“a verse long known in Elven-lore”

[Alcuin]

Frodo … saw fine lines, finer than the finest pen-strokes, running along the ring, outside and inside: lines of fire that seemed to form the letters of a flowing script.
…
“It is only two lines of a verse long known in Elven-lore:

Three Rings for the Elven-kings under the sky,
…Seven for the Dwarf-lords in their halls of stone,
Nine for Mortal Men doomed to die,
…One for the Dark Lord on his dark throne
In the Land of Mordor where the Shadows lie.
…One Ring to rule them all. One Ring to find them,
…One Ring to bring them all and in the darkness bind them
In the Land of Mordor where the Shadows lie.”

Who composed the poem of lore Gandalf quotes?

Sauron engraved two lines upon the Ring:

One Ring to rule them all. One Ring to find them,
One Ring to bring them all and in the darkness bind them

We can safely assume he composed those two lines himself: they are part of the spell he used to bind the Rings of Power to himself in order to dominate and control their wearers.

But the Rings of Power were intended for Elves, at least by their makers, for Sauron did not make them himself: he aided and guided the Noldor of Eregion. They had no intention of giving the Nine or the Seven to Men and Dwarves; though Celebrimbor gave the greatest of the Seven to his friend Durin III of Khazad-dûm, probably as Sauron and his army drew near Eregion and after he dispensed of the Three to Gil-galad and Galadriel.

So my question is: Did Sauron make the eight-line poem of lore, with Celebrimbor and the Elven-smiths hearing and remembering it, passing it along to their kinsfolk to remember? Or did the surviving Noldor -- or possibly even the Dúnedain of Arnor -- wrap the two-line curse into the other six to make recollection easier?