A genuine Jew, born of a Jewish mother? Why is it in the Torah that lineage is traced through the paternal side but modern Jews trace it through the maternal side?
My friend, you are sorely confused. According to Jewish law—from the get-go—the “Jewishness” of a child was determined by the mother being Jewish. There were such things as rape by non-Jews of Jewish women. The only way to be sure of the authentic Jewish lineage was to determine it through the mother.