Consider the following counting problem (or the associated decision problem): Given two positive integers encoded in binary, compute their greatest common divisor (gcd). What is the smallest complexity class this problem is contained in? Can you provide a reference?

In this question I am not primarily interested in asymptotic bounds on the running time, but rather in complexity classes. Is the problem in $AC$? In $AC^1$? Can it be proven not to lie in $AC^0$? What are other complexity classes inside $P$ that are of relevance here?