# The Elevator Problem

Futplex futplex at pseudonym.com
Thu Dec 14 02:46:29 PST 1995

```tallpaul writes:
> Alice says to Bob, in front of all of the other people on the elevator: "I
> have generated a large(ish) amount of large(ish) prime numbers and have
> recorded all of them. I have multipled two of the numbers to get an even
> larger non-prime number. I have done this a large(ish) number of times
> until I have a 'large(ish)/2' set of non-prime numbers. The elements of
> this set are [Alice reads off the set of non-prime numbers and Bob along
> with the other people on the elevator record them.] Bob, go home and pick
> one of the non-prime numbers in the set. Factor it. Use the largest prime
> as a private key in your message to me. Since I know what the numbers all
> are, I'll try all of them to see which one decrypts your message."
>
> Bob has to factor one large(ish) prime.
>
> Alice has to *try* an average of "large(ish)/2" private keys to decrypt
> Bob's message.
>
> The other people on the elevator have to *factor* an average of
> "large(ish)/2/2" number of large(ish) numbers to decrypt the message.
>
> The *relative* security then depends on the number of digits in the
> large(ish) primes and the number of products in the set Alice reads to Bob.
>
[...example with a set of 2 * 10^6 primes...elided]

I think there are two main (related) problems with this protocol:

(1) It does not offer great security. The time required for a brute force
attack is only linear in the time required to execute the protocol. So Alice
will want to start out with an enormous number of primes (the linear factor).
Even so, the attacker's job is relatively easy.

(2) It is rather impractical. The time required to execute the protocol is
prohibitive (assuming Alice uses a huge number of primes).

Consider the numbers in your example. Alice generates N = 2*10^6 large primes
and transmits the 10^6 pair products -- that's on the order of .1 gigabits, or
about 12 megabytes, to transmit (assuming products around 100 bits long, so
Bob can factor one before the heat death of the universe). Bob factors one of
the products, which should take a while for all this to be at all worthwhile.
Let's say it takes Bob approximately an hour to factor.

This will take too long to do online. Alice and Bob won't generate new keys
for each session this way. But to limit the chance that Eve can start to
decrypt their communications in real time to 10%, if Eve has 100
times the computing power of Bob, they'll need to negotiate a new key every
6 weeks or so. This is not so hot.