אלסלאם עליכּם in Judaism and Islam

Someone asked me the following interesting question:

Jews say Shalom Aleichem, and Muslims say "As-Salamu Alaykum" (or something like that). Is it a cognate, or a loan translation? And if the latter, who borrowed from whom?

To rephrase that, the classic Jewish greeting in Hebrew is שלום עליכם and the classic Muslim greeting in Arabic is السلام عليكم / אלסלאם עליכּם . When returning this greeting Jews respond עליכם שלום and Muslims ועליכּם אלסלאם / وعليكم السلام. These are certainly parallel. The question is, cognate or loan?

In truth, I had no idea and I still have no idea. But I thought a good place to start would be to determine how old the Jewish greeting is. I honestly could not recall seeing it in the Talmud (Babylonian, that is) where the greeting Shalom שלום certainly appears.

If it didn't appear before the Islamic era, perhaps that would indicate that it was indeed a loan translation (=Arabic words and custom translated into Hebrew and becoming Jewish custom).

Using my vast bekius and geonus (yes, I am kidding) I discovered that שלום עליכם appears in some of the Massekhtot Ketanot (minor tractates appended to the Talmud which are post-Talmudic, that is, of the Islamic era). But I also discovered that it appears in the Thalmudh Jeruschalmi (presumably redacted at about 400 CE) as well ( three times, actually). Similarly it is found in Genesis Rabbah (now, why is the convention to translate בראשית וכו but not רבה? Things that make you go 'hm.'). But not even once in the Bavli, confirming my prodigious memory of things I didn't see.

This would indicate, if not a Palestinian origin, at least that it was a greeting in use in Talmudic Palestine (unless somehow these specific words are later interpolations, but we obviously shall not take that as a given). However, the reversal greeting (עליכם שלום) is not indicated in these sources. There is still no indication if it was of Jewish origin. It may seem likely, but we shall recall that there was no lack of contact with Arabs, especially in Southern Palestine, and attested amply in the Talmud.

That said, we can probably see that the greeting was in use by Jews, in Hebrew, well before the rise of Islam, which of course spread this Arabic greeting into China to the East and Spain to the West, and deep into North Africa to the South (and of course into post-Talmudic Babylon).

In addition, I looked up the Syriac for Matthew 13:10's "your peace return to you." In the Peshitta it's שלמכון עליכון. Now, I don't know exactly what the date of the Peshitta Matthew is, but it's obviously older than Islam as well (and also warrants discussion about it's place of origin, and its relations to Arabs). Of course this could merely have been a literal translation from the Greek Matthew--or even influenced by the greeting itself.

My questioner told me that he suspects it to be a loan translation (one way or the other) for the simple reason that it is so specific. It isn't merely two words meaning "peace" and "on you," but also the matter of reversal in response.

Finally, I wonder about the geographic distribution of the use of the greeting. Here is description from an 1867 visit to Mountain Jews of the Caucuses by Yehuda Chernyi:

"[When one such man arrives,] the host relieves him of his things, takes him to the guest room, shouls him the best place on the pillow to rest from the road, and then washes his feet. . . . Soon after, elders and respected people will gather in the house and offer the guest their hands with greetings in the biblical language: Shalom aleichem which means "I wish you health," or Barukh-gabo which means "May your stay be happy," and the guest must say Aleichem shalom which means "May you be healthy and happy!"

cited in Sascha L. Goluboff, Are They Jews or Asians? A Cautionary Tale about Mountain Jewish Ethnography, Slavic Review, Vol. 63, No. 1. (Spring, 2004), pp. 113-140.

In any event, what do you say? Note that I have said not a word about the age of the greeting in Arabic. This is because I don't know what it is.