[curves] Ed25519 signatures from Curve25519 keys

David Leon Gil coruus at gmail.com
Wed Jun 18 05:47:53 PDT 2014

Just a quick question: Not reusing keys requires no security proofs at all.
Is there really a benefit that justifies key reuse? (I suppose that if you
really need a verifiable PoP of a Curve25519 key...)

I, personally, am rather uncomfortable with Gap-DH. Especially when CDH is
what you're giving up.

(And as a note to folks who may not be aware, I believe that Adam Langley
has implemented  this: https://github.com/agl/ed25519/tree/master/extra25519

On Monday, June 16, 2014, Trevor Perrin <trevp at trevp.net> wrote:

> Hi,
> I'm wondering about using Curve25519 keys to create and verify Ed25519
> signatures.  For example, imagine you have existing Curve25519
> long-term keys being used for a DH protocol, and you'd like to sign
> with these keys.
> Below is an attempt to provide security analysis, and work out the details.
> I've run parts of this by a few people (Mike Hamburg, Robert Ransom,
> CodesInChaos).  I try to credit them where appropriate, but that
> doesn't imply they endorse all of this.
> Joint security of signatures and DH
> ----
> Schnorr signatures (like Ed25519) have a security proof in the Random
> Oracle Model based on the (Elliptic Curve ) Discrete Log assumption
> [1].
> Many DH protocols have security proofs in a model where the attacker
> has access to a "Decision Diffie-Hellman" oracle.   Often the attacker
> can choose an arbitrary public key, cause a targeted key to perform a
> DH operation with the chosen public key, and then "reveal" the hashed
> output.   By hashing different inputs and seeing if the result equals
> the revealed value, the attacker gains a DDH oracle involving the
> targeted key and her chosen key.   These protocols can often be proven
> secure in the Random Oracle Model based on the "Gap-DH" assumption:
> i.e. the assumption that the Computational DH problem is hard even
> when given a DDH oracle.  Protocols that can be proven secure in such
> a model include ECIES [2], NAXOS [3], Ntor [4], Kudla-Paterson [5],
> HOMQV [6], and others.
> As in [7] section 4.4, it seems fairly easy to argue for "joint
> security" when the same key is used for Schnorr signatures and for a
> protocol with a Gap-DH/ROM security proof, provided the hash function
> is used carefully so that RO queries for the signature can be
> distinguished from RO queries for the DH-protocol.
> In particular:
>  - Giving the DH-protocol adversary a signing oracle doesn't help it,
> as the signing oracle can be simulated in the ROM without knowledge of
> the secret key [1,7].
>  - Giving the signature adversary access to parties running the
> DH-protocol (e.g. an eCK experiment [3] where the secret key is held
> by an honest party) doesn't help if the protocol can be simulated
> without knowledge of the secret key but with a DDH oracle.  If the
> signature adversary + simulator could use the DDH oracle to break the
> DL problem, then the Gap-DH assumption would be violated [7].
> So in principle this seems secure, do people agree?
> Public-key conversion
> ----
> A Curve25519 public-key is encoded as a 255-bit x-coordinate.  An
> Ed25519 public-key is encoded as a 255-bit y-coordinate, plus a "sign"
> bit.
> For a "unified" public-key format, I think it makes sense to stick
> with Curve25519:
>  - Curve25519 has seen more deployment than Ed25519, so existing
> public-keys are more likely to use the Curve25519 format.
>  - The sign bit isn't strictly necessary, since it can be assumed to
> be zero, and the private key can be adjusted to match (see below).
>  - The Curve25519 format is more efficient for DH since it can be used
> directly with the Montgomery ladder.  For signature verification, the
> conversion from Curve25519 format to an Ed25519 point can be combined
> with decompression, so using Curve25519 public keys to verify Ed25519
> signatures can be almost as fast for verification as Ed25519 public
> keys (according to Mike Hamburg).
>  - If code simplicity is more of a concern than speed, it's easy to
> convert the curve25519 x-coordinate into an ed25519 y-coordinate by
> ed_y = (curve_x - 1) / (curve_x + 1) mod 2^255-19 (page 8 of [8], hat
> tip Robert Ransom).  The inversion takes perhaps 10-20% (?) of the
> time for a variable-base scalar mult.  The y-coordinate can then be
> encoded along with a sign bit of 0, allowing existing Ed25519 code to
> be used.
> Private-key conversion
> ----
> If the Ed25519 public-key sign-bit is assumed to be zero, the private
> key may need to be adjusted (per Jivsov [9]).  In other words, if
> multiplying the Curve25519 private key by the Ed25519 base point
> yields an Ed25519 x-coordinate that's "negative" as defined in [8],
> the private key (a) must be negated modulo the order of the base point
> (q), i.e. a = q - a.
> Some existing curve25519 implementations set bit 254 of the private
> key within the scalarmult function, so will interfere with this
> negation (observation due CodesInChaos).   Robert Ransom proposed
> another way to implement the negation that avoids having to modify
> that code:
>  - Before hashing, flip the sign bit of R
>  - Before hashing, encode the sign bit of A as zero
>  - As the last step, negate S, i.e. S = q - S
> Jivsov argues that fixing the sign bit doesn't lose security, even
> against Pollard rho (anyone care to double-check the argument? - [9]
> section 8).  A side-channel that leaks only whether this negation was
> performed (or not) only reveals the sign bit of the original private
> key, so I think also doesn't reduce security.
> Hash inputs
> ----
> Ed25519 calculates SHA512(R || A || M), where:
>  - R is an Ed25519-encoded Schnorr commitment  (= nonce *  base point)
>  - A is an Ed25519-encoded public key
>  - M is the message to sign
> The key-derivation in the DH protocol must hash distinct inputs for
> the ROM security argument to hold.  CodesInChaos suggested beginning
> the key-derivation hash with 32 bytes of 0xFF, as this is an invalid
> Ed25519 point.
> Signature nonce
> ----
> In the original Ed25519 implementation [8], a single master key is
> used to derive both (a) the Ed25519 private scalar and (b) a secret
> key for nonce generation.  Without such a master key, either
>  - the nonce-generation key would have to be explicitly generated and
> stored
>  - the nonce would have to be taken from a (strong!) RNG
>  - the private scalar would have to be used as the nonce-generation key
> All of these seem adequate.  Not sure which is best?
> Anyways, what else have I missed?
> Trevor
> [1] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=
> [2] http://www.cs.ucdavis.edu/~rogaway/papers/dhies.pdf
> [3] http://research.microsoft.com/pubs/81673/strongake-submitted.pdf
> [4] http://cacr.uwaterloo.ca/techreports/2011/cacr2011-11.pdf
> [5] http://www.isg.rhul.ac.uk/~kp/ModularProofs.pdf
> [6] http://eprint.iacr.org/2010/638
> [7] http://eprint.iacr.org/2011/615
> [8] http://ed25519.cr.yp.to/ed25519-20110926.pdf
> [9]
> https://datatracker.ietf.org/doc/draft-jivsov-ecc-compact/?include_text=1
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