safety of Pohlig-Hellman with a common modulus?

[Anton Stiglic]

----- Original Message ----- 
From: "Peter Fairbrother" <zenadsl6186 at zen.co.uk>
To: "David Wagner" <daw-usenet at taverner.cs.berkeley.edu>;
<cryptography at metzdowd.com>
Sent: Saturday, December 06, 2003 7:58 PM
Subject: Re: safety of Pohlig-Hellman with a common modulus?


> David Wagner wrote:
>
> > Steve Bellovin  wrote:
> >> Is it safe to use Pohlig-Hellman encryption with a common modulus?
> >> That is, I want various parties to have their own exponents, but share
> >> the same prime modulus.  In my application, a chosen plaintext attack
> >> will be possible.  (I know that RSA with common modulus is not safe.)
> >
> > Yes, I believe so.  The security of Pohlig-Hellman rests on the
difficulty
> > of the discrete log problem.
>
> Nope. In P-H there is no g. A ciphertext is M^k mod p. An attacker won't
> know k, and usually won't know M, but see below. I don't know what the
> problem is called, but it isn't DLP. Anyone?

If you don`t know M and k, there are several values M', k' such that
M'^k' mod p == M^k mod p.   For example, if M is a generator of the
group mod p, than all other generators M' will have a corresponding k'
that will give you this value.

Think about known plaintext attack or chosen plaintext attack.  A symmetric
cipher should be secure against these attacks and much more...
In these attacks you know the bases of several values...

--Anton

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