# Re: multiplicative group question

From: Khan (khanhvn_at_yahoo.com)
Date: 07/20/05

```Date: 20 Jul 2005 11:24:43 -0700

```

Thanks. The reason I asked that question is that such a (semi)group G
could be useful for Identity-Based (IBC) key distribution. The method
is very simple:

- Let s in G be a system secret known only by a trusted authority (TA).
- Entity A is given its private key = s^A
- To communicate w/ B, A calculates the pair-wise shared key = (s^A)^B.
B does the same (s^B)^A = s^(AB).

The properties (1) and (2) of G will prevent finding s from s^A and
finding s^C from s^A, s^B...

Such a method (if it works) is much simpler than currently known IBC
methods (e.g. Elliptic curves + Weil pairing):

I was thinking about G being the set Cm of complex modular numbers
(i.e. a complex number whose real & img parts are in Zm.) It can be
showed that Cm is a semigroup: multiplication in Cm is associative
(with identity I=1+0i), but multiplicative inverse does not always
exist.

Unfortunately, Scott Fluhrer pointed out that the subgroup generated by
any s in Cm is always a group (inverse does exist).

So the quest for the holy grail continues :)