Editing Multiplication algorithm (section)

===Lower bounds===
There is a trivial lower bound of [[Big O notation#Family of Bachmann–Landau notations|Ω]](''n'') for multiplying two ''n''-bit numbers on a single processor; no matching algorithm (on conventional machines, that is on Turing equivalent machines) nor any sharper lower bound is known. Multiplication lies outside of [[ACC0|AC<sup>0</sup>[''p'']]] for any prime ''p'', meaning there is no family of constant-depth, polynomial (or even subexponential) size circuits using AND, OR, NOT, and MOD<sub>''p''</sub> gates that can compute a product. This follows from a constant-depth reduction of MOD<sub>''q''</sub> to multiplication.<ref>{{cite book |first1=Sanjeev |last1=Arora |first2=Boaz |last2=Barak |title=Computational Complexity: A Modern Approach |publisher=Cambridge University Press |date=2009 |isbn=978-0-521-42426-4 |url={{GBurl|8Wjqvsoo48MC|pg=PR7}}}}</ref> Lower bounds for multiplication are also known for some classes of [[branching program]]s.<ref>{{cite journal |first1=F. |last1=Ablayev |first2=M. |last2=Karpinski |title=A lower bound for integer multiplication on randomized ordered read-once branching programs |journal=Information and Computation |volume=186 |issue=1 |pages=78–89 |date=2003 |doi=10.1016/S0890-5401(03)00118-4 |url=https://core.ac.uk/download/pdf/82445954.pdf}}</ref>