Of course, it didn't occur to me to take a look at the Science section before submitting my own copy of this story (which, since it has several other useful links in it, follows):
I'm also not seeing how this could stop identity theft. If you use this program, aren't you putting your mail in front of the eyeballs of the person that's scanning them?
It seems to me that at this point their attempts to make it difficult to detect would be fairly easily circumvented.
Since as part of the design a Sebek-enabled host will not see any Sebek packets, someone who has rooted the box already could merely send it Sebek packets and observe whether he/she could capture the packets or if they were lost.
This can be prevented to some degree with having the module rewrite outgoing Sebek-like packets which it did not create and similarly recognize these when they arrive and rewrite them as they went out. Of course, one would have to write this in such a way that all packets were still expressible (much like HTML makes it possible to include arbitrary text in a page even though some kinds would have special meaning if left unescaped: &, <, >, etc.).
However, when such schemes start to become more complex as they try to account for all possibilities (radar detector detector detector, anyone?), there is a definite risk that something will be overlooked. Fortunately, with a system like Sebek, you can see what the attacker does up until the point at which they discover Sebek installed and move on. In this way, a system like Sebek provides the tools to better itself if it is not perfect and scripts and techniques that attackers may use to detect it should be short-lived (since I would hope those that create honeynets are actually paying attention to them).
Alright. So, say that x is 0.99999... with an infinite number of nines. You claim that x is a number distinct from 1 (since you claim 1 - x is nonzero). Since the the rationals and irrationals are both dense in the reals, we know that if we pick any two distinct reals, we'll always have numbers between them. That is, it's impossible to pick a number and the "next" real number without skipping over any. Yet that is essentially what you've done by positing that the gap between these numbers is infinitely small.
So let y equal 1 - x. We can show that if y != 0, then 1 - x < y even with a finite number of digits. Just pick n to be the least integer which is at least -log(y)+1 and then 10^n < y just considering a finite number of digits, n.
That is, if you tell me what the nonzero gap between the numbers that you expect to see, one can show that the gap is less than that. Since the gap is a nonnegative number that is less than all nonzero (ie, positive) choices, it is zero.
Therefore, 0.9999999..... = 1.
On an even more off-topic note, has anyone else started to think that trolls that bring up controversial mathematical statements are probably the best way to get responses nowadays? Not to say that the OP was a troll, but if it was, congratulations on all the responses. Also, congratulations to the Monty Hall problem poster some days back that was a troll.
Slashdot Mirror
Of course, it didn't occur to me to take a look at the Science section before submitting my own copy of this story (which, since it has several other useful links in it, follows):
Michael Shafer, a graduate student at Michigan State University, took time out for a "short victory dance" upon learning his computer had discovered the 40th known Mersenne prime as part of The Great Internet Mersenne Prime Search. The number itself is 2**20996011-1 and when expressed in base 10, has 6,320,430 digits (zipped copy). However, this is not necessarily the 40th Mersenne prime; there could be another between the previous largest known prime (M39=2**13466917-1, also discovered by GIMPS) and this one. Also worth noting is the still-standing USD$100,000 EFF prize for the discover of the first prime of at least 10 million (decimal) digits. GIMPS clients are available for various operating systems as well as information on how GIMPS would distribute the prize. A press release on the achievement is available as well as several articles. Of course, this also means there's a new largest known even perfect number in town.
I'm also not seeing how this could stop identity theft. If you use this program, aren't you putting your mail in front of the eyeballs of the person that's scanning them?
It seems to me that at this point their attempts to make it difficult to detect would be fairly easily circumvented.
Since as part of the design a Sebek-enabled host will not see any Sebek packets, someone who has rooted the box already could merely send it Sebek packets and observe whether he/she could capture the packets or if they were lost.
This can be prevented to some degree with having the module rewrite outgoing Sebek-like packets which it did not create and similarly recognize these when they arrive and rewrite them as they went out. Of course, one would have to write this in such a way that all packets were still expressible (much like HTML makes it possible to include arbitrary text in a page even though some kinds would have special meaning if left unescaped: &, <, >, etc.).
However, when such schemes start to become more complex as they try to account for all possibilities (radar detector detector detector, anyone?), there is a definite risk that something will be overlooked. Fortunately, with a system like Sebek, you can see what the attacker does up until the point at which they discover Sebek installed and move on. In this way, a system like Sebek provides the tools to better itself if it is not perfect and scripts and techniques that attackers may use to detect it should be short-lived (since I would hope those that create honeynets are actually paying attention to them).
Alright. So, say that x is 0.99999... with an infinite number of nines. You claim that x is a number distinct from 1 (since you claim 1 - x is nonzero). Since the the rationals and irrationals are both dense in the reals, we know that if we pick any two distinct reals, we'll always have numbers between them. That is, it's impossible to pick a number and the "next" real number without skipping over any. Yet that is essentially what you've done by positing that the gap between these numbers is infinitely small.
Put in other terms,
1 - x = limit(n->infinity, 1 - sum(9*10^(-i), i=1..n)) = limit(n->infinity, 10^(-n))
So let y equal 1 - x. We can show that if y != 0, then 1 - x < y even with a finite number of digits. Just pick n to be the least integer which is at least -log(y)+1 and then 10^n < y just considering a finite number of digits, n.
That is, if you tell me what the nonzero gap between the numbers that you expect to see, one can show that the gap is less than that. Since the gap is a nonnegative number that is less than all nonzero (ie, positive) choices, it is zero.
Therefore, 0.9999999..... = 1.
On an even more off-topic note, has anyone else started to think that trolls that bring up controversial mathematical statements are probably the best way to get responses nowadays? Not to say that the OP was a troll, but if it was, congratulations on all the responses. Also, congratulations to the Monty Hall problem poster some days back that was a troll.