Unformatted text preview: matics. Section 11.3 discusses factoring in more mathematical detail; here I will limit the discussion to how long it takes to factor numbers of various lengths. Factoring large numbers is hard. Unfortunately for algorithm designers, it is getting easier. Even worse, it is getting easier faster than mathematicians expected. In 1976 Richard Guy wrote: “I shall be surprised if anyone regularly factors numbers of size 1080 without special form during the present century” [680]. In 1977 Ron Rivest said that factoring a 125digit number would take 40 quadrillion years [599]. In 1994 a 129digit number was factored [66]. If there is any lesson in all this, it is that making predictions is foolish. Table 7.3 shows factoring records over the past dozen years. The fastest factoring algorithm during the time was the quadratic sieve (see Section 11.3). These numbers are pretty frightening. Today it is not uncommon to see 512bit numbers used in operational systems. Factoring them, and thereby completely compromising their security, is well in the range of possibility: A weekendlong worm on the Internet could do it. Computing power is generally measured in mipsyears: a onemillioninstructionpersecond (mips) computer running for one year, or about 3*1013 instructions. By convention, a 1mips machine is equivalent to the DEC VAX 11/780. Hence, a mipsyear is a VAX 11/780 running for a year, or the equivalent. (A 100 MHz Pentium is about a 50 mips machine; a 1800node Intel Paragon is about 50,000.) The 1983 factorization of a 71digit number required 0.1 mipsyears; the 1994 factorization of a 129digit number required 5000. This dramatic increase in computing power resulted largely from the introduction of distributed computing, using the idle time on a network of workstations. This trend was started by Bob Silverman and fully developed by Arjen Lenstra and Mark Manasse. The 1983 factorization used 9.5 CPU hours on a single Cray XMP; the 1994 factorization took 5000 mipsyears and used the idle time on 1600 computers around the world for about eight months. Modern factoring methods lend themselves to this kind...
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 Fall '10
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 Cryptography, Bruce Schneier, Applied Cryptography, EarthWeb, Search Search Tips

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