What Service Pack, if any, do you have installed on XP? The online installers are signed using a sha256 certificate which is only supported on Windows XP with Service Pack 3. If you don't have Service Pack 3 you can try using the offline installer posted at the bottom of the Installation problems | Flash Player | Windows 7 and earlier page in the 'Still having problems' section, but there is no guarantee that it'll work, due to the same reason. You're best option is to upgrade to a newer supported (by Microsoft) operating system, or at minimum, if not installed, install Service Pack 3 (if you can find it).
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