, such as registry entries, printer objects, and other things. All ACEs are taken into account when a file operation occurs. Deny takes precedence over allow. Windows ACEs support inheritance where you can set an ACE for a directory and have it automatically propagate to lower level directories.

Files in Unix have an owning user (owner) and an owning group (owner-group). There are three fixed "principals" which are owner, members of the owning group, and everyone else (a.k.a world). For each principal there are three "bits" which cover read, write, and execute abilities. (these have different meanings for directories than files, see this). These bits determine who can perform what operations. This is called the file's mode and is built into the file (there are no separate ACEs).