Prime Numbers

130 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
Next we reduce the number P modulo every label in T by creating
a “remainder tree” (see [Bernstein 2004e]). In general, a remainder tree
P mod T for a given integer P and a given product tree T is the corresponding
tree in which each label in T being replaced by its remainder when it is divided
into P . This relabeling is achieved by replacing the label R at the root of T
with P mod R, and then working toward the leaves, each entry is replaced with
the remainder after dividing this entry into the new label of its parent. We
illustrate with the product tree T formed from 1001,...,1008 and the number
P = 9699690 found in our first product tree. We may evidently convert each
asterisk in T to P .
✟❍9699690
✟
✟ ❍❍❍❍❍9699690
✟
✟
✟
9699690
❅
❅
❅
❅
672672
❅
0 ❅330
❅636582 ❅
680❅46 600420
❅
435❅844 ❅564186 ❅
266❅714 Remainder tree P mod T
For each x that we are examining for smoothness, the corresponding leaf
value in the remainder tree is P mod x. Take this residue, sequentially square
modulo x the requisite number of times, and at last take the gcd of the final
result with x. A value of 0 signifies that x is smooth over the primes in P ,
and a nonzero value is itself the largest divisor of x that is smooth over the
primes in P . Here is pseudocode for this beautiful algorithm.
Algorithm 3.3.1 (Batch smoothness test (Bernstein)). We are given a finite
set X of positive integers and a finite set P of primes. For each x ∈X,this
algorithm returns the largest divisor of x composed of primes from P.
1. [Compute product trees]
Compute the product tree for P;
Set P as the product of the members of P;
// We find P at the root of the product tree for P.
Compute the product tree T for X , but only for products at most P ;
2. [Compute remainder tree]
Compute the remainder tree P mod T ; // Notation described in text.
3. [Find smooth parts]
Set e as the least positive integer with max X≤22e; for(x ∈X){
Find P mod x in the remainder tree P mod T ;
// No additional mod calculation is necessary.
r = P mod x;
s = r2e mod x; // Compute s by sequential squaring and reducing.