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Date: Sun, 14 Sep 2014 16:53:33 +0200
From: Thomas Pornin <>
Subject: Re: [PHC] Improving Makwa and arguing points for or against

On Sun, Sep 14, 2014 at 05:13:11AM -0500, Steve Thomas wrote:
> Replacing e=2**n with e=65537**n also require gcd(65537,
> (p-1)*(q-1))=1. This way you can say it's as safe as RSA.

It would not make it "as safe as RSA" but "as unsafe as RSA".

This is an old debate, between RSA and Rabin encryption systems. Rabin
uses squarings modulo a Blum integer N. RSA uses an exponent e which is
relatively prime to phi(N). For both systems, factoring N totally breaks
the system; however, in the case of Rabin, it can be demonstrated that
generic square root extraction modulo N _is_ equivalent to factoring N.
No such demonstration exists for RSA. It is thus possible that there
exists a method which breaks RSA without requiring modulus factorization
(no such method is currently known, but it is not proven that such a
method cannot exist). Thereby, Rabin encryption can be said to be at
least as safe as RSA, and potentially safer.

By relying on squarings instead of RSA-like exponentiations, I am
reusing the safety of Rabin encryption, which is thus better (or at
least no worse) then that of RSA. This has already been put to work: in
a previous email, Bill was worried about "short cycles" in the
permutation (squaring modulo a Blum integer is a permutation of the set
of quadratic residues); thanks to the use of squarings, I could
demonstrate that if short cycles can be found with non-negligible
probability, then they can be turned into an efficient factorization
algorithm, which would imply that both Rabin and RSA encryption, as they
are used today, are broken. This kind of demonstration is precisely why
I want to use squarings instead of RSA-like exponentiations. Squarings
are just better.

Squarings also make for a simpler implementation. All other things being
equal, simplicity is always better: it minimizes risks of bugs and of
data leakage through timing attacks.

Last but not least, using repeated squarings is an instance of the
"time-lock puzzles" of Rivest, Shamir and Wagner (1996). This is a
process which has sustained the collective scrutiny of cryptographers
for quite some time now, and nobody yet found a way to speed up a
sequence of squarings (i.e. to obtain the result faster than doing all
the squarings one after the other), save by factoring the modulus (and
2500+ years of research show that factoring is hard). This is a nice
thing to have: apart from potential mathematical reduction proofs (like
the equivalence between factorization and square root extraction),
extensive review is the only way to know whether a given cryptographic
function is secure or not. By comparison, "repeated RSA-like
exponentiation" is an as-yet unstudied and not-reviewed problem.

> I guess that's not really necessary since post hashing is now required.

As I explain in another mail, this is a misconception. If you fear
that the attacker learns the private factors p and q, then post-hashing
won't stop him; he still gets tremendous attack power from that
knowledge. On the other hand, if he does _not_ know the factors p and q,
then without post-hashing he still faces the Rivest-Shamir-Wagner
time-lock puzzle, which has no known speedup.

Therefore, you must not consider post-hashing as improving security with
regards to the modulus factorization. The role of post-hashing is to
provide unbiased bytes in arbitrary numbers for when Makwa is used as a
password-based KDF (usually for password-based encryption);
incidentally, post-hashing also allows for a reduced storage space in
password verification scenario, but at a price (it prevents smooth
offline work factor increase).

> whether we can prevent the author from saying "you could keep p and q
> for fast path".

Ah, shutting me up... some have tried.

	--Thomas Pornin

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