From: Stuart Painting <stuart@zedtoo.demon.co.uk>
Newsgroups: demon.service
Subject: Re: Spam in an unused mailbox
Date: Fri, 26 Jan 2001 22:18:52 +0000 (UTC)

In article <87vgr361xa.fsf@erlenstar.demon.co.uk>, Andrew Gierth <andrew@erlenstar.demon.co.uk> wrote:

Hmmm. Since the RFCs are silent on how long a mailbox name can be, let's assume a 63-byte maximum. Let us further assume that the mailbox names are restricted to lower-case letters, plus numerics, for a total of 36 possible characters in each of those 63 positions.

In other words, that's 36^63 mailboxes, or 1.14 x 10^98 mailboxes.

With this vast number of mailboxes, 3 questions need to be considered: how long will the messages take to transmit, where will the receiving ISP store them all, and how will the messages reach their destination? Let us consider these questions in turn.

1. How long will the messages take to transmit?

   Were you to transmit a 1-byte message at 100 terabytes a second, you could send 10^14 messages a second (in practice the message would need to be closer to 100 bytes long, but let's keep things simple).

   At 10^14 messages a second, it would take 10^84 seconds to transmit them all. That's 3 x 10^77 years.

   The spammer would grow old and die with the job barely started. The sending equipment would likewise wear out and fail long before every message had been sent. Indeed, the Sun would have expanded to form a red giant (thus destroying the Earth) with only a tiny fraction of the messages having been sent.

   In other words, it's impossible.

2. Where will the receiving ISP store the messages?

   The total mass of the Earth is 5.97 x 10^24 kg.

   Avogadro's constant is 6.02 x 10^23

   Multiplying these numbers gives us a rough estimate (i.e. to within a few orders of magnitude) of the number of atoms in the Earth, which is 3.59 x 10^48. Let us assume that the estimate is 10 orders of magnitude too low, so we shall use 3 x 10^58 instead.

   (10^98) divided by (3 x 10^58) gives 3 x 10^39.

   So, assuming that the receiving ISP could store 1 message in a single atom, it would have to consume all of the matter in the Earth, plus a further 10^39 planets of similar size, to store all of the messages.

   In other words, it's impossible.

3. How will the messages reach their destination?

   You might argue that the above two problems could be circumvented in some fashion (for example, 50 years from now we might be able to transmit messages at unimaginable speeds) so now we come to the real clincher.

   You see, these messages have to reach their destination, and that in turn means there needs to be a mechanism by which this happens.

   Let us suppose that you use photons to transmit the messages. Photons travel at the speed of light, 3 x 10^8 metres per second. The sender and receiver are a finite distance apart, say 2 km. So, a photon will take 0.000006 seconds to make the journey.

   These photons have to come from somewhere. According to Einstein's famous equation (e=mc^2) you can work out the total energy available if the Earth were completely converted into energy. That is, (5.97 x 10^24) x (3 x 10^8) x (3 x 10^8) = 5.37 x 10^41 joules.

   Aside: You don't actually have to use the Earth itself. Venus is almost the same mass, so would produce roughly the same energy.

   The energy of a photon is the product of its frequency and Planck's Constant, which is 6.62559 x 10^(-34) joule seconds.

   Let us suppose that we were able to use photons with a frequency of one hertz, so you would get 6.62 x 10^34 photons per joule. Converting the entire Earth into these photons would give you 3.55 x 10^76 photons, so you would need 3.2 x 10^21 round trips (i.e. reusing the photons after they had reached their destination, and assuming that each message required only one photon) which equates to 3.84 x 10^16 seconds.

   That's still 1,240,000,000 years.

   And there's the small matter of gravitational distortion. Introducing an energy equivalent to the mass of the Earth into a region only 2km across would result in the almost instantaneous collapse of any buildings used to house the sending and/or receiving equipment, and would also upset the Earth's axis of rotation, with devastating consequences for everyone on the planet.

   In other words, it's impossible.

Of course, if you've managed to do three impossible things before breakfast, you might like to visit Milliways, the Restaurant at the End of the Universe...