12M Digit Prime Number Sets Record, Nets $100,000 232
coondoggie writes "A 12-million-digit prime number, the largest such number ever discovered, has landed a voluntary math research group a $100,000 prize from the Electronic Frontier Foundation (EFF). The number, known as a Mersenne prime, is the 45th known Mersenne prime, written shorthand as 2 to the power of 43,112,609, minus 1 . A Mersenne number is a positive integer that is one less than a power of two, the group stated. The computing project called the Great Internet Mersenne Prime Search (GIMPS) made the discovery on a computer at the University of California, Los Angeles (UCLA) Mathematics Department."
Re:I haven't tested this thoroughly, but... (Score:3, Interesting)
Re:so? (Score:3, Interesting)
I can run PGP and have it generate primes for me within a few seconds that are sufficient for decent encryption, and they don't need to be a gazillion digits long.
I agree with you entirely. The debate, however, is in the definitions of "sufficient" and "decent". For most needs, you need far fewer than a gazillion digits. For national security type stuff, a gazillion digits may be appropriate. For a math/crypto nerd, even a bazillion gazillion digits will never be enough.
Re:so? (Score:2, Interesting)
Indeed. Although it should be noted that Mersenne primes are sort of useless for this. If you know one of the factors is a Mersenne Prime then there are only 47 candidates.
Re:Actually the 47th (Score:1, Interesting)
It's the 47th by size; 45th by order of discovery. Two smaller Mersenne primes were discovered after the current record-holder. One was discovered within a couple of days of the discovery of the prize-winner-- and it was, itself, large enough to claim the EFF prize ($100k for 10M digits).
Re:Sounds cool, but... (Score:3, Interesting)
...who works for EFF.
So what's their interest in this? Is it cryptography related?
Re:A Question (Score:3, Interesting)
I think $100,000 is roughly how much, in electricity costs, it costs to run the many, many computers for 5-10 years needed to grind out this particular number. Plus maintenance and taxes. You could pretty much say this research was done "at cost".
Re:so? (Score:4, Interesting)
Strictly speaking, large primes are useful but not specific large primes. Indeed, this Mersenne prime is much too large to be useful for practical cryptography.
The main reason that primes are useful for cryptography is that there are functions associated with primes where the functions are easy to calculate but have inverses that are very difficult to calculate unless one has extra information. The classic example of this is the discrete log problem. Essentially, if one is doing modular arithmetic with a given prime (that is arithmetic where you only look at the remainders when divided by that prime. This is essentially like a clock with p numbers on it. On a normal clock is arithmetic mod 12) then it turns out that for any prime p, there is a number k such that k^1, k^2, k^3... k^(p-1) taken mod pruns through all the possible remainders 1,2,3... p-1 (obviously not in order). Such a k is called a primitive root (or a generator) Now, it is very easy given a k and an n to calculate k^n (mod p). However, given the equation k^x=m (mod p) it is very hard to find the right value of x without doing a lot of work (essentially, you have to more or less list out k^1,k^2,k^3... until you get to m). The difficulty in this process is used in a number of public key crypto systems and other systems. For example, the Diffie-Hellman algorithm which was the first modern crypto algorithm discovered uses the difficulty of this process to make it so that two people can without ever having talked to each other before have a conversation and at the end of it have a secret code that no one else can figure out without impractical levels of computation. This is true even if one has access to all the communication that goes on between the two parties. The RSA algorithm which is more practical than DH for a variety of reasons also works off of a similar procedure.
The above is assuming that the discrete log is actually a difficult problem which everyone believes but is not proven. Proving this would imply that P != NP which is one of the Clay Millennium Problems (so million dollar prize and all that fun stuff). Mersenne primes are very interesting but not for any crypto related reason.
Various useful details (Score:4, Interesting)
The project GIMPS that is mentioned in the title uses a distributed computing system to search for Mersenne primes. They use a modified form of the Lucas-Lehmer test http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test [wikipedia.org] where they use a Fast Fourier Transform to be able to do the large multiplications efficiently.
We care about Mersenne primes because they correspond to even perfect numbers. If one has a Mersenne prime 2^p -1 then (2^p-1)(2^(p-1)) is an even perfect number. This was proven by the ancient Greeks. Euler then proved much later that every even perfect number is of this form. The oldest two unsolved problems in mathematics are whether there are infinitely many even perfect numbers and whether there are any odd perfect numbers. Thus, every time we discover a new Mersenne prime we get a new even perfect number. And if we can ever get enough insight to resolve whether or not there are infinitely many Mersenne primes then we can resolve one of the oldest unsolved problems in all of mathematics.
Re:so? (Score:3, Interesting)
I was going by this description of RSA [wikipedia.org]. Quote:
In real life situations the primes selected would be much larger, however in our example it would be relatively trivial to factor n, 3233, obtained from the freely available public key back to the primes p and q. Given e, also from the public key, we could then compute d and so acquire the private key.
I didn't go through all the math myself, but if this description is true, then knowing one of the primes makes factoring n becomes unnecessary, allowing us to "compute d and so acquire the private key."
So at least the RSA encryption is easily broken if you have a good guess as to one of the primes used.