 # @ Loup's

Impossible? Like that would stop me.

# XCKDF: X25519 Chacha20 Key Derivation Function

We describe here a way to derive keys from X25519 shared secrets, using nothing but the Chacha20 stream cipher (and its related hash HChacha20).

This is a cascading scheme, intended to be used with up to 4 levels (we could go beyond, but the protocols I intend to use it for don't go beyond 4 key exchanges).

``````   DH1       DH2       DH3       DH4
┌─┴─┐     ┌─┴─┐     ┌─┴─┐     ┌─┴─┐
0╶┤ H │   0╶┤ H │   0╶┤ H │   0╶┤ H │
└─┬─┘     └─┬─┘     └─┬─┘     └─┬─┘
│   ┌────╴+   ┌────╴+   ┌────╴+
┌─┴─┐ │   ┌─┴─┐ │   ┌─┴─┐ │   ┌─┴─┐
p╶┤ H │ │ p╶┤ H │ │ p╶┤ H │ │ p╶┤ H │
└─┬─┘ │   └─┬─┘ │   └─┬─┘ │   └─┬─┘
┌─┴─┐ │   ┌─┴─┐ │   ┌─┴─┐ │   ┌─┴─┐
1╶┤ C ├─┘ 1╶┤ C ├─┘ 1╶┤ C ├─┘ 1╶┤ C │
└┬┬┬┘     └┬┬┬┘     └┬┬┬┘     └┬┬┬┘
││└╴AK1   ││└╴AK2   ││└╴AK3   ││└╴AK4
│└╴EK1    │└╴EK2    │└╴EK3    │└╴EK4
└╴PK1     └╴PK2     └╴PK3     └╴PK4
``````

DH1, DH2, DH3, and DH4, are X25519 shared secrets. "+" means XOR, and p is an arbitrary constant (typically the protocol name, used for domain separation) The boxes mean the following:

``````       key
┌─┴─┐
input╶┤ H │       out = HChacha20(key, input)
└─┬─┘
out

key
┌─┴─┐       out1 = Chacha20(key, nonce)[ 0: 31]
nonce╶┤ C ├╴out1  out2 = Chacha20(key, nonce)[32: 63]
└┬┬┬┘       out3 = Chacha20(key, nonce)[64: 95]
││└╴out2   out4 = Chacha20(key, nonce)[96:127]
│└╴out3
└╴out4
``````

Keys span 32 bytes. The HChacha20 input spans 16 bytes. All outputs span 32 bytes. Chacha20 technically outputs a stream, we just divide the first 128 bytes of that stream among the four outputs.

## Security Goals

• If DH1 is secure, then:

• AK1, AK2, AK3, and AK4 are independent random strings.
• EK1, EK2, EK3, and EK4 are independent random strings.
• PK1, PK2, PK3, and PK4 are independent random strings.
• If DH2 is secure, then:

• AK2, AK3, and AK4 are independent random strings.
• EK2, EK3, and EK4 are independent random strings.
• PK2, PK3, and PK4 are independent random strings.
• If DH3 is secure, then:

• AK3 and AK4 are independent random strings.
• EK3 and EK4 are independent random strings.
• PK3 and PK4 are independent random strings.
• If DH4 is secure, then:

• AK4 is an independent random string.
• EK4 is an independent random string.
• PK4 is an independent random string.

Being a "random string" means being indistinguishable from uniform random string in a feasible amount of computation. Being "secure" for a DH shared secret means the private keys involved in the key exchange are independent random strings, unknown to anyone but their rightful owner.

## Analysis of a single stage

A single stage of the cascade looks like this:

``````         DH      (1) hash_s  = HChacha20(DH, 0)
┌─┴─┐    (2) x       = CK_prev XOR hash_s
0╶┤ H │    (3) hash_x  = HChacha20(x, p)
└─┬─┘    (4) CK_next = Chacha20(hash_x, 1)[ 0: 31]
CK_prev╶─╴+      (5) AK      = Chacha20(hash_x, 1)[32: 63]
┌─┴─┐    (6) EK      = Chacha20(hash_x, 1)[64: 95]
p╶┤ H │    (7) PK      = Chacha20(hash_x, 1)[96:127]
└─┬─┘
┌─┴─┐
1╶┤ C ├╴CK_next
└┬┬┬┘
││└╴AK
│└╴EK
└╴PK
``````

The security goals of a single step are:

• If CK_prev is an independent uniform random string, then

• CK_next, AK, EK, and PK are independent, uniform random strings.
• If DH is secure, then

• CK_next, AK, EK, and PK are independent, uniform random strings.

### Assumptions

• Assumption 1: Chacha20 is a PRF: given a random string and nonce, it outputs an independent random stream. (This is the security goal of Chacha20. It holds as long as Chacha20 is secure.)

• Assumption 2: HChacha20 is a PRF: given a random string and an input, it outputs an independent random string. (This is the security goal of HChacha20. There's a security reduction proving that it holds as long as Chacha20 is secure.)

• Assumption 3: The HChacha20 of a secure X25519 shared secret and input zero is random. (This comes from a security conjecture made by Daniel Bernstein in his Curve25519 paper, and applied in NaCl and Libsodium. The only difference is that he used HSalsa20, and we use HChacha20.)

• Theorem 4: the XOR of two independent inputs, at least one of which is random, is random. (We consider this theorem "obvious", and won't prove it here.)

### Preliminary lemmas

• Lemma 5: If DH is secure, then hash_s is random. (From equation (1) and assumption 3.)

• Lemma 6: If CK_prev and hash_s are independent, and at least one of them is random, then x is random. (From equation (2) and theorem 4)

• Lemma 7: If x is random, then hash_x is an independent random string. Note that hash_x also depends on p. Changing p would produce a different independent random string. (From equation (3) and assumption 2)

• Lemma 8: If hash_x is random, then CK_next, AK, EK, and PK are independent random strings. (From equations (4) to (7), and assumption 1.)

### Induction lemmas

• Lemma 9: If both CK_prev and DH are known, then CK_next is known (Obvious, from equations 1 to 7.)

• Lemma 10: If CK_prev is known and DH is secure, then CK_next, AK, EK, and PK are independent random strings. (From lemma 5 to 8.)

• Lemma 11: If CK_prev is an independent random string, then CK_next, AK, EK, and PK are independent random strings. (From lemma 5 to 8.)

## Analysis of the whole chain

Notes: CK_prev(0) is zero. CK_next(n) = CK_prev(n+1) for all n > 0.

• Step zero: If DH(i) is known, for 0 < i ≤ n, then CK_prev(n+1) is known.

• Initialisation step: If CK_prev(n) is known and DH(n) is secure, then AK(n), EK(n), PK(n), and CK_prev(n+1) are independent random strings. (From lemma 10.)

• Induction step: If CK_prev(n) is an independent random string, then AK(n), EK(n), PK(n), and CK_prev(n+1) are independent random strings. (From lemma 11.)

• Conclusion:

• If DH(1) is secure, then AK(n), EK(n), and PK(n) are independent random strings, for all n > 0.
• If DH(1) is known, let i be the smallest integer such that DH(i) is known, and DH(i+1) is secure. Then AK(n), EK(n), and PK(n), for all n > i.

QED.

## Design rationale

This key derivation is intended to be used in protocols that already rely on Chacha20 for encryption. Using it instead of an actual hash saves a primitive, allowing implementations to be simpler and smaller.

All output keys are, when secure, independent random strings. This independence is important because:

• The AK keys are intended to be used with Poly1305, a one time authenticator that partially reveals the key. Thus, revealing an AK key must not leak any other secret.

• The EK keys are intended to be used as one time pads (a 32 byte key is enough to encrypt 32 bytes with XOR), and are thus revealed to the attacker if they happen to know the plaintext. Thus, revealing an EK key must not leak any other secret.

• The PK keys are intended to be used as payload keys for reduced round trip messages, or session keys (for the last one). Keeping them independent from everything else removes all restrictions on nonce or counter usage (This is in contrast to the Noise protocol framework, which reserves the maximum value for its own use.)

The intermediate HChacha20 call with p has been added to provide domain separation across protocols, even if users end up reusing keys between protocols. This can happen if users perform an exchanges between their long term keys, or if they (misguidedly) reuse an ephemeral key for several protocols to save key exchanges during a protocol negotiation. Users who don't need such domain separation could arguably remove it, and save a tiny bit of computation in the process. I personally think the flexibility is worth the cost.

The call to Chacha20 has a nonce of 1, to avoid collision with the HChacha20 of the DH key exchanges, even if the attacker controls one an knows the other. For instance, an attacker knowing the value of DH1 and having control over DH2 could have DH2 be HChacha20(HChacha20(DH1, 0), p), Causing half of CK_prev(2) to be correlated with HChacha20(DH2). Changing the nonce basically forces the attacker to perform a preimage attack on HChacha20. (This is a minor concern: the attacker can only has limited control over the the value of a DH key exchange to begin with, and having the opportunity to exercise that control means they know the relevant keys anyway, so they don't need to control anything.)