Original text:
The ciphersuites are defined in section mls-ciphersuites. <del> Depending on the Diffie-Hellman group of the ciphersuite, different rules apply to private key derivation and public key verification. For all ciphersuites defined in this document, the Derive-Key-Pair function begins by deriving a "key pair secret" of appropriate length, then converting it to a private key in the required group. The ciphersuite specifies the required length and the conversion. 6.1.1. For X25519, the key pair secret is 32 octets long.  No conversion is required, since any 32-octet string is a valid X25519 private key. The corresponding public key is X25519(SHA-256(X), 9). For X448, the key pair secret is 56 octets long.  No conversion is required, since any 56-octet string is a valid X448 private key.  The corresponding public key is X448(SHA-256(X), 5). Implementations MUST use the approach specified in RFC7748 to calculate the Diffie-Hellman shared secret.  Implementations MUST check whether the computed Diffie-Hellman shared secret is the all- zero value and abort if so, as described in Section 6 of RFC7748.  If implementers use an alternative implementation of these elliptic curves, they MUST perform the additional checks specified in Section 7 of 6.1.2. For P-256, the key pair secret is 32 octets long.  For P-521, the key pair secret is 66 octets long.  In either case, the private key derived from a key pair secret is computed by interpreting the key pair secret as a big-endian integer. ECDH calculations for these curves (including parameter and key generation as well as the shared secret calculation) are performed according to IEEE1363 using the ECKAS-DH1 scheme with the identity map as key derivation function (KDF), so that the shared secret is the x-coordinate of the ECDH shared secret elliptic curve point represented as an octet string.  Note that this octet string (Z in IEEE 1363 terminology) as output by FE2OSP, the Field Element to Octet String Conversion Primitive, has constant length for any given field; leading zeros found in this octet string MUST NOT be truncated. (Note that this use of the identity KDF is a technicality.  The complete picture is that ECDH is employed with a non-trivial KDF because MLS does not directly use this secret for anything other than for computing other secrets.) Clients MUST validate remote public values by ensuring that the point is a valid point on the elliptic curve.  The appropriate validation procedures are defined in Section 4.3.7 of X962 and alternatively in Section 5.6.2.3 of keyagreement.  This process consists of three steps: (1) verify that the value is not the point at infinity (O), (2) verify that for Y = (x, y) both integers are in the correct interval, (3) ensure that (x, y) is a correct solution to the elliptic curve equation.  For these curves, implementers do not need to verify membership in the correct subgroup. </del> 6.2. A member of a group authenticates the identities of other

Reviewer comments:
The first commit is editorial. I removed all the DH terminology I could find and replaced it with KEM terminology. The second commit is related to URL It would massively reduce complexity of implementations if people were able to rely on HPKE to do all private key parsing, key validation, public key computation, etc. Currently, all the information in the deleted CFRG and NIST sections is either redundant (performed by HPKE) or outdated and performed correctly by HPKE (e.g., old P-256 DH secret representation). Instead of having all this duplicated, it would be easier to keep it in one place. An issue with the new definition of : Repeatedly calling can be computationally expensive. Each call requires computing the group hash (which, okay, it could be cached) and serializing a struct. It would be nice to do something that's cheaper but still ties context into the secret key generation.
Update: This now uses HPKE's function to do all the heavy lifting

Revised text:
The ciphersuites are defined in section mls-ciphersuites.  6.2. A member of a group authenticates the identities of other