Lower Bounds
There is a trivial lower bound of Ω(n) for multiplying two n-bit numbers on a single processor; no matching algorithm (on conventional Turing machines) nor any better lower bound is known. Multiplication lies outside of AC0 for any prime p, meaning there is no family of constant-depth, polynomial (or even subexponential) size circuits using AND, OR, NOT, and MODp gates that can compute a product. This follows from a constant-depth reduction of MODq to multiplication. Lower bounds for multiplication are also known for some classes of branching programs.
Read more about this topic: Multiplication Algorithm
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