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The libsodium documentation contains a function crypto_sign_ed25519_pk_to_curve25519 that converts an Ed25519 key into a Curve25519 one, so it can be used for both key exchange/encryption and signing:

"Ed25519 keys can be converted to Curve25519 keys, so that the same key pair can be used both for authenticated encryption (crypto_box) and for signatures (crypto_sign)."

The same page ends "Notes: If you can afford it, using distinct keys for signing and for encryption is still highly recommended."

My question is whether reusing an Ed25519 keypair for both uses has been proven secure anywhere, for example in the sense of Joint Security of Encryption and Signature or a similar paper?

The two constructions seem to be:

  • EdDSA signature (i.e. "Schnorr signature done properly")
  • Ephemeral DH key exchange using a one-time keypair on the sender side and the recipient's public key (mapped to Curve25519).

Both involve a one-time ephemeral keypair and a hash function, I'm fine with the random oracle model (so we can assume the EdDSA nonce, which is actually a function of the key and message, is really random) and one can get domain separation by including a separation tag in the key exchange, i.e. when Alice wants to send a file to Bob whose public key is Q, she picks a pair (r, r.P) and sets the key as k = H(r.Q, "this is a key not a signature") or something similar.

Intuitively this should be fine in the random oracle model; has someone actually (dis)proved it though?

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    $\begingroup$ That's kind of unfair. I think it is not recommended because security would be hard to proof :) $\endgroup$
    – Maarten Bodewes ♦
    Commented Jul 22, 2016 at 11:07
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    $\begingroup$ @MaartenBodewes I don't expect it to be hard to prove if you're willing to assume that XSalsa20 and SHA512 are independent hash functions modeled as random oracles. $\endgroup$
    – CodesInChaos
    Commented Jul 22, 2016 at 14:44
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    $\begingroup$ I was hoping DH+EdDSA was such a standard combination that someone had done it in the appendix of their paper somewhere as part of a larger work ... on its own it's only really worth an eprint, I agree with Maarten here. But considering that libsodium has a function for doing this in practice, it's probably worth someone trying just to be sure. It would make a good first proof for a PhD student perhaps. $\endgroup$
    – user2552
    Commented Jul 24, 2016 at 8:47

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Update (2022): See On using the same key pair for Ed25519 and an X25519 based KEM.

The paper On the Joint Security of Encryption and Signature in EMV is very close. Section 4.4, "On the Joint Security of ECIES and EC-Schnorr", states the following result:

Theorem 2. In the random oracle model ECIES-KEM and EC-Schnorr are jointly secure if the gap-DLP problem and gap-DH problem are both hard.

Obviously, this does not apply to EdDSA/DH out of the box: ECIES is a certain application of Diffie–Hellman, and EdDSA is a slightly modified variant of elliptic-curve Schnorr signatures. However, the proof sketches in the paper should be easily adaptable to EdDSA combined with any other reasonable Diffie–Hellman incarnation. (The details of this depend a little on how the DH shared secret is used for symmetric (authenticated) encryption, but it seems possible to modify the proof for all realistic settings.)

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    $\begingroup$ That's interesting! I am working in a commercial company with both ed25519 and x25519 and we would love to use just one key pair, but so far we haven't. Actually, we would be willing to pay for someone to prove this. But who can do it? Finding such expertise seems hard. $\endgroup$
    – Frans Lundberg
    Commented Feb 7, 2018 at 20:53
  • $\begingroup$ I think a summary of the 2021 paper would be a great addition as it could be more readable. $\endgroup$
    – samuel-lucas6
    Commented Jul 31, 2022 at 13:58
  • $\begingroup$ In the paper it seems to indicate this is OK to do. But it acquires X25519 shared secret from the Ed25519 key through a DH key exchange. Then it does an HKDF-Extract on the shared secret, the resulting derived key is then used (for KEM?). My question is why is HKDF-Extract needed? And if it is needed, why should a salt be needed especially since its unlikely to have a rainbow attack here. $\endgroup$
    – CMCDragonkai
    Commented Sep 16, 2022 at 5:16
  • $\begingroup$ The salt in HKDF plays a role in the security proof, and it should be there to fulfill the semantic and therefore claim the proven security. Without it, the construction looks secure, but the security is no longer proven, that is, we do not know if it is really secure now, though it looks like a secure one. $\endgroup$
    – Tan
    Commented Oct 5, 2023 at 6:36

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