The real problem, if we can fit it in, probably explained much better by someone more knowledgeable than me but...
White stuff is salts being transported through the floor from underneath by moisture transmission. It's in solution till it gets to the surface, at which point the water evaporates and deposits the salt crystals
In short, you won't fix it until the source of moisture has been eliminated. If a general cleaning product and good scrub & rinse doesn't sort it, something phosphoric based might have a better effect but needs testing thoroughly first.
