InternetDraft  Key Blinding for Signature Schemes  January 2024 
Denis, et al.  Expires 25 July 2024  [Page] 
 Workgroup:
 WG Working Group
 InternetDraft:
 draftirtfcfrgsignaturekeyblinding05
 Published:
 Intended Status:
 Informational
 Expires:
Key Blinding for Signature Schemes
Abstract
This document describes extensions to existing digital signature schemes for key blinding. The core property of signing with key blinding is that a blinded public key and all signatures produced using the blinded key pair are independent of the unblinded key pair. Moreover, signatures produced using blinded key pairs are indistinguishable from signatures produced using unblinded key pairs. This functionality has a variety of applications, including Tor onion services and privacypreserving airdrop for bootstrapping cryptocurrency systems.¶
About This Document
This note is to be removed before publishing as an RFC.¶
The latest revision of this draft can be found at https://cfrg.github.io/draftirtfcfrgsignaturekeyblinding/draftirtfcfrgsignaturekeyblinding.html. Status information for this document may be found at https://datatracker.ietf.org/doc/draftirtfcfrgsignaturekeyblinding/.¶
Discussion of this document takes place on the CFRG Working Group mailing list (mailto:cfrg@irtf.org), which is archived at https://mailarchive.ietf.org/arch/browse/cfrg/. Subscribe at https://www.ietf.org/mailman/listinfo/cfrg/.¶
Source for this draft and an issue tracker can be found at https://github.com/cfrg/draftirtfcfrgsignaturekeyblinding.¶
Status of This Memo
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This InternetDraft will expire on 25 July 2024.¶
Copyright Notice
Copyright (c) 2024 IETF Trust and the persons identified as the document authors. All rights reserved.¶
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1. Introduction
Digital signature schemes allow a signer to sign a message using a private signing key and produce a digital signature such that anyone can verify the digital signature over the message with the public verification key corresponding to the signing key. Digital signature schemes typically consist of three functions:¶

KeyGen: A function for generating a private signing key
skS
and the corresponding public verification keypkS
.¶ 
Sign(skS, msg): A function for signing an input message
msg
using a private signing keyskS
, producing a digital signaturesig
.¶ 
Verify(pkS, msg, sig): A function for verifying the digital signature
sig
over input messagemsg
against a public verification keypkS
, yielding true if the signature is valid and false otherwise.¶
In some applications, it's useful for a signer to produce digital signatures using the same longterm private signing key such that a verifier cannot link any two signatures to the same signer. In other words, the signature produced is independent of the longterm privatesigning key, and the public verification key for verifying the signature is independent of the longterm public verification key. This type of functionality has a number of practical applications, including, for example, in the Tor onion services protocol [TORDIRECTORY] and privacypreserving airdrop for bootstrapping cryptocurrency systems [AIRDROP]. It is also necessary for a variant of the Privacy Pass issuance protocol [RATELIMITED].¶
One way to accomplish this is by signing with a private key which is a function of the longterm private signing key and a freshly chosen blinding key, and similarly by producing a public verification key which is a function of the longterm public verification key and same blinding key. A signature scheme with this functionality is referred to as signing with key blinding.¶
A signature scheme with key blinding aims to achieve unforgeability and unlinkability. Informally, unforgeability means that one cannot produce a valid (message, signature) pair for any blinding key without access to the private signing key. Similarly, unlinkability means that one cannot distinguish between two signatures produced from two separate key signing keys, and two signatures produced from the same signing key but with different blinding keys.¶
This document describes extensions to EdDSA [RFC8032] and ECDSA [ECDSA] to enable signing with key blinding. Security analysis of these extensions is currently underway; see Section 8 for more details.¶
This functionality is also possible with other signature schemes, including some postquantum signature schemes [ESS21], though such extensions are not specified here.¶
1.1. DISCLAIMER
This document is a work in progress and is still undergoing security analysis. As such, it MUST NOT be used for real world applications. See Section 8 for additional information.¶
2. Conventions and Definitions
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.¶
The following terms are used throughout this document to describe the blinding modification.¶

G
: The standard base point.¶ 
sk
: A signature scheme private key. For EdDSA, this is a a randomly generated private seed of length 32 bytes or 57 bytes according to [RFC8032], Section 5.1.5 or [RFC8032], Section 5.2.5, respectively. For [ECDSA],sk
is a random scalar in the primeorder elliptic curve group.¶ 
pk(sk)
: The public key corresponding to the private keysk
.¶ 
concat(x0, ..., xN)
: Concatenation of byte strings.concat(0x01, 0x0203, 0x040506) = 0x010203040506
.¶ 
ScalarMult(pk, k): Multiply the public key pk by scalar k, producing a new public key as a result.¶

ModInverse(x, L): Compute the multiplicative inverse of x modulo L.¶
In pseudocode descriptions below, integer multiplication of two scalar values is denoted
by the * operator. For example, the product of two scalars x
and y
is denoted as x * y
.¶
3. Key Blinding
At a high level, a signature scheme with key blinding allows signers to blind their private signing key such that any signature produced with a private signing key and blinding key is independent of the private signing key. Similar to the signing key, the blinding key is also a private key. For example, the blind is a 32byte or 57byte random seed for Ed25519 or Ed448 variants, respectively, whereas the blind for ECDSA over P256 is a random value in the scalar field for the P256 elliptic curve group.¶
In more detail, consider first the basic digital signature syntax, which is a combination of the following functionalities:¶

KeyGen: A function for generating a private and public key pair
(skS, pkS)
.¶ 
Sign(skS, msg): A function for signing a message
msg
with the given private keyskS
, producing a signaturesig
.¶ 
Verify(pkS, msg, sig): A function for verifying a signature
sig
over messagemsg
against the public keypkS
, which returns 1 upon success and 0 otherwise.¶
Key blinding introduces three new functionalities for the signature scheme syntax:¶

BlindKeyGen: A function for generating a private blind key.¶

BlindPublicKey(pkS, bk, ctx): Blind the public verification key
pkS
using the private blinding keybk
and contextctx
, yielding a blinded public keypkR
.¶ 
BlindKeySign(skS, bk, ctx, msg): Sign a message
msg
using the private signing keyskS
with the private blind keybk
and contextctx
.¶
For a given bk
produced from BlindKeyGen, key pair (skS, pkS)
produced from
KeyGen, a context value ctx
, and message msg
, correctness requires the following
equivalence to hold with overwhelming probability:¶
Verify(BlindKeySign(skS, bk, ctx), msg, BlindPublicKey(pkS, bk, ctx)) = 1¶
Security requires that signatures produced using BlindKeySign are unlinkable from signatures produced using the standard signature generation function with the same private key.¶
When the context value is known, a signature scheme with key blinding may also support the ability to unblind public keys. This is represented with the following function.¶

UnblindPublicKey(pkR, bk, ctx): Unblind the public verification key
pkR
using the private blinding keybk
and contextctx
.¶
For a given bk
produced from BlindKeyGen, (skS, pkS)
produced from KeyGen, and context
value ctx
, correctness of this function requires the following equivalence to hold:¶
UnblindPublicKey(BlindPublicKey(pkS, bk, ctx), bk, ctx) = pkS¶
Considerations for choosing context strings are discussed in Section 7.¶
4. Ed25519ph, Ed25519ctx, and Ed25519
This section describes implementations of BlindPublicKey, UnblindPublicKey, and BlindKeySign as
modifications of routines in [RFC8032], Section 5.1. BlindKeyGen invokes the key generation
routine specified in [RFC8032], Section 5.1.5 and outputs only the private key. This section
assumes a context value ctx
has been configured or otherwise chosen by the application.¶
4.1. BlindPublicKey and UnblindPublicKey
BlindPublicKey transforms a private blind bk into a scalar for the edwards25519 group and then multiplies the target key by this scalar. UnblindPublicKey performs essentially the same steps except that it multiplies the target public key by the multiplicative inverse of the scalar, where the inverse is computed using the order of the group L, described in [RFC8032], Section 5.1.¶
More specifically, BlindPublicKey(pk, bk, ctx) works as follows.¶

Construct the blind_ctx as concat(bk, 0x00, ctx), where bk is a 32byte octet string, hash the result using SHA512(blind_ctx), and store the digest in a 64octet large buffer, denoted b. Interpret the lower 32 bytes buffer as a littleendian integer, forming a secret scalar s. Note that this explicitly skips the buffer pruning step in [RFC8032], Section 5.1.¶

Perform a scalar multiplication ScalarMult(pk, s), and output the encoding of the resulting point as the public key.¶
UnblindPublicKey(pkR, bk, ctx) works as follows.¶

Compute the secret scalar s from bk and ctx as in BlindPublicKey.¶

Compute the sInv = ModInverse(s, L), where L is as defined in [RFC8032], Section 5.1.¶

Perform a scalar multiplication ScalarMult(pk, sInv), and output the encoding of the resulting point as the public key.¶
4.2. BlindKeySign
BlindKeySign transforms a private key bk into a scalar for the edwards25519 group and a message prefix to blind both the signing scalar and the prefix of the message used in the signature generation routine.¶
More specifically, BlindKeySign(skS, bk, msg) works as follows:¶

Hash the private key skS, 32 octets, using SHA512. Let h denote the resulting digest. Construct the secret scalar s1 from the first half of the digest, and the corresponding public key A1, as described in [RFC8032], Section 5.1.5. Let prefix1 denote the second half of the hash digest, h[32],...,h[63].¶

Construct the blind_ctx as concat(bk, 0x00, ctx), where bk is a 32byte octet string, hash the result using SHA512(blind_ctx), and store the digest in a 64octet large buffer, denoted b. Interpret the lower 32 bytes buffer as a littleendian integer, forming a secret scalar s2. Note that this explicitly skips the buffer pruning step in [RFC8032], Section 5.1.5. Let prefix2 denote the second half of the hash digest, b[32],...,b[63].¶

Compute the signing scalar s = s1 * s2 (mod L) and the signing public key A = ScalarMult(G, s).¶

Compute the signing prefix as concat(prefix1, prefix2).¶

Run the rest of the Sign procedure in [RFC8032], Section 5.1.6 from step (2) onwards using the modified scalar s, public key A, and string prefix.¶
5. Ed448ph and Ed448
This section describes implementations of BlindPublicKey, UnblindPublicKey, and BlindKeySign as
modifications of routines in [RFC8032], Section 5.2. BlindKeyGen invokes the key generation
routine specified in [RFC8032], Section 5.1.5 and outputs only the private key. This section
assumes a context value ctx
has been configured or otherwise chosen by the application.¶
5.1. BlindPublicKey and UnblindPublicKey
BlindPublicKey and UnblindPublicKey for Ed448ph and Ed448 are implemented just as these routines are for Ed25519ph, Ed25519ctx, and Ed25519, except that SHAKE256 is used instead of SHA512 for hashing the secret blind context, i.e., the concatenation of blind key bk and context ctx, to a 114byte buffer (and using the lower 57bytes for the secret), and the order of the edwards448 group L is as defined in [RFC8032], Section 5.2.1. Note that this process explicitly skips the buffer pruning step in [RFC8032], Section 5.2.5.¶
5.2. BlindKeySign
BlindKeySign for Ed448ph and Ed448 is implemented just as this routine for Ed25519ph, Ed25519ctx, and Ed25519, except in how the scalars (s1, s2), public keys (A1, A2), and message strings (prefix1, prefix2) are computed. More specifically, BlindKeySign(skS, bk, msg) works as follows:¶

Hash the private key skS, 57 octets, using SHAKE256(skS, 117). Let h1 denote the resulting digest. Construct the secret scalar s1 from the first half of h1, and the corresponding public key A1, as described in [RFC8032], Section 5.2.5. Let prefix1 denote the second half of the hash digest, h1[57],...,h1[113].¶

Construct the blind_ctx as concat(bk, 0x00, ctx), where bk is a 57byte octet string, hash the result using SHAKE256(blind_ctx, 117), and store the digest in a 117octet digest, denoted h2. Interpret the lower 57 bytes buffer as a littleendian integer, forming a secret scalar s2. Note that this explicitly skips the buffer pruning step in [RFC8032], Section 5.2. Let prefix2 denote the second half of the hash digest, h2[57],...,h2[113].¶

Compute the signing scalar s = s1 * s2 (mod L) and the signing public key A = ScalarMult(A1, s2).¶

Compute the signing prefix as concat(prefix1, prefix2).¶

Run the rest of the Sign procedure in [RFC8032], Section 5.2.6 from step (2) onwards using the modified scalar s, public key A, and string prefix.¶
6. ECDSA
[[DISCLAIMER: Multiplicative blinding for ECDSA is known to be NOT be SUFCMAsecure in the presence of an adversary that controls the blinding value. [MSMHI15] describes this in the context of relatedkey attacks. This variant may likely be removed in followup versions of this document based on further analysis.]]¶
This section describes implementations of BlindPublicKey, UnblindPublicKey, and BlindKeySign as functions implemented on top of an existing [ECDSA] implementation. BlindKeyGen invokes the key generation routine specified in [ECDSA] and outputs only the private key. In the descriptions below, let p be the order of the corresponding elliptic curve group used for ECDSA. For example, for P256, p = 115792089210356248762697446949407573529996955224135760342422259061068512044369.¶
This section assumes a context value ctx
has been configured or otherwise chosen by the application.¶
6.1. BlindPublicKey and UnblindPublicKey
BlindPublicKey multiplies the public key pkS by an augmented private key bk yielding a new public key pkR. UnblindPublicKey inverts this process by multiplying the input public key by the multiplicative inverse of the augmented bk. Augmentation here maps the private key bk to another scalar using hash_to_field as defined in Section 5 of [H2C], with DST set to "ECDSA Key Blind", L set to the value corresponding to the target curve, e.g., 48 for P256 and 72 for P384, expand_message_xmd with a hash function matching that used for the corresponding digital signature algorithm, and prime modulus equal to the order p of the corresponding curve. Letting HashToScalar denote this augmentation process, and blind_ctx = concat(bk, 0x00, ctx), BlindPublicKey and UnblindPublicKey are then implemented as follows:¶
BlindPublicKey(pk, bk, ctx) = ScalarMult(pk, HashToScalar(blind_ctx)) UnblindPublicKey(pkR, bk, ctx) = ScalarMult(pkR, ModInverse(HashToScalar(blind_ctx), p))¶
6.2. BlindKeySign
BlindKeySign transforms the signing key skS by the private key bk along with context ctx into a new signing key, skR, and then invokes the existing ECDSA signing procedure. More specifically, skR = skS * HashToScalar(blind_ctx) (mod p), where blind_ctx = concat(bk, 0x00, ctx).¶
7. Application Considerations
Choice of the context string ctx
is applicationspecific. For example, in Tor [TORDIRECTORY],
the context string is set to the concatenation of the longterm signer public key and an
integer epoch. This makes it so that unblinding a blinded public key requires knowledge of
the longterm public key as well as the blinding key. Similarly, in a ratelimited version
of Privacy Pass [RATELIMITED], the context is empty, thereby allowing unblinding by anyone
in possession of the blinding key.¶
Applications are RECOMMENDED to choose context strings that are distinct from other protocols as a way of enforcing domain separation. See Section 2.2.5 of [HASHTOCURVE] for additional discussion around the construction of suitable domain separation values.¶
8. Security Considerations
The signature scheme extensions in this document aim to achieve unforgeability and unlinkability. Informally, unforgeability means that one cannot produce a valid (message, signature) pair for any blinding key without access to the private signing key. Similarly, unlinkability means that one cannot distinguish between two signatures produced from two independent key signing keys, and two signatures produced from the same signing key but with different blinds. Security analysis of the extensions in this document with respect to these two properties is currently underway.¶
Preliminary analysis has been done for a variant of these extensions used for identity key blinding routine used in Tor's Hidden Service feature [TORBLINDING]. For EdDSA, further analysis is needed to ensure this is compliant with the signature algorithm described in [RFC8032].¶
The constructions in this document assume that both the signing and blinding keys are private, and, as such, not controlled by an attacker. [MSMHI15] demonstrate that ECDSA with attackercontrolled multiplicative blinding for producing related keys can be abused to produce forgeries. In particular, if an attacker can control the private blinding key used in BlindKeySign, they can construct a forgery over a different message that validates under a different public key. One mitigation to this problem is to change BlindKeySign such that the signature is computed over the input message as well as the blind public key. However, this would require verifiers to treat both the blind public key and message as input to their verification interface. The construction in Section 6 does not require this change. However, further analysis is needed to determine whether or not this construction is safe.¶
9. IANA Considerations
This document has no IANA actions.¶
10. Test Vectors
This section contains test vectors for a subset of the signature schemes covered in this document.¶
10.1. Ed25519 Test Vectors
This section contains test vectors for Ed25519 as described in [RFC8032]. Each test vector lists the private key and blind seeds, denoted skS and bk and encoded as hexadecimal strings, along with the public key pkS corresponding to skS encoded has hexadecimal strings according to [RFC8032], Section 5.1.2. Each test vector also includes the blinded public key pkR computed from skS and bk, denoted pkR and encoded has a hexadecimal string. Finally, each vector includes the message and signature values, each encoded as hexadecimal strings.¶
// Randomly generated private key and blind seed, empty context skS: d142b3b1d532b0a516353a0746a6d43a86cee8efaf6b14ae85c2199072f47d93 pkS: cd875d3f46a8e8742cf4a6a9f9645d4153a394a5a0a8028c9041cd455d093cd5 bk: bb58c768d9b16571f553efd48207e64391e16439b79fe9409e70b38040c81302 pkR: 666443ce8f03fa09240db73a584efad5462ffe346b14fd78fb666b25db29902f message: 68656c6c6f20776f726c64 context: signature: 5458111c708ce05cb0a1608b08dc649937dc22cf1da045eb866f2face50be 930e79b44d57e5215a82ac227bdccccca52bfe509b96efe8e723cb42b5f14be5f0e¶
// Randomly generated private key seed and zero blind seed, empty context skS: aa69e9cb50abf39b05ebc823242c4fd13ccadd0dadc1b45f6fcbf7be4f30db5d pkS: 5c9a9e271f204c931646aa079e2e66f0783ab3d29946eff37bd3b569e9c8e009 bk: 0000000000000000000000000000000000000000000000000000000000000000 pkR: 23eb5eccb9448ee8403c36595ccfd5edd7257ae70da69aa22282a0a7cd97e443 message: 68656c6c6f20776f726c64 context: signature: 4e9f3ad2b14cf2f9bbf4b88a8832358a568bd69368b471dfabac594e8a8b3 3ab54978ecf902560ed754f011186c4c4dda65d158b96c1e6b99a8e150a26e51e03¶
// Randomly generated private key and blind seed, nonempty context skS: d1e5a0f806eb3c491566cef6d2d195e6bbf0a54c9de0e291a7ced050c63ea91c pkS: 8b37c949d39cddf4d2a0fc0da781ea7f85c7bfbdfeb94a3c9ecb5e8a3c24d65f bk: 05b235297dff87c492835d562c6e03c0f36b9c306f2dcb3b5038c2744d4e8a70 pkR: 019b0a06107e01361facdad39ec16a9647c86c0086bc38825eb664b97d9c514d message: 68656c6c6f20776f726c64 context: d6bbaa0646f5617d3cbd1e22ef05e714d1ec7812efff793999667648b2cc54bc signature: f54214acb3c695c46b1e7aa2da947273cb19ec33d8215dde0f43a8f7250fe bb508f4a5007e3c96be6402074ec843d40358a281ff969c66c1724016208650dd09¶
// Randomly generated private key seed and zero blind seed, nonempty context skS: 89e3e3acef6a6c2d9b7c062199bf996f9ae96b662c73e2b445636f9f22d5012e pkS: 3f667a2305a8baf328a1d8e9ed726f278229607d28fb32d9933da7379947ac44 bk: 0000000000000000000000000000000000000000000000000000000000000000 pkR: 90a543dd29c6e6cd08ef85c43618f2d314139db5baed802383cf674310294e40 message: 68656c6c6f20776f726c64 context: 802def4d21c7c7d0fa4b48af5e85f8ebfc4119a04117c14d961567eaef2859f2 signature: ce305a0f40a3270a84d2d9403617cdb89b7b4edf779b4de27f9acaadf1716 84b162e752c95f17b16aaca7c2662e69ba9696bdd230a107ecab973886e8d5bf00e¶
10.2. ECDSA(P384, SHA384) Test Vectors
This section contains test vectors for ECDSA with P384 and SHA384, as described in [ECDSA]. Each test vector lists the signing and blinding keys, denoted skS and bk, each serialized as a bigendian integers and encoded as hexadecimal strings. Each test vector also blinded public key pkR, encoded as compressed elliptic curve points according to [ECDSA]. Finally, each vector lists message and signature values, where the message is encoded as a hexadecimal string, and the signature value is serialized as the concatenation of scalars (r, s) and encoded as a hexadecimal string.¶
// Randomly generated signing and blind private keys, empty context skS: fcc8217ec4c89862d069a6679026c8042a74a513ba5b4a63da58488643132afaf35 9c3645dcc99c11862d9606370b9b7 pkS: 02582e4108018f9657f8bb55192838ff057442c8f7dc265f195dc1e4aa2cff2ec10 e2f2220dbeb300125d46b00dff747f1 bk: 1d3b48eec849b9d0e7376be1eca90369663939d140a8f3418ebc2221159402647a9e 283a78694377915b2894bc38cfe5 pkR: 03031c9914e4aa550605ded5c8b2604a2910c7c4d7e1e8608d81152a2ed3b8eb85a c8c7896107c91875090b651f43d2f31 message: 68656c6c6f20776f726c64 context: signature: 0ca279fba24a47ef2dded3f3171f805779d41ff0c3b13af260977d26f9df8 a0993591b34e84f954149a478408abc685cb88ca32e482ffb9ea2f377ac949cb37468f18 4b8f03ce4c7da06c024a38e3d8f2a9eea84493288627a13f317cc6d8457¶
// Randomly generated signing and blind private keys, nonempty context skS: 5f9ed9f16ac74cb510689321cbd6a0a9602f50a96cb17ff479ec46fff130afcd9fe d3766c6d98fe4b4f1c2fa275f58ed pkS: 03e690b68b39c0bfb0be6a7f7f0ab49a930437b427dbf588c7acbf3fc8e3e221c83 03e2d38c7bfe735d2d8afaecfacec8c bk: 7c65bba8e98f1f75eb9748ccc4a85b7d5d9523522d02909958e0e2fc81693dbb4d10 460355eec3a3af54184ced97697a pkR: 0280a5180793a1c8155face304fea93783514124cdf7f0fedab11da05289e192da3 6a9f0e3ab4544d75f8eaa8ef9987554 message: 68656c6c6f20776f726c64 context: 327a0a52fa1c01d376cfc259925555920d89f15b509bb84e7385ff7207dcb93d signature: 240e49a4dc681e3cedb241f2cf97f7c86f215902c03e38838e1d23d127c61 debca8af590ebb0fd7f1dd58a51a63aa45e5991fda32da0e7e9bb56b9374be6fed60c672 2de2689f6a969af5c78b78e5dcc353d8a47a71f337586f737b020e541c1¶
11. References
11.1. Normative References
 [ECDSA]
 American National Standards Institute, "Public Key Cryptography for the Financial Services Industry  The Elliptic Curve Digital Signature Algorithm (ECDSA)", ANSI ANS X9.622005, .
 [HASHTOCURVE]
 FazHernandez, A., Scott, S., Sullivan, N., Wahby, R. S., and C. A. Wood, "Hashing to Elliptic Curves", Work in Progress, InternetDraft, draftirtfcfrghashtocurve16, , <https://datatracker.ietf.org/doc/html/draftirtfcfrghashtocurve16>.
 [RFC2119]
 Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfceditor.org/rfc/rfc2119>.
 [RFC8032]
 Josefsson, S. and I. Liusvaara, "EdwardsCurve Digital Signature Algorithm (EdDSA)", RFC 8032, DOI 10.17487/RFC8032, , <https://www.rfceditor.org/rfc/rfc8032>.
 [RFC8174]
 Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfceditor.org/rfc/rfc8174>.
11.2. Informative References
 [AIRDROP]
 Wahby, R. S., Boneh, D., Jeffrey, C., and J. Poon, "An airdrop that preserves recipient privacy", n.d., <https://eprint.iacr.org/2020/676.pdf>.
 [ESS21]
 Eaton, E., Stebila, D., and R. Stracovsky, "PostQuantum KeyBlinding for Authentication in Anonymity Networks", , <https://eprint.iacr.org/2021/963>.
 [H2C]
 FazHernandez, A., Scott, S., Sullivan, N., Wahby, R. S., and C. A. Wood, "Hashing to Elliptic Curves", Work in Progress, InternetDraft, draftirtfcfrghashtocurve16, , <https://datatracker.ietf.org/doc/html/draftirtfcfrghashtocurve16>.
 [MSMHI15]
 Morita, H., Schuldt, J., Matsuda, T., Hanaoka, G., and T. Iwata, "On the Security of the Schnorr Signature Scheme and DSA Against RelatedKey Attacks", Springer International Publishing, Information Security and Cryptology  ICISC 2015 pp. 2035, DOI 10.1007/9783319308401_2, ISBN ["9783319308395", "9783319308401"], , <https://doi.org/10.1007/9783319308401_2>.
 [RATELIMITED]
 Hendrickson, S., Iyengar, J., Pauly, T., Valdez, S., and C. A. Wood, "RateLimited Token Issuance Protocol", Work in Progress, InternetDraft, draftprivacypassratelimittokens03, , <https://datatracker.ietf.org/doc/html/draftprivacypassratelimittokens03>.
 [TORBLINDING]
 Hopper, N., "Proving Security of Tor’s Hidden Service Identity Blinding Protocol", , <https://wwwusers.cse.umn.edu/~hoppernj/basicproof.pdf>.
 [TORDIRECTORY]
 "Tor directory protocol, version 3", n.d., <https://gitweb.torproject.org/torspec.git/tree/rendspecv3.txt>.
Acknowledgments
The authors would like to thank Dennis Jackson for helpful discussions that informed the development of this draft.¶