Diffie-Hellman for Matchmaking?

[Dimitris Tsapakidis]

Ok, I am reposting this, with more detail this time. Hope this
answers the two questions I received.

I want to design/find some matchmaking protocols. I define matchmaking
as follows:
Bob must find out whether Alice has declared (commited) her interest
in him, if and only if he has declared (commited) his interest in her.
Before he does so, he can at most know that a girl is interested in him.
Another description: Bob and Alice can have a date if they both commit
to each other. If only one commits, nobody will ever find out about it.

Below are the protocols I came up with. They all depend on the
Diffie-Hellman "common key" as derived in the DH key exchange.

- T is the trusted third party.
- hash_k() is a keyed hash function with key k.
- pseudo(Alice) is a pseudonym for Alice.
- n is a large prime.
- g is a primitive element mod n.
- A is Alice's secret exponent. Her public key would be g^A mod n.
- Alice's and Bob's "common key" would be g^(AB)mod n

----

1. The mediated off-line one:

- T selects a secret k (which he uses for the duration of a month,
  say).

- Alice is interested in Bob, so she calculates a=g^(AB)mod n and
  anonymously and securely sends it to T along with pseudo(Alice).

- T calculates c=hash_k(a) and broadcasts c and pseudo(A) to the
  planet or puts them on a bulletin board.

- Bob is ignorant of Alice's actions. If he ever decides that he
  likes her, he calculates and sends a=g^(AB)mod n to T (plus
  pseudo(Bob)). T will calculate hash_k(a) which equals c. T broadcasts
  c plus pseudo(B) so both Alice and Bob will know they have a match.
  If Bob is not actually interested in Alice, he doesn't perform this
  step at all, so nothing happens, and Alice just assumes Bob is not
  interested.

Replay is not an issue here.

2. The mediated on-line one:

- Alice is interested in Bob, so she calculates a=g^(AB)mod n and
  ANONYMOUSLY approaches Bob.

- Bob calculates b=g^(BX)mod n, where X may or may not be Alice.
  If Bob is not interested in anyone, he could tell Alice to leave,
  or select a random X while calculating b.

- Alice and Bob want to compare a and b. They generate a random k using
  some coin-flipping protocol. Then they send (possibly pseudonymously)
  hash_k(a) and hash_k(b) to T who compares them and announces the
  result.

Using a keyed hash function reduces the trust on T, compared to
protocol (1), I hope. Now, T has to conspire with one of the
parties to get the "common key". I suspect this can be improved
using the Digital Envelopes protocol as described by Fagin,
Naor and Winkler in "Comparing information without leaking it"
to replace the hashing.

3. Non-mediated on-line one: This third protocol would remove T in
(2). Alice and Bob would compare their "common keys" directly. The
problem here is fairness i.e. to ensure nobody finds the result
of the comparison first. If this was possible, people could
"probe" other people's interests and terminate the protocol
as soon as they find the answer (i.e. the result of the
"common key" comparison). I think ZK protocols would be of use
here. I am still working on this one.

----

Assume that in any of these protocols Alice calculates her "common key"
with X: a=g^(AX)mod n which means that she is interested in X. The
reason a must remain sercet is that if X learns it, he could calculate
all possible b(Y)=g^(YX)mod n until he finds a Y such that b(Y)=a,
in which case he will find out Alice is interested in him. Note
that X will not try all possible values for Y, but will use the
public keys of all the girls, instead.

I would appreciate any comments on these protocols and on the
use of DH "common key".

  Dimitris


-- 
Dimitris Tsapakidis               PGP keyID: 735590D5
dimitrt at dcs.rhbnc.ac.uk           MSc in Information Security,
This space reserved               Royal Holloway, University of London
for future use.                   Origin: Thessaloniki, Macedonia, Hellas