Prime Numbers

Chapter 5
EXPONENTIAL FACTORING ALGORITHMS
For almost all of the multicentury history of factoring, the only algorithms
available were exponential, namely, the running time was, in the worst case,
a fixed positive power of the number being factored. But in the early 1970s,
subexponential factoring algorithms began to come “on line.” These methods,
discussed in the next chapter, have their running time to factor n bounded
by an expression of the form n o(1) . One might wonder, then, why the current
chapter exists in this book. We have several reasons for including it.
(1) An exponential factoring algorithm is often the algorithm of choice for
small inputs. In particular, in some subexponential methods, smallish
auxiliary numbers are factored in a subroutine, and such a subroutine
might invoke an exponential factoring method.
(2) In some cases, an exponential algorithm is a direct ancestor of a
subexponential algorithm. For example, the subexponential elliptic curve
method grew out of the exponential p − 1 method. One might think of the
exponential algorithms as possible raw material for future developments,
much as various wild strains of agricultural cash crops are valued for their
possible future contributions to the plant gene pool.
(3) It is still the case that the fastest, rigorously analyzed, deterministic
factoring algorithm is exponential.
(4) Some factoring algorithms, both exponential and subexponential, are
the basis for analogous algorithms for discrete logarithm computations.
For some groups the only discrete logarithm algorithms we have are
exponential.
(5) Many of the exponential algorithms are pure delights.
We hope then that the reader is convinced that this chapter is worth it!
5.1 Squares
An old strategy to factor a number is to express it as the difference of two
nonconsecutive squares. Let us now expand on this theme.
5.1.1 Fermat method
If one can write n in the form a 2 − b 2 ,wherea, b are nonnegative integers,
then one can immediately factor n as (a + b)(a − b). If a − b>1, then the