Prime Numbers

9.5 Large-integer multiplication 507
know that the complexity must be
O(D ln D)
operations, and as we have said, these are usually, in practice, floatingpoint
operations (both adds and multiplies are bounded in this fashion).
Now the bit complexity is not O((n/b)ln(n/b))—that is, we cannot just
substitute D = n/b in the operation-complexity estimate—because floatingpoint
arithmetic on larger digits must, of course, be more expensive. When
these notions are properly analyzed we obtain the Strassen bound of
O(n(C ln n)(C ln ln n)(C ln ln ln n) ···)
bit operations for the basic FFT multiply, where C is a constant and the
ln ln ··· chain is understood to terminate when it falls below 1. Before we
move ahead with other estimates, we must point out that even though this bit
complexity is not asymptotically optimal, some of the greatest achievements
in the general domain of large-integer arithmetic have been achieved with this
basic Schönhage–Strassen FFT, and yes, using floating-point operations.
Now, the Schönhage Algorithm 9.5.23 gets neatly around the problem
that for a fixed number of signal digits D, the digit operations (small
multiplications) must get more complex for larger operands. Analysis of
the recursion within the algorithm starts with the observation that at top
recursion level, there are two DFTs (but very simple ones—only shifting and
adding occur) and the dyadic multiply. Detailed analysis yields the best-known
complexity bound of
O(n(ln n)(ln ln n))
bit operations, although the Nussbaumer method’s complexity, which we
discuss next, is asymptotically equivalent.
Next, one can see that (as seen in Exercise 9.67) the complexity of
Nussbaumer convolution is
O(D ln D)
operations in the R ring. This is equivalent to the complexity of floating-point
FFT methods, if ring operations are thought of as equivalent to floating-point
operations. However, with the Nussbaumer method there is a difference: One
may choose the digit base B with impunity. Consider a base B ∼ n, sothat
b ∼ ln n, in which case one is effectively using D = n/ ln n digits. It turns out
that the Nussbaumer method for integer multiplication then takes O(n ln ln n)
additions and O(n) multiplications of numbers each having O(ln n) bits.It
follows that the complexity of the Nussbaumer method is asymptotically that
of the Schönhage method, i.e., O(n ln n ln ln n) bit operations. Such complexity
issues for both Nussbaumer and the original Schönhage–Strassen algorithm
are discussed in [Bernstein 1997].