Prime Numbers

510 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
9.6.1 Polynomial multiplication
We have seen that polynomial multiplication is equivalent to acyclic
convolution. Therefore, the product of two polynomials can be effected via
a cyclic and a negacyclic. One simply constructs respective signals having the
polynomial coefficients, and invokes Theorem 9.5.10. An alternative is simply
to zero-pad the signals to twice their lengths and perform a single cyclic (or
single negacyclic).
But there exist interesting—and often quite efficient—means of multiplying
polynomials if one has a general integer multiply algorithm. The method
amounts to placing polynomial coefficients strategically within certain large
integers, and doing all the arithmetic with one high-precision integer multiply.
We give the algorithm for the case that all polynomial coefficients are nonnegative,
although this constraint is irrelevant for multiplication in polynomial
rings (mod p):
Algorithm 9.6.1 (Fast polynomial multiplication: Binary segmentation).
Given two polynomials x(t) = D−1 j=0 xjtj and y(t) = E−1 k=0 yktk with all coefficients
integral and nonnegative, this algorithm returns the polynomial product
z(t) =x(t)y(t) in the form of a signal having the coefficients of z.
1. [Initialize]
Choose b such that 2 b > max{D, E} max{xj} max{yk};
2. [Create binary segmentation integers]
X = x 2 b ;
Y = y 2 b ;
// These X, Y can be constructed by arranging a binary array of
sufficiently many 0’s, then writing in the bits of each coefficient,
justified appropriately.
3. [Multiply]
u = XY ; // Integer multiplication.
4. [Reassemble coefficients into signal]
for(0 ≤ l