Prime Numbers
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Prime Numbers

Prime Numbers Prime Numbers

thales.doa.fmph.uniba.sk
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2 Chapter 1 PRIMES! where exponents ai are positive integers and p1

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

2 Chapter 1 PRIMES!<br />

where exponents ai are positive integers and p1

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