From: Hal <hfinney@shell.portal.com>
Date: Sat, 16 Dec 1995 02:01:58 +0800
To: cypherpunks@toad.com
Subject: Re: Kocher's RSA attack
Message-ID: <199512151506.HAA27760@jobe.shell.portal.com>
MIME-Version: 1.0
Content-Type: text/plain


From: Derek Atkins <warlord@MIT.EDU>
> I must disagree, software implementations of RSA can and probably do
> allow the timing attacks.  It all depends on the modexp implementation.
> Most implementations that I know of, when performing an x^y mod n will
> require a squarings and b multiplies, where a is the number of bits in
> y and b is the number of 1-bits in y.

This is not enough - Paul Kocher's attack depends on the individual
modular multiplies taking different times.  (Actually, that is for his
attack on Diffie Hellman.  The RSA CRT decryption attack uses a
completely different principle, but I guess we are ignoring that for
now.)  The fact that timing a modular exponentiation would give
information about the density of 1 bits in the exponent is not
particularly new or surprising, as has been mentioned here.  What is
new is that you can actually figure out the specific exponent value.
But that requires variable-timing modmult, not just variable-timing
modexp.

PGP is somewhat unique in having a multiplicity of modmult algorithms
which can be selected at compile time.  I am not sure which of these
might be variable time and which might be fixed.  The most likely place
for time variation IMO is in the modular reduction rather than the
multiply; the multiply is generally deterministic with no variation due
to data values (although as was pointed out here, on some processors a
hardware multiply instruction may take variable time depending on its
inputs).  Some modular reductions involve trial division to some extent
or other, with different numbers of iterations possible depending on
certain (maybe unusual) values.  However I believe at least one of the
PGP modular reductions consists of multiplying by the reciprocal of the
modulus, followed by a fixed shift, and this one should be constant time
on a machine which has constant-time multiplies and shifts.  (This is
just going from memory, I haven't looked at the algorithm in several
years.)

Hal