How does Bilbo no that his small sword was made by elves in The Hobbit?

In the 3rd Chapter of The Hobbit 'A Short Rest', Gandalf and Thorin show their sWords to Elrond at RivenDELL. He tells them they were made by the High Elves of the West.

In Riddles in the Dark, Bilbo pulls out his sWord and it glows in the dark. He thought it rather splendid to be wearing a sWord that glowed the way Gandalf and Thorin's sWords did. Bilbo's little sWord, which he later gives the name 'Sting', is of the same design and has the same magic.