| RFC 8696 | Using PSK in the CMS | December 2019 |
| Housley | Standards Track | [Page] |
The invention of a large-scale quantum computer would pose a serious challenge for the cryptographic algorithms that are widely deployed today. The Cryptographic Message Syntax (CMS) supports key transport and key agreement algorithms that could be broken by the invention of such a quantum computer. By storing communications that are protected with the CMS today, someone could decrypt them in the future when a large-scale quantum computer becomes available. Once quantum-secure key management algorithms are available, the CMS will be extended to support the new algorithms if the existing syntax does not accommodate them. This document describes a mechanism to protect today's communication from the future invention of a large-scale quantum computer by mixing the output of key transport and key agreement algorithms with a pre-shared key.¶
This is an Internet Standards Track document.¶
This document is a product of the Internet Engineering Task Force (IETF). It represents the consensus of the IETF community. It has received public review and has been approved for publication by the Internet Engineering Steering Group (IESG). Further information on Internet Standards is available in Section 2 of RFC 7841.¶
Information about the current status of this document, any errata, and how to provide feedback on it may be obtained at https://www.rfc-editor.org/info/rfc8696.¶
Copyright (c) 2019 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License.¶
The invention of a large-scale quantum computer would pose a serious challenge for the cryptographic algorithms that are widely deployed today [S1994]. It is an open question whether or not it is feasible to build a large-scale quantum computer and, if so, when that might happen [NAS2019]. However, if such a quantum computer is invented, many of the cryptographic algorithms and the security protocols that use them would become vulnerable.¶
The Cryptographic Message Syntax (CMS) [RFC5652][RFC5083] supports key transport and key agreement algorithms that could be broken by the invention of a large-scale quantum computer [C2PQ]. These algorithms include RSA [RFC8017], Diffie-Hellman [RFC2631], and Elliptic Curve Diffie-Hellman (ECDH) [RFC5753]. As a result, an adversary that stores CMS-protected communications today could decrypt those communications in the future when a large-scale quantum computer becomes available.¶
Once quantum-secure key management algorithms are available, the CMS will be extended to support them if the existing syntax does not already accommodate the new algorithms.¶
In the near term, this document describes a mechanism to protect today's communication from the future invention of a large-scale quantum computer by mixing the output of existing key transport and key agreement algorithms with a pre-shared key (PSK). Secure communication can be achieved today by mixing a strong PSK with the output of an existing key transport algorithm, like RSA [RFC8017], or an existing key agreement algorithm, like Diffie-Hellman [RFC2631] or Elliptic Curve Diffie-Hellman (ECDH) [RFC5753]. A security solution that is believed to be quantum resistant can be achieved by using a PSK with sufficient entropy along with a quantum-resistant key derivation function (KDF), like an HMAC-based key derivation function (HKDF) [RFC5869], and a quantum-resistant encryption algorithm, like 256-bit AES [AES]. In this way, today's CMS-protected communication can be resistant to an attacker with a large-scale quantum computer.¶
In addition, there may be other reasons for including a strong PSK besides protection against the future invention of a large-scale quantum computer. For example, there is always the possibility of a cryptoanalytic breakthrough on one or more classic public key algorithms, and there are longstanding concerns about undisclosed trapdoors in Diffie-Hellman parameters [FGHT2016]. Inclusion of a strong PSK as part of the overall key management offers additional protection against these concerns.¶
Note that the CMS also supports key management techniques based on symmetric key-encryption keys and passwords, but they are not discussed in this document because they are already quantum resistant. The symmetric key-encryption key technique is quantum resistant when used with an adequate key size. The password technique is quantum resistant when used with a quantum-resistant key derivation function and a sufficiently large password.¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
CMS values are generated using ASN.1 [X680], which uses the Basic Encoding Rules (BER) and the Distinguished Encoding Rules (DER) [X690].¶
The major data structures include a version number as the first item in the data structure. The version number is intended to avoid ASN.1 decode errors. Some implementations do not check the version number prior to attempting a decode; then, if a decode error occurs, the version number is checked as part of the error-handling routine. This is a reasonable approach; it places error processing outside of the fast path. This approach is also forgiving when an incorrect version number is used by the sender.¶
Whenever the structure is updated, a higher version number will be assigned. However, to ensure maximum interoperability, the higher version number is only used when the new syntax feature is employed. That is, the lowest version number that supports the generated syntax is used.¶
The CMS enveloped-data content type [RFC5652] and the CMS authenticated-enveloped-data content type [RFC5083] support both key transport and key agreement public key algorithms to establish the key used to encrypt the content. No restrictions are imposed on the key transport or key agreement public key algorithms, which means that any key transport or key agreement algorithm can be used, including algorithms that are specified in the future. In both cases, the sender randomly generates the content-encryption key, and then all recipients obtain that key. All recipients use the sender-generated symmetric content-encryption key for decryption.¶
This specification defines two quantum-resistant ways to establish a symmetric key-encryption key, which is used to encrypt the sender-generated content-encryption key. In both cases, the PSK is used as one of the inputs to a key-derivation function to create a quantum-resistant key-encryption key. The PSK MUST be distributed to the sender and all of the recipients by some out-of-band means that does not make it vulnerable to the future invention of a large-scale quantum computer, and an identifier MUST be assigned to the PSK. It is best if each PSK has a unique identifier; however, if a recipient has more than one PSK with the same identifier, the recipient can try each of them in turn. A PSK is expected to be used with many messages, with a lifetime of weeks or months.¶
The content-encryption key or content-authenticated-encryption key is quantum resistant, and the sender establishes it using these steps:¶
When using a key transport algorithm:¶
When using a key agreement algorithm:¶
As specified in Section 6.2.5 of [RFC5652], recipient information for additional key management techniques is represented in the OtherRecipientInfo type. Two key management techniques are specified in this document, and they are each identified by a unique ASN.1 object identifier.¶
The first key management technique, called "keyTransPSK" (see Section 3), uses a key transport algorithm to transfer the key-derivation key from the sender to the recipient, and then the key-derivation key is mixed with the PSK using a KDF. The output of the KDF is the key-encryption key, which is used for the encryption of the content-encryption key or content-authenticated-encryption key.¶
The second key management technique, called "keyAgreePSK" (see Section 4), uses a key agreement algorithm to establish a pairwise key-encryption key. This pairwise key-encryption key is then mixed with the PSK using a KDF to produce a second pairwise key-encryption key, which is then used to encrypt the content-encryption key or content-authenticated-encryption key.¶
Per-recipient information using keyTransPSK is represented in the KeyTransPSKRecipientInfo type, which is indicated by the id-ori-keyTransPSK object identifier. Each instance of KeyTransPSKRecipientInfo establishes the content-encryption key or content-authenticated-encryption key for one or more recipients that have access to the same PSK.¶
The id-ori-keyTransPSK object identifier is:¶
id-ori OBJECT IDENTIFIER ::= { iso(1) member-body(2) us(840)
rsadsi(113549) pkcs(1) pkcs-9(9) smime(16) 13 }
id-ori-keyTransPSK OBJECT IDENTIFIER ::= { id-ori 1 } ¶
The KeyTransPSKRecipientInfo type is:¶
KeyTransPSKRecipientInfo ::= SEQUENCE {
version CMSVersion, -- always set to 0
pskid PreSharedKeyIdentifier,
kdfAlgorithm KeyDerivationAlgorithmIdentifier,
keyEncryptionAlgorithm KeyEncryptionAlgorithmIdentifier,
ktris KeyTransRecipientInfos,
encryptedKey EncryptedKey }
PreSharedKeyIdentifier ::= OCTET STRING
KeyTransRecipientInfos ::= SEQUENCE OF KeyTransRecipientInfo ¶
The fields of the KeyTransPSKRecipientInfo type have the following meanings:¶
Per-recipient information using keyAgreePSK is represented in the KeyAgreePSKRecipientInfo type, which is indicated by the id-ori-keyAgreePSK object identifier. Each instance of KeyAgreePSKRecipientInfo establishes the content-encryption key or content-authenticated-encryption key for one or more recipients that have access to the same PSK.¶
The id-ori-keyAgreePSK object identifier is:¶
id-ori-keyAgreePSK OBJECT IDENTIFIER ::= { id-ori 2 }
The KeyAgreePSKRecipientInfo type is:
KeyAgreePSKRecipientInfo ::= SEQUENCE {
version CMSVersion, -- always set to 0
pskid PreSharedKeyIdentifier,
originator [0] EXPLICIT OriginatorIdentifierOrKey,
ukm [1] EXPLICIT UserKeyingMaterial OPTIONAL,
kdfAlgorithm KeyDerivationAlgorithmIdentifier,
keyEncryptionAlgorithm KeyEncryptionAlgorithmIdentifier,
recipientEncryptedKeys RecipientEncryptedKeys } ¶
The fields of the KeyAgreePSKRecipientInfo type have the following meanings:¶
Many KDFs internally employ a one-way hash function. When this is the case, the hash function that is used is indirectly indicated by the KeyDerivationAlgorithmIdentifier. HKDF [RFC5869] is one example of a KDF that makes use of a hash function.¶
Other KDFs internally employ an encryption algorithm. When this is the case, the encryption that is used is indirectly indicated by the KeyDerivationAlgorithmIdentifier. For example, AES-128-CMAC can be used for randomness extraction in a KDF as described in [NIST2018].¶
A KDF has several input values. This section describes the conventions for using the KDF to compute the key-encryption key for KeyTransPSKRecipientInfo and KeyAgreePSKRecipientInfo. For simplicity, the terminology used in the HKDF specification [RFC5869] is used here.¶
The KDF inputs are:¶
CMSORIforPSKOtherInfo ::= SEQUENCE {
psk OCTET STRING,
keyMgmtAlgType ENUMERATED {
keyTrans (5),
keyAgree (10) },
keyEncryptionAlgorithm KeyEncryptionAlgorithmIdentifier,
pskLength INTEGER (1..MAX),
kdkLength INTEGER (1..MAX) } ¶
The fields of type CMSORIforPSKOtherInfo have the following meanings:¶
The KDF output is:¶
An acceptable KDF MUST accept IKM, L, and info inputs; an acceptable KDF MAY also accept salt and other inputs. All of these inputs MUST influence the output of the KDF. If the KDF requires a salt or other inputs, then those inputs MUST be provided as parameters of the KeyDerivationAlgorithmIdentifier.¶
This section contains the ASN.1 module for the two key management techniques defined in this document. This module imports types from other ASN.1 modules that are defined in [RFC5912] and [RFC6268].¶
<CODE BEGINS>
CMSORIforPSK-2019
{ iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-9(9)
smime(16) modules(0) id-mod-cms-ori-psk-2019(69) }
DEFINITIONS EXPLICIT TAGS ::=
BEGIN
-- EXPORTS All
IMPORTS
AlgorithmIdentifier{}, KEY-DERIVATION
FROM AlgorithmInformation-2009 -- [RFC5912]
{ iso(1) identified-organization(3) dod(6) internet(1)
security(5) mechanisms(5) pkix(7) id-mod(0)
id-mod-algorithmInformation-02(58) }
OTHER-RECIPIENT, OtherRecipientInfo, CMSVersion,
KeyTransRecipientInfo, OriginatorIdentifierOrKey,
UserKeyingMaterial, RecipientEncryptedKeys, EncryptedKey,
KeyDerivationAlgorithmIdentifier, KeyEncryptionAlgorithmIdentifier
FROM CryptographicMessageSyntax-2010 -- [RFC6268]
{ iso(1) member-body(2) us(840) rsadsi(113549)
pkcs(1) pkcs-9(9) smime(16) modules(0)
id-mod-cms-2009(58) } ;
--
-- OtherRecipientInfo Types (ori-)
--
SupportedOtherRecipInfo OTHER-RECIPIENT ::= {
ori-keyTransPSK |
ori-keyAgreePSK,
... }
--
-- Key Transport with Pre-Shared Key
--
ori-keyTransPSK OTHER-RECIPIENT ::= {
KeyTransPSKRecipientInfo IDENTIFIED BY id-ori-keyTransPSK }
id-ori OBJECT IDENTIFIER ::= { iso(1) member-body(2) us(840)
rsadsi(113549) pkcs(1) pkcs-9(9) smime(16) 13 }
id-ori-keyTransPSK OBJECT IDENTIFIER ::= { id-ori 1 }
KeyTransPSKRecipientInfo ::= SEQUENCE {
version CMSVersion, -- always set to 0
pskid PreSharedKeyIdentifier,
kdfAlgorithm KeyDerivationAlgorithmIdentifier,
keyEncryptionAlgorithm KeyEncryptionAlgorithmIdentifier,
ktris KeyTransRecipientInfos,
encryptedKey EncryptedKey }
PreSharedKeyIdentifier ::= OCTET STRING
KeyTransRecipientInfos ::= SEQUENCE OF KeyTransRecipientInfo
--
-- Key Agreement with Pre-Shared Key
--
ori-keyAgreePSK OTHER-RECIPIENT ::= {
KeyAgreePSKRecipientInfo IDENTIFIED BY id-ori-keyAgreePSK }
id-ori-keyAgreePSK OBJECT IDENTIFIER ::= { id-ori 2 }
KeyAgreePSKRecipientInfo ::= SEQUENCE {
version CMSVersion, -- always set to 0
pskid PreSharedKeyIdentifier,
originator [0] EXPLICIT OriginatorIdentifierOrKey,
ukm [1] EXPLICIT UserKeyingMaterial OPTIONAL,
kdfAlgorithm KeyDerivationAlgorithmIdentifier,
keyEncryptionAlgorithm KeyEncryptionAlgorithmIdentifier,
recipientEncryptedKeys RecipientEncryptedKeys }
--
-- Structure to provide 'info' input to the KDF,
-- including the Pre-Shared Key
--
CMSORIforPSKOtherInfo ::= SEQUENCE {
psk OCTET STRING,
keyMgmtAlgType ENUMERATED {
keyTrans (5),
keyAgree (10) },
keyEncryptionAlgorithm KeyEncryptionAlgorithmIdentifier,
pskLength INTEGER (1..MAX),
kdkLength INTEGER (1..MAX) }
END
<CODE ENDS>¶
The security considerations related to the CMS enveloped-data content type in [RFC5652] and the security considerations related to the CMS authenticated-enveloped-data content type in [RFC5083] continue to apply.¶
Implementations of the key derivation function must compute the entire result, which, in this specification, is a key-encryption key, before outputting any portion of the result. The resulting key-encryption key must be protected. Compromise of the key-encryption key may result in the disclosure of all content-encryption keys or content-authenticated-encryption keys that were protected with that keying material; this, in turn, may result in the disclosure of the content. Note that there are two key-encryption keys when a PSK with a key agreement algorithm is used, with similar consequences for the compromise of either one of these keys.¶
Implementations must protect the PSK, key transport private key, agreement private key, and key-derivation key. Compromise of the PSK will make the encrypted content vulnerable to the future invention of a large-scale quantum computer. Compromise of the PSK and either the key transport private key or the agreement private key may result in the disclosure of all contents protected with that combination of keying material. Compromise of the PSK and the key-derivation key may result in the disclosure of all contents protected with that combination of keying material.¶
A large-scale quantum computer will essentially negate the security provided by the key transport algorithm or the key agreement algorithm, which means that the attacker with a large-scale quantum computer can discover the key-derivation key. In addition, a large-scale quantum computer effectively cuts the security provided by a symmetric key algorithm in half. Therefore, the PSK needs at least 256 bits of entropy to provide 128 bits of security. To match that same level of security, the key derivation function needs to be quantum resistant and produce a key-encryption key that is at least 256 bits in length. Similarly, the content-encryption key or content-authenticated-encryption key needs to be at least 256 bits in length.¶
When using a PSK with a key transport or a key agreement algorithm, a key-encryption key is produced to encrypt the content-encryption key or content-authenticated-encryption key. If the key-encryption algorithm is different than the algorithm used to protect the content, then the effective security is determined by the weaker of the two algorithms. If, for example, content is encrypted with 256-bit AES and the key is wrapped with 128-bit AES, then, at most, 128 bits of protection are provided. Implementers must ensure that the key-encryption algorithm is as strong or stronger than the content-encryption algorithm or content-authenticated-encryption algorithm.¶
The selection of the key-derivation function imposes an upper bound on the strength of the resulting key-encryption key. The strength of the selected key-derivation function should be at least as strong as the key-encryption algorithm that is selected. NIST SP 800-56C Revision 1 [NIST2018] offers advice on the security strength of several popular key-derivation functions.¶
Implementers should not mix quantum-resistant key management algorithms with their non-quantum-resistant counterparts. For example, the same content should not be protected with KeyTransRecipientInfo and KeyTransPSKRecipientInfo. Likewise, the same content should not be protected with KeyAgreeRecipientInfo and KeyAgreePSKRecipientInfo. Doing so would make the content vulnerable to the future invention of a large-scale quantum computer.¶
Implementers should not send the same content in different messages, one using a quantum-resistant key management algorithm and the other using a non-quantum-resistant key management algorithm, even if the content-encryption key is generated independently. Doing so may allow an eavesdropper to correlate the messages, making the content vulnerable to the future invention of a large-scale quantum computer.¶
This specification does not require that PSK be known only by the sender and recipients. The PSK may be known to a group. Since confidentiality depends on the key transport or key agreement algorithm, knowledge of the PSK by other parties does not inherently enable eavesdropping. However, group members can record the traffic of other members and then decrypt it if they ever gain access to a large-scale quantum computer. Also, when many parties know the PSK, there are many opportunities for theft of the PSK by an attacker. Once an attacker has the PSK, they can decrypt stored traffic if they ever gain access to a large-scale quantum computer in the same manner as a legitimate group member.¶
Sound cryptographic key hygiene is to use a key for one and only one purpose. Use of the recipient's public key for both the traditional CMS and the PSK-mixing variation specified in this document would be a violation of this principle; however, there is no known way for an attacker to take advantage of this situation. That said, an application should enforce separation whenever possible. For example, a purpose identifier for use in the X.509 extended key usage certificate extension [RFC5280] could be identified in the future to indicate that a public key should only be used in conjunction with or without a PSK.¶
Implementations must randomly generate key-derivation keys as well as content-encryption keys or content-authenticated-encryption keys. Also, the generation of public/private key pairs for the key transport and key agreement algorithms rely on random numbers. The use of inadequate pseudorandom number generators (PRNGs) to generate cryptographic keys can result in little or no security. An attacker may find it much easier to reproduce the PRNG environment that produced the keys, searching the resulting small set of possibilities, rather than brute-force searching the whole key space. The generation of quality random numbers is difficult. [RFC4086] offers important guidance in this area.¶
Implementers should be aware that cryptographic algorithms become weaker with time. As new cryptanalysis techniques are developed and computing performance improves, the work factor to break a particular cryptographic algorithm will be reduced. Therefore, cryptographic algorithm implementations should be modular, allowing new algorithms to be readily inserted. That is, implementers should be prepared for the set of supported algorithms to change over time.¶
The security properties provided by the mechanisms specified in this document can be validated using formal methods. A ProVerif proof in [H2019] shows that an attacker with a large-scale quantum computer that is capable of breaking the Diffie-Hellman key agreement algorithm cannot disrupt the delivery of the content-encryption key to the recipient and that the attacker cannot learn the content-encryption key from the protocol exchange.¶
An observer can see which parties are using each PSK simply by watching the PSK key identifiers. However, the addition of these key identifiers does not really weaken the privacy situation. When key transport is used, the RecipientIdentifier is always present, and it clearly identifies each recipient to an observer. When key agreement is used, either the IssuerAndSerialNumber or the RecipientKeyIdentifier is always present, and these clearly identify each recipient.¶
One object identifier for the ASN.1 module in Section 6 was assigned in the "SMI Security for S/MIME Module Identifier (1.2.840.113549.1.9.16.0)" registry [IANA]:¶
id-mod-cms-ori-psk-2019 OBJECT IDENTIFIER ::= {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1)
pkcs-9(9) smime(16) mod(0) 69 } ¶
One new entry has been added in the "SMI Security for S/MIME Mail Security (1.2.840.113549.1.9.16)" registry [IANA]:¶
id-ori OBJECT IDENTIFIER ::= { iso(1) member-body(2) us(840)
rsadsi(113549) pkcs(1) pkcs-9(9) smime(16) 13 } ¶
A new registry titled "SMI Security for S/MIME Other Recipient Info Identifiers (1.2.840.113549.1.9.16.13)" has been created.¶
Updates to the new registry are to be made according to the Specification Required policy as defined in [RFC8126]. The expert is expected to ensure that any new values identify additional RecipientInfo structures for use with the CMS. Object identifiers for other purposes should not be assigned in this arc.¶
Two assignments were made in the new "SMI Security for S/MIME Other Recipient Info Identifiers (1.2.840.113549.1.9.16.13)" registry [IANA] with references to this document:¶
id-ori-keyTransPSK OBJECT IDENTIFIER ::= {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1)
pkcs-9(9) smime(16) id-ori(13) 1 }
id-ori-keyAgreePSK OBJECT IDENTIFIER ::= {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1)
pkcs-9(9) smime(16) id-ori(13) 2 } ¶
This example shows the establishment of an AES-256 content-encryption key using:¶
In real-world use, the originator would encrypt the key-derivation key in their own RSA public key as well as the recipient's public key. This is omitted in an attempt to simplify the example.¶
The pre-shared key known to Alice and Bob, in hexadecimal, is:¶
c244cdd11a0d1f39d9b61282770244fb0f6befb91ab7f96cb05213365cf95b15¶
The identifier assigned to the pre-shared key is:¶
ptf-kmc:13614122112¶
Alice obtains Bob's public key:¶
-----BEGIN PUBLIC KEY----- MIIBojANBgkqhkiG9w0BAQEFAAOCAY8AMIIBigKCAYEA3ocW14cxncPJ47fnEjBZ AyfC2lqapL3ET4jvV6C7gGeVrRQxWPDwl+cFYBBR2ej3j3/0ecDmu+XuVi2+s5JH Keeza+itfuhsz3yifgeEpeK8T+SusHhn20/NBLhYKbh3kiAcCgQ56dpDrDvDcLqq vS3jg/VO+OPnZbofoHOOevt8Q/roahJe1PlIyQ4udWB8zZezJ4mLLfbOA9YVaYXx 2AHHZJevo3nmRnlgJXo6mE00E/6qkhjDHKSMdl2WG6mO9TCDZc9qY3cAJDU6Ir0v SH7qUl8/vN13y4UOFkn8hM4kmZ6bJqbZt5NbjHtY4uQ0VMW3RyESzhrO02mrp39a uLNnH3EXdXaV1tk75H3qC7zJaeGWMJyQfOE3YfEGRKn8fxubji716D8UecAxAzFy FL6m1JiOyV5acAiOpxN14qRYZdHnXOM9DqGIGpoeY1UuD4Mo05osOqOUpBJHA9fS whSZG7VNf+vgNWTLNYSYLI04KiMdulnvU6ds+QPz+KKtAgMBAAE= -----END PUBLIC KEY-----¶
Bob's RSA public key has the following key identifier:¶
9eeb67c9b95a74d44d2f16396680e801b5cba49c¶
Alice randomly generates a content-encryption key:¶
c8adc30f4a3e20ac420caa76a68f5787c02ab42afea20d19672fd963a5338e83¶
Alice randomly generates a key-derivation key:¶
df85af9e3cebffde6e9b9d24263db31114d0a8e33a0d50e05eb64578ccde81eb¶
Alice encrypts the key-derivation key in Bob's public key:¶
52693f12140c91dea2b44c0b7936f6be46de8a7bfab072bcb6ecfd56b06a9f65 1bd4669d336aef7b449e5cd9b151893b7c7a3b8e364394840b0a5434cbf10e1b 5670aefd074faf380665d204fb95153543346f36c2125dba6f4d23d2bc61434b 5e36ff72b3eafe57c6cf7f74924c309f174b0b8753554b58ed33a8848d707a98 c0c2b1ddcfd09e31fe213ca0a48dd157bd7d842e85cc76f77710d58efeaa0525 c651bcd1410fb47534ecabaf5ab7daabed809d4b97220caf6d4929c5fb684f7b b8692e6e70332ff9b3f7c11d6cac51d4a35593173d48f80ca843b89789d625e7 997ad7d674d25a2a7d165a5f39b3cb6358e937bdb02ac8a524ac93113cedd9ad c68263025c0bb0997d716e58d4d7b69739bf591f3e71c7678dc0df96f3df9e8a a5738f4f9ce21489f300e040891b20b2ab6d9051b3c2e68efa2fa9799a706878 d5f462018c021d6669ed649f9acdf78476810198bfb8bd41ffedc585eafa957e ea1d3625e4bed376e7ae49718aee2f575c401a26a29941d8da5b7ee9aca36471¶
Alice produces a 256-bit key-encryption key with HKDF using SHA-384; the secret value is the key-derivation key; and the 'info' is the DER-encoded CMSORIforPSKOtherInfo structure with the following values:¶
0 56: SEQUENCE {
2 32: OCTET STRING
: C2 44 CD D1 1A 0D 1F 39 D9 B6 12 82 77 02 44 FB
: 0F 6B EF B9 1A B7 F9 6C B0 52 13 36 5C F9 5B 15
36 1: ENUMERATED 5
39 11: SEQUENCE {
41 9: OBJECT IDENTIFIER aes256-wrap (2 16 840 1 101 3 4 1 45)
: }
52 1: INTEGER 32
55 1: INTEGER 32
: } ¶
The DER encoding of CMSORIforPSKOtherInfo produces 58 octets:¶
30380420c244cdd11a0d1f39d9b61282770244fb0f6befb91ab7f96cb0521336 5cf95b150a0105300b060960864801650304012d020120020120¶
The HKDF output is 256 bits:¶
f319e9cebb35f1c6a7a9709b8760b9d0d3e30e16c5b2b69347e9f00ca540a232¶
Alice uses AES-KEY-WRAP to encrypt the 256-bit content-encryption key with the key-encryption key:¶
ea0947250fa66cd525595e52a69aaade88efcf1b0f108abe291060391b1cdf59 07f36b4067e45342¶
Alice encrypts the content using AES-256-GCM with the content-encryption key. The 12-octet nonce used is:¶
cafebabefacedbaddecaf888¶
The content plaintext is:¶
48656c6c6f2c20776f726c6421¶
The resulting ciphertext is:¶
9af2d16f21547fcefed9b3ef2d¶
The resulting 12-octet authentication tag is:¶
a0e5925cc184e0172463c44c¶
Alice encodes the AuthEnvelopedData and the ContentInfo and sends the result to Bob. The resulting structure is:¶
0 650: SEQUENCE {
4 11: OBJECT IDENTIFIER
: authEnvelopedData (1 2 840 113549 1 9 16 1 23)
17 633: [0] {
21 629: SEQUENCE {
25 1: INTEGER 0
28 551: SET {
32 547: [4] {
36 11: OBJECT IDENTIFIER
: keyTransPSK (1 2 840 113549 1 9 16 13 1)
49 530: SEQUENCE {
53 1: INTEGER 0
56 19: OCTET STRING 'ptf-kmc:13614122112'
77 13: SEQUENCE {
79 11: OBJECT IDENTIFIER
: hkdf-with-sha384 (1 2 840 113549 1 9 16 3 29)
: }
92 11: SEQUENCE {
94 9: OBJECT IDENTIFIER
: aes256-wrap (2 16 840 1 101 3 4 1 45)
: }
105 432: SEQUENCE {
109 428: SEQUENCE {
113 1: INTEGER 2
116 20: [0]
: 9E EB 67 C9 B9 5A 74 D4 4D 2F 16 39 66 80 E8 01
: B5 CB A4 9C
138 13: SEQUENCE {
140 9: OBJECT IDENTIFIER
: rsaEncryption (1 2 840 113549 1 1 1)
151 0: NULL
: }
153 384: OCTET STRING
: 52 69 3F 12 14 0C 91 DE A2 B4 4C 0B 79 36 F6 BE
: 46 DE 8A 7B FA B0 72 BC B6 EC FD 56 B0 6A 9F 65
: 1B D4 66 9D 33 6A EF 7B 44 9E 5C D9 B1 51 89 3B
: 7C 7A 3B 8E 36 43 94 84 0B 0A 54 34 CB F1 0E 1B
: 56 70 AE FD 07 4F AF 38 06 65 D2 04 FB 95 15 35
: 43 34 6F 36 C2 12 5D BA 6F 4D 23 D2 BC 61 43 4B
: 5E 36 FF 72 B3 EA FE 57 C6 CF 7F 74 92 4C 30 9F
: 17 4B 0B 87 53 55 4B 58 ED 33 A8 84 8D 70 7A 98
: C0 C2 B1 DD CF D0 9E 31 FE 21 3C A0 A4 8D D1 57
: BD 7D 84 2E 85 CC 76 F7 77 10 D5 8E FE AA 05 25
: C6 51 BC D1 41 0F B4 75 34 EC AB AF 5A B7 DA AB
: ED 80 9D 4B 97 22 0C AF 6D 49 29 C5 FB 68 4F 7B
: B8 69 2E 6E 70 33 2F F9 B3 F7 C1 1D 6C AC 51 D4
: A3 55 93 17 3D 48 F8 0C A8 43 B8 97 89 D6 25 E7
: 99 7A D7 D6 74 D2 5A 2A 7D 16 5A 5F 39 B3 CB 63
: 58 E9 37 BD B0 2A C8 A5 24 AC 93 11 3C ED D9 AD
: C6 82 63 02 5C 0B B0 99 7D 71 6E 58 D4 D7 B6 97
: 39 BF 59 1F 3E 71 C7 67 8D C0 DF 96 F3 DF 9E 8A
: A5 73 8F 4F 9C E2 14 89 F3 00 E0 40 89 1B 20 B2
: AB 6D 90 51 B3 C2 E6 8E FA 2F A9 79 9A 70 68 78
: D5 F4 62 01 8C 02 1D 66 69 ED 64 9F 9A CD F7 84
: 76 81 01 98 BF B8 BD 41 FF ED C5 85 EA FA 95 7E
: EA 1D 36 25 E4 BE D3 76 E7 AE 49 71 8A EE 2F 57
: 5C 40 1A 26 A2 99 41 D8 DA 5B 7E E9 AC A3 64 71
: }
: }
541 40: OCTET STRING
: EA 09 47 25 0F A6 6C D5 25 59 5E 52 A6 9A AA DE
: 88 EF CF 1B 0F 10 8A BE 29 10 60 39 1B 1C DF 59
: 07 F3 6B 40 67 E4 53 42
: }
: }
: }
583 55: SEQUENCE {
585 9: OBJECT IDENTIFIER data (1 2 840 113549 1 7 1)
596 27: SEQUENCE {
598 9: OBJECT IDENTIFIER
: aes256-GCM (2 16 840 1 101 3 4 1 46)
609 14: SEQUENCE {
611 12: OCTET STRING
: CA FE BA BE FA CE DB AD DE CA F8 88
: }
: }
625 13: [0]
: 9A F2 D1 6F 21 54 7F CE FE D9 B3 EF 2D
: }
640 12: OCTET STRING A0 E5 92 5C C1 84 E0 17 24 63 C4 4C
: }
: }
: } ¶
Bob's private key is:¶
-----BEGIN RSA PRIVATE KEY----- MIIG5AIBAAKCAYEA3ocW14cxncPJ47fnEjBZAyfC2lqapL3ET4jvV6C7gGeVrRQx WPDwl+cFYBBR2ej3j3/0ecDmu+XuVi2+s5JHKeeza+itfuhsz3yifgeEpeK8T+Su sHhn20/NBLhYKbh3kiAcCgQ56dpDrDvDcLqqvS3jg/VO+OPnZbofoHOOevt8Q/ro ahJe1PlIyQ4udWB8zZezJ4mLLfbOA9YVaYXx2AHHZJevo3nmRnlgJXo6mE00E/6q khjDHKSMdl2WG6mO9TCDZc9qY3cAJDU6Ir0vSH7qUl8/vN13y4UOFkn8hM4kmZ6b JqbZt5NbjHtY4uQ0VMW3RyESzhrO02mrp39auLNnH3EXdXaV1tk75H3qC7zJaeGW MJyQfOE3YfEGRKn8fxubji716D8UecAxAzFyFL6m1JiOyV5acAiOpxN14qRYZdHn XOM9DqGIGpoeY1UuD4Mo05osOqOUpBJHA9fSwhSZG7VNf+vgNWTLNYSYLI04KiMd ulnvU6ds+QPz+KKtAgMBAAECggGATFfkSkUjjJCjLvDk4aScpSx6+Rakf2hrdS3x jwqhyUfAXgTTeUQQBs1HVtHCgxQd+qlXYn3/qu8TeZVwG4NPztyi/Z5yB1wOGJEV 3k8N/ytul6pJFFn6p48VM01bUdTrkMJbXERe6g/rr6dBQeeItCaOK7N5SIJH3Oqh 9xYuB5tH4rquCdYLmt17Tx8CaVqU9qPY3vOdQEOwIjjMV8uQUR8rHSO9KkSj8AGs Lq9kcuPpvgJc2oqMRcNePS2WVh8xPFktRLLRazgLP8STHAtjT6SlJ2UzkUqfDHGK q/BoXxBDu6L1VDwdnIS5HXtL54ElcXWsoOyKF8/ilmhRUIUWRZFmlS1ok8IC5IgX UdL9rJVZFTRLyAwmcCEvRM1asbBrhyEyshSOuN5nHJi2WVJ+wSHijeKl1qeLlpMk HrdIYBq4Nz7/zXmiQphpAy+yQeanhP8O4O6C8e7RwKdpxe44su4Z8fEgA5yQx0u7 8yR1EhGKydX5bhBLR5Cm1VM7rT2BAoHBAP/+e5gZLNf/ECtEBZjeiJ0VshszOoUq haUQPA+9Bx9pytsoKm5oQhB7QDaxAvrn8/FUW2aAkaXsaj9F+/q30AYSQtExai9J fdKKook3oimN8/yNRsKmhfjGOj8hd4+GjX0qoMSBCEVdT+bAjjry8wgQrqReuZnu oXU85dmb3jvv0uIczIKvTIeyjXE5afjQIJLmZFXsBm09BG87Ia5EFUKly96BOMJh /QWEzuYYXDqOFfzQtkAefXNFW21Kz4Hw2QKBwQDeiGh4lxCGTjECvG7fauMGlu+q DSdYyMHif6t6mx57eS16EjvOrlXKItYhIyzW8Kw0rf/CSB2j8ig1GkMLTOgrGIJ1 0322o50FOr5oOmZPueeR4pOyAP0fgQ8DD1L3JBpY68/8MhYbsizVrR+Ar4jM0f96 W2bF5Xj3h+fQTDMkx6VrCCQ6miRmBUzH+ZPs5n/lYOzAYrqiKOanaiHy4mjRvlsy mjZ6z5CG8sISqcLQ/k3Qli5pOY/v0rdBjgwAW/UCgcEAqGVYGjKdXCzuDvf9EpV4 mpTWB6yIV2ckaPOn/tZi5BgsmEPwvZYZt0vMbu28Px7sSpkqUuBKbzJ4pcy8uC3I SuYiTAhMiHS4rxIBX3BYXSuDD2RD4vG1+XM0h6jVRHXHh0nOXdVfgnmigPGz3jVJ B8oph/jD8O2YCk4YCTDOXPEi8Rjusxzro+whvRR+kG0gsGGcKSVNCPj1fNISEte4 gJId7O1mUAAzeDjn/VaS/PXQovEMolssPPKn9NocbKbpAoHBAJnFHJunl22W/lrr ppmPnIzjI30YVcYOA5vlqLKyGaAsnfYqP1WUNgfVhq2jRsrHx9cnHQI9Hu442PvI x+c5H30YFJ4ipE3eRRRmAUi4ghY5WgD+1hw8fqyUW7E7l5LbSbGEUVXtrkU5G64T UR91LEyMF8OPATdiV/KD4PWYkgaqRm3tVEuCVACDTQkqNsOOi3YPQcm270w6gxfQ SOEy/kdhCFexJFA8uZvmh6Cp2crczxyBilR/yCxqKOONqlFdOQKBwFbJk5eHPjJz AYueKMQESPGYCrwIqxgZGCxaqeVArHvKsEDx5whI6JWoFYVkFA8F0MyhukoEb/2x 2qB5T88Dg3EbqjTiLg3qxrWJ2OxtUo8pBP2I2wbl2NOwzcbrlYhzEZ8bJyxZu5i1 sYILC8PJ4Qzw6jS4Qpm4y1WHz8e/ElW6VyfmljZYA7f9WMntdfeQVqCVzNTvKn6f hg6GSpJTzp4LV3ougi9nQuWXZF2wInsXkLYpsiMbL6Fz34RwohJtYA== -----END RSA PRIVATE KEY-----¶
Bob decrypts the key-derivation key with his RSA private key:¶
df85af9e3cebffde6e9b9d24263db31114d0a8e33a0d50e05eb64578ccde81eb¶
Bob produces a 256-bit key-encryption key with HKDF using SHA-384; the secret value is the key-derivation key; and the 'info' is the DER-encoded CMSORIforPSKOtherInfo structure with the same values as shown in Appendix A.1. The HKDF output is 256 bits:¶
f319e9cebb35f1c6a7a9709b8760b9d0d3e30e16c5b2b69347e9f00ca540a232¶
Bob uses AES-KEY-WRAP to decrypt the content-encryption key with the key-encryption key; the content-encryption key is:¶
c8adc30f4a3e20ac420caa76a68f5787c02ab42afea20d19672fd963a5338e83¶
Bob decrypts the content using AES-256-GCM with the content-encryption key and checks the received authentication tag. The 12-octet nonce used is:¶
cafebabefacedbaddecaf888¶
The 12-octet authentication tag is:¶
a0e5925cc184e0172463c44c¶
The received ciphertext content is:¶
9af2d16f21547fcefed9b3ef2d¶
The resulting plaintext content is:¶
48656c6c6f2c20776f726c6421¶
This example shows the establishment of an AES-256 content-encryption key using:¶
In real-world use, the originator would treat themselves as an additional recipient by performing key agreement with their own static public key and the ephemeral private key generated for this message. This is omitted in an attempt to simplify the example.¶
The pre-shared key known to Alice and Bob, in hexadecimal, is:¶
4aa53cbf500850dd583a5d9821605c6fa228fb5917f87c1c078660214e2d83e4¶
The identifier assigned to the pre-shared key is:¶
ptf-kmc:216840110121¶
Alice randomly generates a content-encryption key:¶
937b1219a64d57ad81c05cc86075e86017848c824d4e85800c731c5b7b091033¶
Alice obtains Bob's static ECDH public key:¶
-----BEGIN PUBLIC KEY----- MHYwEAYHKoZIzj0CAQYFK4EEACIDYgAEScGPBO9nmUwGrgbGEoFY9HR/bCo0WyeY /dePQVrwZmwN2yMJmO2d1kWCvLTz8U7atinxyIRe9CV54yau1KWU/wbkhPDnzuSM YkcpxMGo32z3JetEloW5aFOja13vv/W5 -----END PUBLIC KEY-----¶
It has a key identifier of:¶
e8218b98b8b7d86b5e9ebdc8aeb8c4ecdc05c529¶
Alice generates an ephemeral ECDH key pair on the same curve:¶
-----BEGIN EC PRIVATE KEY----- MIGkAgEBBDCMiWLG44ik+L8cYVvJrQdLcFA+PwlgRF+Wt1Ab25qUh8OB7OePWjxp /b8P6IOuI6GgBwYFK4EEACKhZANiAAQ5G0EmJk/2ks8sXY1kzbuG3Uu3ttWwQRXA LFDJICjvYfr+yTpOQVkchm88FAh9MEkw4NKctokKNgpsqXyrT3DtOg76oIYENpPb GE5lJdjPx9sBsZQdABwlsU0Zb7P/7i8= -----END EC PRIVATE KEY-----¶
Alice computes a shared secret called "Z" using Bob's static ECDH public key and her ephemeral ECDH private key; Z is:¶
3f015ed0ff4b99523a95157bbe77e9cc0ee52fcffeb7e41eac79d1c11b6cc556 19cf8807e6d800c2de40240fe0e26adc¶
Alice computes the pairwise key-encryption key, called "KEK1", from Z using the X9.63 KDF with the ECC-CMS-SharedInfo structure with the following values:¶
0 21: SEQUENCE {
2 11: SEQUENCE {
4 9: OBJECT IDENTIFIER aes256-wrap (2 16 840 1 101 3 4 1 45)
: }
15 6: [2] {
17 4: OCTET STRING 00 00 00 20
: }
: } ¶
The DER encoding of ECC-CMS-SharedInfo produces 23 octets:¶
3015300b060960864801650304012da206040400000020¶
The X9.63 KDF output is the 256-bit KEK1:¶
27dc25ddb0b425f7a968ceada80a8f73c6ccaab115baafcce4a22a45d6b8f3da¶
Alice produces the 256-bit KEK2 with HKDF using SHA-384; the secret value is KEK1; and the 'info' is the DER-encoded CMSORIforPSKOtherInfo structure with the following values:¶
0 56: SEQUENCE {
2 32: OCTET STRING
: 4A A5 3C BF 50 08 50 DD 58 3A 5D 98 21 60 5C 6F
: A2 28 FB 59 17 F8 7C 1C 07 86 60 21 4E 2D 83 E4
36 1: ENUMERATED 10
39 11: SEQUENCE {
41 9: OBJECT IDENTIFIER aes256-wrap (2 16 840 1 101 3 4 1 45)
: }
52 1: INTEGER 32
55 1: INTEGER 32
: } ¶
The DER encoding of CMSORIforPSKOtherInfo produces 58 octets:¶
303804204aa53cbf500850dd583a5d9821605c6fa228fb5917f87c1c07866021 4e2d83e40a010a300b060960864801650304012d020120020120¶
The HKDF output is the 256-bit KEK2:¶
7de693ee30ae22b5f8f6cd026c2164103f4e1430f1ab135dc1fb98954f9830bb¶
Alice uses AES-KEY-WRAP to encrypt the content-encryption key with the KEK2; the wrapped key is:¶
229fe0b45e40003e7d8244ec1b7e7ffb2c8dca16c36f5737222553a71263a92b de08866a602d63f4¶
Alice encrypts the content using AES-256-GCM with the content-encryption key. The 12-octet nonce used is:¶
dbaddecaf888cafebabeface¶
The plaintext is:¶
48656c6c6f2c20776f726c6421¶
The resulting ciphertext is:¶
fc6d6f823e3ed2d209d0c6ffcf¶
The resulting 12-octet authentication tag is:¶
550260c42e5b29719426c1ff¶
Alice encodes the AuthEnvelopedData and the ContentInfo and sends the result to Bob. The resulting structure is:¶
0 327: SEQUENCE {
4 11: OBJECT IDENTIFIER
: authEnvelopedData (1 2 840 113549 1 9 16 1 23)
17 310: [0] {
21 306: SEQUENCE {
25 1: INTEGER 0
28 229: SET {
31 226: [4] {
34 11: OBJECT IDENTIFIER
: keyAgreePSK (1 2 840 113549 1 9 16 13 2)
47 210: SEQUENCE {
50 1: INTEGER 0
53 20: OCTET STRING 'ptf-kmc:216840110121'
75 85: [0] {
77 83: [1] {
79 19: SEQUENCE {
81 6: OBJECT IDENTIFIER
: ecdhX963KDF-SHA256 (1 3 132 1 11 1)
89 9: OBJECT IDENTIFIER
: aes256-wrap (2 16 840 1 101 3 4 1 45)
: }
100 60: BIT STRING, encapsulates {
103 57: OCTET STRING
: 1B 41 26 26 4F F6 92 CF 2C 5D 8D 64 CD BB 86 DD
: 4B B7 B6 D5 B0 41 15 C0 2C 50 C9 20 28 EF 61 FA
: FE C9 3A 4E 41 59 1C 86 6F 3C 14 08 7D 30 49 30
: E0 D2 9C B6 89 0A 36 0A 6C
: }
: }
: }
162 13: SEQUENCE {
164 11: OBJECT IDENTIFIER
: hkdf-with-sha384 (1 2 840 113549 1 9 16 3 29)
: }
177 11: SEQUENCE {
179 9: OBJECT IDENTIFIER
: aes256-wrap (2 16 840 1 101 3 4 1 45)
: }
190 68: SEQUENCE {
192 66: SEQUENCE {
194 22: [0] {
196 20: OCTET STRING
: E8 21 8B 98 B8 B7 D8 6B 5E 9E BD C8 AE B8 C4 EC
: DC 05 C5 29
: }
218 40: OCTET STRING
: 22 9F E0 B4 5E 40 00 3E 7D 82 44 EC 1B 7E 7F FB
: 2C 8D CA 16 C3 6F 57 37 22 25 53 A7 12 63 A9 2B
: DE 08 86 6A 60 2D 63 F4
: }
: }
: }
: }
: }
260 55: SEQUENCE {
262 9: OBJECT IDENTIFIER data (1 2 840 113549 1 7 1)
273 27: SEQUENCE {
275 9: OBJECT IDENTIFIER
: aes256-GCM (2 16 840 1 101 3 4 1 46)
286 14: SEQUENCE {
288 12: OCTET STRING
: DB AD DE CA F8 88 CA FE BA BE FA CE
: }
: }
302 13: [0]
: FC 6D 6F 82 3E 3E D2 D2 09 D0 C6 FF CF
: }
317 12: OCTET STRING 55 02 60 C4 2E 5B 29 71 94 26 C1 FF
: }
: }
: } ¶
Bob obtains Alice's ephemeral ECDH public key from the message:¶
-----BEGIN PUBLIC KEY----- MHYwEAYHKoZIzj0CAQYFK4EEACIDYgAEORtBJiZP9pLPLF2NZM27ht1Lt7bVsEEV wCxQySAo72H6/sk6TkFZHIZvPBQIfTBJMODSnLaJCjYKbKl8q09w7ToO+qCGBDaT 2xhOZSXYz8fbAbGUHQAcJbFNGW+z/+4v -----END PUBLIC KEY-----¶
Bob's static ECDH private key is:¶
-----BEGIN EC PRIVATE KEY----- MIGkAgEBBDAnJ4hB+tTUN9X03/W0RsrYy+qcptlRSYkhaDIsQYPXfTU0ugjJEmRk NTPj4y1IRjegBwYFK4EEACKhZANiAARJwY8E72eZTAauBsYSgVj0dH9sKjRbJ5j9 149BWvBmbA3bIwmY7Z3WRYK8tPPxTtq2KfHIhF70JXnjJq7UpZT/BuSE8OfO5Ixi RynEwajfbPcl60SWhbloU6NrXe+/9bk= -----END EC PRIVATE KEY-----¶
Bob computes a shared secret called "Z" using Alice's ephemeral ECDH public key and his static ECDH private key; Z is:¶
3f015ed0ff4b99523a95157bbe77e9cc0ee52fcffeb7e41eac79d1c11b6cc556 19cf8807e6d800c2de40240fe0e26adc¶
Bob computes the pairwise key-encryption key, KEK1, from Z using the X9.63 KDF with the ECC-CMS-SharedInfo structure with the values shown in Appendix B.1. The X9.63 KDF output is the 256-bit KEK1:¶
27dc25ddb0b425f7a968ceada80a8f73c6ccaab115baafcce4a22a45d6b8f3da¶
Bob produces the 256-bit KEK2 with HKDF using SHA-384; the secret value is KEK1; and the 'info' is the DER-encoded CMSORIforPSKOtherInfo structure with the values shown in Appendix B.1. The HKDF output is the 256-bit KEK2:¶
7de693ee30ae22b5f8f6cd026c2164103f4e1430f1ab135dc1fb98954f9830bb¶
Bob uses AES-KEY-WRAP to decrypt the content-encryption key with the KEK2; the content-encryption key is:¶
937b1219a64d57ad81c05cc86075e86017848c824d4e85800c731c5b7b091033¶
Bob decrypts the content using AES-256-GCM with the content-encryption key and checks the received authentication tag. The 12-octet nonce used is:¶
dbaddecaf888cafebabeface¶
The 12-octet authentication tag is:¶
550260c42e5b29719426c1ff¶
The received ciphertext content is:¶
fc6d6f823e3ed2d209d0c6ffcf¶
The resulting plaintext content is:¶
48656c6c6f2c20776f726c6421¶
Many thanks to Roman Danyliw, Ben Kaduk, Burt Kaliski, Panos Kampanakis, Jim Schaad, Robert Sparks, Sean Turner, and Daniel Van Geest for their review and insightful comments. They have greatly improved the design, clarity, and implementation guidance.¶