[Cfrg] Weak Diffie-Hellman Primes

Michael D'Errico <mike-list@pobox.com> Sat, 10 October 2020 22:01 UTC

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Date: Sat, 10 Oct 2020 18:01:11 -0400
From: "Michael D'Errico" <mike-list@pobox.com>
To: cfrg@irtf.org
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Subject: [Cfrg] Weak Diffie-Hellman Primes
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Hi,

I'm not a member of this list, but was encouraged to
start a discussion here about a discovery I made
w.r.t. the published Diffie-Hellman prime numbers in
RFC's 2409, 3526, and 7919.  These primes all have
a very interesting property where you get 64 or more
bits (the least significant bits of 2^X mod P for some
secret X and prime P) detailing how the modulo
operation was done.  These 64 bits probably reduce
the security of Diffie-Hellman key exchanges though
I have not tried to figure out how.

The number 2^X is going to be a single bit with value
1 followed by a lot of zeros.  All of the primes in the
above mentioned RFC's have 64 bits of 1 in the most
and least significant positions.  The 2's complement
of these primes will have a one in the least significant
bit and at least 63 zeros to the left.

When you think about how a modulo operation is done
manually, you compare a shifted version of P against
the current value of the operand (which is initially 2^X)
and if it's larger than the (shifted) P, you subtract P at
that position and shift P to the right, or if the operand
is smaller than (the shifted) P, you just shift P to the
right without subtracting.

Instead of subtracting, you can add the 2's complement
I mentioned above.  Because of the fact that there are
63 zeros followed by a 1 in the lowest position, you will
see a record of when the modulo operation performed
a subtraction (there's a one) and when it didn't (there's
a zero).

You can use the value of the result you were given by
your peer (which is 2^X mod P) and then add back the
various 2^j * P's detailed wherever the lowest 64 bits
had a value of 1 to find the state of the mod  P operation
when it wasn't yet finished.  This intermediate result is
likely going to make it easier to determine X than just a
brute force search.

I don't plan to join this list, though I am flattered to have
been asked to do so.  I'm not a cryptographer.

Mike