[curves] Costs of cofactor > 1
trevp at trevp.net
Wed Dec 24 03:20:53 PST 2014
Mike's considering an encoding that allows dealing with cofactor > 1
curves as if the cofactor = 1.
Below I try to list the costs of cofactor > 1 for common algorithms.
The goal is to see how annoying the cofactor actually is, so we can
weigh whether proposals like Mike's are worthwhile.
There's no need for public-key validation with twist-secure curves and
single-coordinate or compressed encoding. So there's nothing for the
cofactor to affect.
For DH it seems typical and easy to clear the cofactor (as done in
Curve25519 by setting the private key to 0 mod cofactor). This means
a public key could be tampered into a few different-but-equivalent
values by adding points of low order.
But a similar DH tampering is possible regardless of cofactor
(multiply both public keys by the same scalar). This sort of
tampering doesn't accomplish much - at worst, you could imagine Alice
depositing something with Bob over a secure channel that she plans to
retrieve later based on her long-term DH public key. Alice's naive
assumption that Bob must be seeing the "correct" encoding for her
public-key could be violated by tampering.
All this is easily prevented by the common practice of hashing encoded
DH public keys into the session key, or binding them with signatures
Ed25519 verification can clear the cofactor (fast batch verification)
or not (fast single-signature verification).
In an anonymity situation an attacker might be able to use a signature
that passes the first implementation but fails the second to learn
which choice was taken.
I suggested standardizing the stricter check (don't clear cofactor) since:
* it's the common implementation
* batch verification seems rarely (if ever?) used
* most verifiers won't worry about leaking this tiny bit of information
Robert suggested standardizing the other way, since clearing the
cofactor has a very small impact on single-signature performance, but
allows ~2x optimization for batch verification.
I'm interested in other opinions. Also, are there real uses for batch
"Contributory" key agreement
The term "contributory" pops up in discussions of 25519. The idea
seems to be that some protocols are insecure if someone can force
different DH shared secrets to the same value, so with 25519 you'd
need to either check for low-order inputs, or check for the identity
These checks are easy, but easier with cofactor=1 because there's no
low-order inputs so nothing to check.
However: do protocols that require this property actually exist?
AFAIK the term "contributory" comes from Group Key Agreements to
describe calculating a shared key based on DH public keys from several
parties. But this doesn't seem to be either a security property or a
requirement on the DH - a malicious insider to the GKA doesn't need to
force the session key, she just executes the GKA and learns the
session key. George Danezis has a good analysis .
One could argue that cofactor=1 is more robust because forcing the
same key in different sessions might assist attacks if the protocol
has other mistakes. That actually happened with TLS triple handshake,
so can't be dismissed, but it wouldn't be relevant to a well-designed
I'm not seeing cofactor>1 as a big deal, or worth much effort to change.
It seems like cofactor=1 might be a tiny bit more robust if your
protocol has other flaws, and a smidgen more anonymous if you want to
Is that it? What am I missing?
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