InternetDraft  Key Blinding for Signature Schemes  May 2022 
Denis, et al.  Expires 4 November 2022  [Page] 
 Workgroup:
 WG Working Group
 InternetDraft:
 draftdewcfrgsignaturekeyblinding02
 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://chriswood.github.io/draftdewcfrgsignaturekeyblinding/draftdewcfrgsignaturekeyblinding.html. Status information for this document may be found at https://datatracker.ietf.org/doc/draftdewcfrgsignaturekeyblinding/.¶
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/.¶
Source for this draft and an issue tracker can be found at https://github.com/chriswood/draftdewcfrgsignaturekeyblinding.¶
Status of This Memo
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Copyright Notice
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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 extends a basic digital scheme with four new functions:¶
 BlindKeyGen: A function for generating a private blind key.¶
 BlindPublicKey(pkS, bk): Blind the public verification key
pkS
using the private blinding keybk
, yielding a blinded public keypkR
.¶  UnblindPublicKey(pkR, bk): Unblind the public verification key
pkR
using the private blinding keybk
.¶  BlindKeySign(skS, bk, msg): Sign a message
msg
using the private signing keyskS
with the private blind keybk
.¶
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 7 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 7 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 that remains secret. 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 scalar in the P256 group. Key blinding introduces four new functionalities for the signature scheme:¶
 BlindKeyGen: A function for generating a private blind key.¶
 BlindPublicKey(pkS, bk): Blind the public verification key
pkS
using the private blinding keybk
, yielding a blinded public keypkR
.¶  UnblindPublicKey(pkR, bk): Unblind the public verification key
pkR
using the private blinding keybk
.¶  BlindKeySign(skS, bk, msg): Sign a message
msg
using the private signing keyskS
with the private blind keybk
.¶
For a given bk
produced from BlindKeyGen, correctness requires the following equivalence to hold:¶
UnblindPublicKey(BlindPublicKey(pkS, bk), bk) = pkS¶
Security requires that signatures produced using BlindKeySign are unlinkable from signatures produced using the standard signature generation function with the same private key.¶
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.¶
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) works as follows.¶
 Hash the 32byte private key bk using SHA512, storing 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) works as follows.¶
 Compute the secret scalar s from bk 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].¶
 Hash the 32byte private key bk using SHA512, storing the digest in a 64octet large buffer, denoted b. Interpret the lower 32 bytes buffer as a littleendian integer, forming a secret scalar s2. 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.¶
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 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.¶
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 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 5.2.5. Let prefix1 denote the second half of the hash digest, h[57],...,h[113].¶
 Perform the same routine to transform the secret blind bk into a secret scalar s2, public key A2, and prefix2.¶
 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.¶
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, BlindPublicKey and UnblindPublicKey are then implemented as follows:¶
BlindPublicKey(pk, bk) = ScalarMult(pk, HashToScalar(bk)) UnblindPublicKey(pk, bk) = ScalarMult(pk, ModInverse(HashToScalar(bk), p))¶
6.2. BlindKeySign
BlindKeySign transforms the signing key skS by the private key bk into a new signing key, skR, and then invokes the existing ECDSA signing procedure. More specifically, skR = skS * HashToScalar(bk) (mod p).¶
7. 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 separate 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.¶
8. IANA Considerations
This document has no IANA actions.¶
9. Test Vectors
This section contains test vectors for a subset of the signature schemes covered in this document.¶
9.1. 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 skS: 875532ab039b0a154161c284e19c74afa28d5bf5454e99284bbcffaa71eebf45 pkS: 3b5983605b277cd44918410eb246bb52d83adfc806ccaa91a60b5b2011bc5973 bk: c461e8595f0ac41d374f878613206704978115a226f60470ffd566e9e6ae73bf pkR: e52bbb204e72a816854ac82c7e244e13a8fcc3217cfdeb90c8a5a927e741a20f message: 68656c6c6f20776f726c64 signature: f35d2027f14250c07b3b353359362ec31e13076a547c749a981d0135fce06 7a361ad6522849e6ed9f61d93b0f76428129b9eb3f9c3cd0bfa1bc2a086a5eebd09¶
// Randomly generated private key seed and zero blind seed skS: f3348942e77a83943a6330d372e7531bb52203c2163a728038388ea110d1c871 pkS: ada4f42be4b8fa93ddc7b41ca434239a940b4b18d314fe04d5be0b317a861ddf bk: 0000000000000000000000000000000000000000000000000000000000000000 pkR: 7b8dcabbdfce4f8ad57f38f014abc4a51ac051a4b77b345da45ee2725d9327d0 message: 68656c6c6f20776f726c64 signature: b38b9d67cb4182e91a86b2eb0591e04c10471c1866202dd1b3b076fb86a61 c7c4ab5d626e5c5d547a584ca85d44839c13f6c976ece0dcba53d82601e6737a400¶
9.2. 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 skS: 0e1e4fcc2726e36c5a24be3d30dc6f52d61e6614f5c57a1ec7b829d8adb7c85f456 c30c652d9cd1653cef4ce4da9008d pkS: 03c66e61f5e12c35568928d9a0ffbc145ee9679e17afea3fba899ed3f878f9e82a8 859ce784d9ff43fea2bc8e726468dd3 bk: 865b6b7fc146d0f488854932c93128c3ab3572b7137c4682cb28a2d55f7598df467 e890984a687b22c8bc60a986f6a28 pkR: 038defb9b698b91ee7f3985e54b57b519be237ced2f6f79408558ff7485bf2d60a2 4dc986b9145e422ea765b56de7c5956 message: 68656c6c6f20776f726c64 signature: 5e5643a8c22b274ec5f776e63ed23ff182c8c87642e35bd5a5f7455ae1a19 a9956795df33e2f8b30150904ef6ba5e7ee4f18cef026f594b4d21fc157552ce3cf6d7ef c3226b8d8194fc93df1c7f5facafc96daab7c5a0d840fbd3b9342f2ddad¶
10. References
10.1. 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, .
 [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>.
10.2. 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, draftirtfcfrghashtocurve14, , <https://datatracker.ietf.org/doc/html/draftirtfcfrghashtocurve14>.
 [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", Information Security and Cryptology  ICISC 2015 pp. 2035, DOI 10.1007/9783319308401_2, , <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, draftprivacypassratelimittokens02, , <https://datatracker.ietf.org/doc/html/draftprivacypassratelimittokens02>.
 [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/dirspec.txt>.
Acknowledgments
The authors would like to thank Dennis Jackson for helpful discussions that informed the development of this draft.¶