safety of Pohlig-Hellman with a common modulus?

[Peter Fairbrother]

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?

> Knowing the discrete log of g^y doesn't help
> me learn the discrete log of g^x (assuming x,y are picked independently).
> This is not like RSA, where using a common modulus allows devastating
> attacks.
> 
> There is a small caveat, but it is pretty minor.  There are some
> precomputation attacks one can do which depend only on the prime p; after
> a long precomputation, one can compute discrete logs mod p fairly quickly.
> The more people who use the same modulus, the more attractive such a
> precomputation effort will be.  So the only reason (that I know of)
> for using different modulii with Pohlig-Hellman is to avoid putting all
> your eggs in one basket.

Not usually.  In general index calculus attacks don't work on P-H, except in
chosen plaintext attacks (where the chosen plaintext sort-of substitutes for
g).

When using P-H I usually pre-encrypt data in any old symmetric cipher with a
random IV and any old key, to avoid known plaintext attacks. There are
several other attacks to be aware of, including some nasty
adaptive-chosen-plaintext and chosen-ciphertext attacks.

-- 
Peter Fairbrother

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