Prime Numbers

1.1 Problems and progress 3
(QS) algorithm (see Chapter 6) by D. Atkins, M. Graff, A. Lenstra, and
P. Leyland. RSA129 was factored as
3490529510847650949147849619903898133417764638493387843990820577
×
32769132993266709549961988190834461413177642967992942539798288533,
and the secret message was decrypted to reveal: “THE MAGIC WORDS ARE
SQUEAMISH OSSIFRAGE.”
Over the last decade, many other factoring and related milestones have
been achieved. For one thing, the number field sieve (NFS) is by now
dominant: As of this 2nd book edition, NFS has factored RSA-576 (174
decimal digits), and the “special” variant SNFS has reached 248 decimal digits.
The elliptic curve method (ECM) has now reached 59 decimal digits (for a
prime factor that is not the largest in the number). Such records can be found
in [Zimmermann 2000], a website that is continually updated. We provide a
more extensive list of records below.
Another interesting achievement has been the discovery of factors of
various Fermat numbers Fn =22n+ 1 discussed in Section 1.3.2. Some of
the lower-lying Fermat numbers such as F9,F10,F11 have been completely
factored, while impressive factors of some of the more gargantuan Fn have
been uncovered. Depending on the size of a Fermat number, either the number
field sieve (NFS) (for smaller Fermat numbers, such as F9) or the elliptic curve
method (ECM) (for larger Fermat numbers) has been brought to bear on the
problem (see Chapters 6 and 7). Factors having 30 or 40 or more decimal
digits have been uncovered in this way. Using methods covered in various
sections of the present book, it has been possible to perform a primality test
on Fermat numbers as large as F24, a number with more than five million
decimal digits. Again, such achievements are due in part to advances in
machinery and software, and in part to algorithmic advances. One possible
future technology—quantum computation—may lead to such a tremendous
machinery advance that factoring could conceivably end up being, in a few
decades, say, unthinkably faster than it is today. Quantum computation is
discussed in Section 8.5.
We have indicated that prime numbers figure into modern cryptography—
the science of encrypting and decrypting secret messages. Because many
cryptographic systems depend on prime-number studies, factoring, and related
number-theoretical problems, technological and algorithmic advancement
have become paramount. Our ability to uncover large primes and prove
them prime has outstripped our ability to factor, a situation that gives some
comfort to cryptographers. As of this writing, the largest number ever to
have been proved prime is the gargantuan Mersenne prime 225964951 − 1,
which can be thought of, roughly speaking, as a “thick book” full of decimal
digits. The kinds of algorithms that make it possible to do speedy arithmetic
with such giant numbers is discussed in Chapter 8.8. But again, alongside
such algorithmic enhancements come machine improvements. To convey an
idea of scale, the current hardware and algorithm marriage that found each