From: Hal <hfinney@shell.portal.com>
Date: Wed, 13 Dec 1995 11:13:07 +0800
To: cypherpunks@toad.com
Subject: Re: Time-based cryptanalysis: How to defeat it?
Message-ID: <199512130144.RAA08950@jobe.shell.portal.com>
MIME-Version: 1.0
Content-Type: text/plain


From: futplex@pseudonym.com (Futplex)
> I don't understand why Kocher's point is correct. For example, why do the
> times diverge with the following modification of the modexp algorithm on
> pg.2 of the abstract ?
> 
> 	Algorithm to compute R = y^x mod n:
> 		Let R_0 = 1.
> 		Let y_0 = y.
> 		For i = 0 upto (bits_in_x - 1):
> 			Let M = (R_i * y_i) mod n.
> 			Let R_(i+1) = (bit i of x) * M  +
> 					(1 - (bit i of x)) * R_i.
> 			Let y_(i+1) = (y_i)^2 mod n.
> 		End.

I posted a similar idea on sci.crypt, but later I realized that Paul Kocher
is right.

Your algorithm works OK for the first iteration.  The amount of work is
pretty much constant regardless of whether bit 0 of x is 0 or 1.
However, at the end of that iteration R_1 will have one of two
different values depending on that bit 0 value.  And, the attacker can
know these two values, and if he controls y he can even choose them
(they will be either y or 1).

Now, on the next iteration, the time it takes will be different
depending on bit 0 of x.  It won't depend on the bit 1 value, but
different bit 0 values will cause R_1 to be different.  So the time of
this iteration will depend on the value of the bit used in the previous
iteration, and likewise for the following iterations.

If the attacker can choose y, he can arrange that the two different R_1
values will take different times on average for the rest of the
calculation.  So he finds out bit 0 as before, and from there he can go
on and find the other bits.

Hal