This guide refers to downloading and verifying Litecoin Core in Ubuntu 16.04. The guidelines should be pretty much the same for any Debian-based Linux distro.
Download the Litecoin Core Client & Signatures Document
Download the relevant Litecoin Core .tar file and the GPG signatures document.
To keep things organised, create a dedicated subdirectory and move into it:
Verify the Signatures Document
Download a copy of the Litecoin Core PGP key, as shown on the main website:
Now you can import the Litecoin public key:
Once the key is installed, you can verify the document that contains the hash signatures:
Checksum
You can then safely use the signatures document to verify that the checksum of the download matches the expected checksum provided by the project:
Glitch in Signatures Document
The Signatures document contains Windows line endings, which causes the sha256sum check to fail (on Ubuntu, likely also on Mac):
Fortunately this is easily remedied. Open the signatures file in Gedit, and ‘Save As’ with the same filename. You’ll be given an option for ‘Line Ending’, for which you should choose Unix/Linux.