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The name Chein actually comes from the Hebrew name "Chane" חן (the "Ch is gutteral and is pronouned as if you are clearing your throat)
In any case, my father, Isidor, asserted that because the name was a Hebrew name in Russia as well as for several other reasons, the family is probably descendant from the great KAZAR conversion.
The Chein family from Russia, is originally from Spain. The patriarch of the family was Mar Shaltiel Chein the son of the last Babylonian exile-arch in Iraq Mar Chizkiah Who was a descendent father to son from king David. Mar shaltiel fled Iraq after the Caliph of Baghdad killed his father, and arrived in Barcelona in 1058. There are descendants who go by other surnames, like Shaltiel, Saltiel... many of his descendants fled Spain in 1392 and found shelter in Candia (modern Heraklion) Crete. From there the family spread throughout Italy and Europe. Some members of the Chein family lived in shottland a suburb of Danzig (modern Gdansk). Circa 1780 Rabbi Elchonon Chein then a child of 7 was brought to Lithuania to learn Torah, He was a Chossid of the Chabad movement. Later his descendants moved to Russia.