Internet-Draft RSA Blind Signatures March 2021
Denis, et al. Expires 9 September 2021 [Page]
Workgroup:
Network Working Group
Internet-Draft:
draft-wood-cfrg-rsa-blind-signatures-00
Published:
Intended Status:
Informational
Expires:
Authors:
F. Denis
Fastly Inc.
F. Jacobs
Apple Inc.
C.A. Wood
Cloudflare

RSA Blind Signatures

Abstract

This document specifies the RSA-based blind signature scheme with appendix (RSA-BSSA). RSA blind signatures were first introduced by Chaum for untraceable payments [Chaum83]. It extends RSA-PSS encoding specified in [RFC8017] to enable blind signature support.

Discussion Venues

This note is to be removed before publishing as an RFC.

Source for this draft and an issue tracker can be found at https://github.com/chris-wood/draft-wood-cfrg-blind-signatures.

Status of This Memo

This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.

Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.

Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress."

This Internet-Draft will expire on 9 September 2021.

Table of Contents

1. Introduction

Originally introduced in the context of digital cash systems by Chaum for untraceable payments [Chaum83], RSA blind signatures turned out to have a wide range of applications ranging from electric voting schemes to authentication mechanisms.

Recently, interest in blind signatures has grown to address operational shortcomings from VOPRFs such as [I-D.irtf-cfrg-voprf]. Specifically, VOPRF evaluation requires access to the private key, and is therefore required for both issuance and redemption of tokens in anonymous authentication protocols such as Privacy Pass [I-D.davidson-pp-protocol]. This limitation complicates deployments where it is not desirable to distribute secret keys entities performing token verification. Additionally, if the private key is kept in a Hardware Security Module, the number of operations on the key are doubled compared to a scheme where the private key is only required for issuance of the tokens.

In contrast, cryptographic signatures provide a primitive that is publicly verifiable and does not require access to the private key for verification. Moreover, [JKK14] shows that one can realize a VOPRF in the Random Oracle Model by hashing a (deterministic) blind signature-message pair.

This document specifies the RSA Blind Signature Scheme with Appendix (RSABSSA). In order to facilitate deployment, we define it in such a way that the resulting (unblinded) signature can be verified with a standard RSA-PSS library.

2. Requirements Notation

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.

3. Notation

The following terms are used throughout this document to describe the protocol operations in this document:

4. Blind Signature Protocol Overview

In this section, we sketch the blind signature protocol wherein a client and server interact to compute sig = Sign(skS, msg), where msg is the private message to be signed, and skS is the server's private key. In this protocol, the server learns nothing of msg, whereas the client learns sig and nothing of skS.

The core issuance protocol runs as follows:

   Client(pkS, msg)                      Server(skS, pkS)
  -------------------------------------------------------
  blinded_msg, inv = Blind(pkS, msg)

                        blinded_msg
                        ---------->

                 blind_sig = BlindSign(skS, blinded_msg)

                         blind_sig
                        <----------

  sig = Finalize(pkS, msg, blind_sig, inv)

Upon completion, correctness requires that clients can verify signature sig over private input message msg using the server public key pkS by invoking the RSASSA-PSS-VERIFY routine defined in [RFC3447]. The finalization function performs that check before returning the signature.

5. RSABSSA Signature Instantiation

Section 8.1 of [RFC8017] defines RSASSA-PSS RSAE, which is a signature algorithm using RSASSA-PSS [RFC8017] with mask generation function 1. In this section, we define RSABSSA, a blinded variant of this algorithm.

5.1. Signature Generation

As outlined in Section 4, signature generation involves three subroutines: Blind, BlindSign, and Finalize. The output from Finalize is a signature over the input to Blind. A specification of these subroutines is below.

5.1.1. Blind

rsabssa_blind encodes an input message and blinds it with the server's public key. It outputs the blinded message to be sent to the server and the corresponding inverse, both encoded as octet strings. RSAVP1 and EMSA-PSS-ENCODE are as defined in [RFC3447].

rsabssa_blind(pkS, msg)

Parameters:
- kLen, the length in octets of the RSA modulus n
- kBits, the length in bits of the RSA modulus n

Inputs:
- pkS, server public key (n, e)
- msg, message to be signed, an octet string
- HF, the hash function used to hash the message
- MGF, the mask generation function

Outputs:
- blinded_msg, an octet string of length kLen
- inv, an octet string of length kLen

Errors:
- "message too long": Raised when the input message is too long.
- "encoding error": Raised when the input message fails encoding.
- "invalid blind": Raised when the inverse of r cannot be found.

Steps:
1. encoded_message = EMSA-PSS-ENCODE(msg, kBits - 1)
   with MGF and HF as defined in the parameters
2. If EMSA-PSS-ENCODE raises an error, raise the error and stop
3. m = OS2IP(encoded_message)
4. r = random_integer_uniform(1, n)
5. r_inv = inverse_mod(r, n)
6. If finding the inverse fails, raise an "invalid blind" error
   and stop
7. x = RSAVP1(pkS, r)
8. z = m * x mod n
9. blinded_msg = I2OSP(z, kLen)
10. inv = I2OSP(r_inv, kLen)
11. output blinded_msg, inv

5.1.2. BlindSign

rsabssa_blind_sign performs the RSA private key operation on the client's blinded message input and returns the output encoded as an octet string. RSASP1 is as defined in [RFC3447].

rsabssa_blind_sign(skS, blinded_msg)

Parameters:
- kLen, the length in octets of the RSA modulus n

Inputs:
- blinded_msg, encoded and blinded message to be signed, an
  octet string

Outputs:
- blind_sig, an octet string of length kLen

Errors:
- "unexpected input size": Raised when a byte string input doesn't
  have the expected length.

Steps:
1. If len(blinded_msg) != kLen, raise "unexpected input size"
   and stop
2. m = OS2IP(blinded_msg)
3. s = RSASP1(skS, m)
4. blind_sig = I2OSP(s, kLen)
5. output blind_sig

5.1.3. Finalize

rsabssa_finalize validates the server's response, unblinds the message to produce a signature, verifies it for correctness, and outputs the signature upon success. Note that this function will internally hash the input message as is done in rsabssa_blind.

rsabssa_finalize(pkS, msg, blind_sig, inv)

Parameters:
- kLen, the length in octets of the RSA modulus n

Inputs:
- pkS, server public key
- msg, message to be signed, an octet string
- blind_sig, signed and blinded element, an octet string of
  length kLen
- inv, inverse of the blind, an octet string of length kLen

Outputs:
- sig, an octet string of length kLen

Errors:
- "invalid signature": Raised when the signature is invalid
- "unexpected input size": Raised when a byte string input doesn't
  have the expected length.

Steps:
1. If len(blind_sig) != kLen, raise "unexpected input size" and stop
2. If len(inv) != kLen, raise "unexpected input size" and stop
3. z = OS2IP(blind_sig)
4. r_inv = OS2IP(inv)
5. s = z * r_inv mod n
6. sig = I2OSP(s, kLen)
7. result = RSASSA-PSS-VERIFY(pkS, msg, sig)
8. If result = "valid signature", output sig, else
   raise "invalid signature" and stop

5.2. Encoding Options

The RSASSA-PSS parameters are defined as in [RFC8230]. Implementations MUST support PS384-encoding, using SHA-384 as hash function for the message and mask generation function with a 48-byte salt.

The RSA-PSS encoding functions take the following optional parameters:

  • Hash: hash function (hLen denotes the length in octets of the hash function output)
  • MGF: mask generation function
  • sLen: intended length in octets of the salt

The blinded functions above are orthogonal to the choice of these options.

6. Public Key Certification

If the server public key is carried in an X.509 certificate, it MUST use the RSASSA-PSS OID [RFC5756]. It MUST NOT use the rsaEncryption OID [RFC5280].

7. Security Considerations

Bellare et al. [BNPS03] proved security of Chaum's original blind signature scheme based on RSA-FDH based on "one-more-RSA-inversion." Note that the design in this document differs only in message encoding, i.e., using PSS instead of FDH.

[[OPEN ISSUE: confirm that results from BNPS03 apply to this construction]]

7.1. Timing Side Channels

rsabssa_blind_sign is functionally a remote procedure call for applying the RSA private key operation. As such, side channel resistance is paramount to protect the private key from exposure [RemoteTiming]. Implementations MUST include side channel attack mitigations, such as RSA blinding, to avoid leaking information about the private key through timing side channels.

7.2. Message Robustness

An essential property of blind signature schemes is that signer learns nothing of the message being signed. In some circumstances, this may raise concerns of arbitrary signing oracles. Applications using blind signature schemes should take precautions to ensure that such oracles do not cause cross-protocol attacks. This can be done, for example, by keeping blind signature keys distinct from signature keys used for other protocols, such as TLS.

An alternative solution to this problem of message blindness is to give signers proof that the message being signed is well-structured. Depending on the application, zero knowledge proofs could be useful for this purpose. Defining such a proof is out of scope for this document.

7.3. Salt State

The PSS salt is a randomly generated string chosen when a message is encoded. If the salt is not generated randomly, or is otherwise constructed maliciously, it might be possible for the salt to carry client information to the server. For example, the salt might be maliciously constructed to encode the local IP address of the client. Implementations MUST ensure that the salt is generated correctly to mitigate such issues.

7.4. Key Substitution Attacks

RSA is well known to permit key substitution attacks, wherein an attacker generates a key pair (skA, pkA) that verify some known (message, signature) pair produced under a different (skS, pkS) key pair [WM99]. This means it may be possible for an attacker to use a (message, signature) pair from one context in another. Entities that verify signatures must take care to ensure a (message, signature) pair verifies with the expected public key.

7.5. Alternative RSA Encoding Functions

This document document uses PSS encoding as specified in [RFC3447] for a number of reasons. First, it is recommended in recent standards, including TLS 1.3 [RFC8446], X.509v3 [RFC4055], and even PKCS#1 itself. According to [RFC3447], "Although no attacks are known against RSASSA-PKCS#1 v1.5, in the interest of increased robustness, RSA-PSS is recommended for eventual adoption in new applications." While RSA-PSS is more complex than RSASSA-PKCS#1 v1.5 encoding, ubiquity of RSA-PSS support influenced the design decision in this draft, despite PKCS#1 v1.5 having equivalent security properties for digital signatures [JKM18]

Full Domain Hash (FDH) [RSA-FDH] encoding is also possible, and this variant has equivalent security to PSS [KK18]. However, FDH is less standard and not used widely in related technologies. Moreover, FDH is deterministic, whereas PSS is probabilistic.

7.6. Alternative Blind Signature Schemes

There are a number of blind signature protocols beyond RSA. This section summarizes these at a high level, and discusses why an RSA-based variant was chosen for the basis of this specification.

  • Blind Schnorr [Sch01]: This is a three-message protocol based on the classical Schnorr signature scheme over elliptic curve groups. Although simple, the hardness problem upon which this is based - Random inhomogeneities in a Overdetermined Solvable system of linear equations, or ROS - can be broken in polynomial time when a small number of concurrent signing sessions are invoked [PolytimeROS]. This can lead to signature forgeries in practice. Signers can enforce concurrent sessions, though the limit (approximately 256) for reasonably secure elliptic curve groups is small enough to make large-scale signature generation prohibitive. In contrast, the variant in this specification has no such concurrency limit.
  • Clause Blind Schnorr [FPS20]: This is a three-message protocol based on a variant of the blind Schnorr signature scheme. This variant of the protocol is not known to be vulnerable to the attack in [PolytimeROS], though the protocol is still new and under consideration. In the future, this may be a candidate for future blind signatures based on blind signatures. However, the three-message flow necessarily requires two round trips between the client and server, which may be prohibitive for large-scale signature generation. Further analysis and experimentation with this scheme is needed.
  • BSA [Abe01]: This is a three-message protocol based on elliptic curve groups similar to blind Schnorr. It is also not known to be vulnerable to the ROS attack in [PolytimeROS]. Kastner et al. [KLRX20] proved concurrent security with a polynomial number of sessions. For similar reasons to the clause blind Schnorr scheme above, the additional number of round trips requires further analysis and experimentation.
  • Blind BLS [BLS-Proposal]: The Boneh-Lynn-Shacham [I-D.irtf-cfrg-bls-signature] scheme can incorporate message blinding when properly instantiated with Type III pairing group. This is a two-message protocol similar to the RSA variant, though it requires pairing support, which is not common in widely deployed cryptographic libraries backing protocols such as TLS. In contrast, the specification in this document relies upon widely deployed cryptographic primitives.

7.7. Post-Quantum Readiness

The blind signature scheme specified in this document is not post-quantum ready since it is based on RSA. (Shor's polynomial-time factorization algorithm readily applies.)

8. IANA Considerations

This document makes no IANA requests.

9. References

9.1. Normative References

[RFC2119]
Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfc-editor.org/info/rfc2119>.
[RFC3447]
Jonsson, J. and B. Kaliski, "Public-Key Cryptography Standards (PKCS) #1: RSA Cryptography Specifications Version 2.1", RFC 3447, DOI 10.17487/RFC3447, , <https://www.rfc-editor.org/info/rfc3447>.
[RFC5756]
Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk, "Updates for RSAES-OAEP and RSASSA-PSS Algorithm Parameters", RFC 5756, DOI 10.17487/RFC5756, , <https://www.rfc-editor.org/info/rfc5756>.
[RFC8017]
Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch, "PKCS #1: RSA Cryptography Specifications Version 2.2", RFC 8017, DOI 10.17487/RFC8017, , <https://www.rfc-editor.org/info/rfc8017>.
[RFC8174]
Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfc-editor.org/info/rfc8174>.
[RFC8230]
Jones, M., "Using RSA Algorithms with CBOR Object Signing and Encryption (COSE) Messages", RFC 8230, DOI 10.17487/RFC8230, , <https://www.rfc-editor.org/info/rfc8230>.

9.2. Informative References

[Abe01]
Abe, M., "A Secure Three-Move Blind Signature Scheme for Polynomially Many Signatures", DOI 10.1007/3-540-44987-6_9, Lecture Notes in Computer Science pp. 136-151, , <https://doi.org/10.1007/3-540-44987-6_9>.
[BLS-Proposal]
"[Privacy-pass] External verifiability: a concrete proposal", n.d., <https://mailarchive.ietf.org/arch/msg/privacy-pass/BDOOhSLwB3uUJcfBiss6nUF5sUA/>.
[BNPS03]
Bellare, ., Namprempre, ., Pointcheval, ., and . Semanko, "The One-More-RSA-Inversion Problems and the Security of Chaum's Blind Signature Scheme", DOI 10.1007/s00145-002-0120-1, Journal of Cryptology Vol. 16, pp. 185-215, , <https://doi.org/10.1007/s00145-002-0120-1>.
[Chaum83]
"Blind Signatures for Untraceable Payments", <http://sceweb.sce.uhcl.edu/yang/teaching/csci5234WebSecurityFall2011/Chaum-blind-signatures.PDF>.
[FPS20]
Fuchsbauer, G., Plouviez, A., and Y. Seurin, "Blind Schnorr Signatures and Signed ElGamal Encryption in the Algebraic Group Model", DOI 10.1007/978-3-030-45724-2_3, Advances in Cryptology - EUROCRYPT 2020 pp. 63-95, , <https://doi.org/10.1007/978-3-030-45724-2_3>.
[I-D.davidson-pp-protocol]
Davidson, A., "Privacy Pass: The Protocol", Work in Progress, Internet-Draft, draft-davidson-pp-protocol-01, , <http://www.ietf.org/internet-drafts/draft-davidson-pp-protocol-01.txt>.
[I-D.irtf-cfrg-bls-signature]
Boneh, D., Gorbunov, S., Wahby, R., Wee, H., and Z. Zhang, "BLS Signatures", Work in Progress, Internet-Draft, draft-irtf-cfrg-bls-signature-04, , <http://www.ietf.org/internet-drafts/draft-irtf-cfrg-bls-signature-04.txt>.
[I-D.irtf-cfrg-voprf]
Davidson, A., Faz-Hernandez, A., Sullivan, N., and C. Wood, "Oblivious Pseudorandom Functions (OPRFs) using Prime-Order Groups", Work in Progress, Internet-Draft, draft-irtf-cfrg-voprf-05, , <http://www.ietf.org/internet-drafts/draft-irtf-cfrg-voprf-05.txt>.
[JKK14]
"Round-Optimal Password-Protected Secret Sharing and T-PAKE in the Password-Only model", <https://eprint.iacr.org/2014/650>.
[JKM18]
Jager, T., Kakvi, S., and A. May, "On the Security of the PKCS#1 v1.5 Signature Scheme", DOI 10.1145/3243734.3243798, Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, , <https://doi.org/10.1145/3243734.3243798>.
[KK18]
Kakvi, S. and E. Kiltz, "Optimal Security Proofs for Full Domain Hash, Revisited", DOI 10.1007/s00145-017-9257-9, Journal of Cryptology Vol. 31, pp. 276-306, , <https://doi.org/10.1007/s00145-017-9257-9>.
[KLRX20]
"On Pairing-Free Blind Signature Schemes in the Algebraic Group Model", n.d., <https://eprint.iacr.org/2020/1071>.
[PolytimeROS]
"On the (in)security of ROS", n.d., <https://eprint.iacr.org/2020/945.pdf>.
[RemoteTiming]
"Remote Timing Attacks are Practical", , <https://crypto.stanford.edu/~dabo/papers/ssl-timing.pdf>.
[RFC4055]
Schaad, J., Kaliski, B., and R. Housley, "Additional Algorithms and Identifiers for RSA Cryptography for use in the Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile", RFC 4055, DOI 10.17487/RFC4055, , <https://www.rfc-editor.org/info/rfc4055>.
[RFC5280]
Cooper, D., Santesson, S., Farrell, S., Boeyen, S., Housley, R., and W. Polk, "Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile", RFC 5280, DOI 10.17487/RFC5280, , <https://www.rfc-editor.org/info/rfc5280>.
[RFC8446]
Rescorla, E., "The Transport Layer Security (TLS) Protocol Version 1.3", RFC 8446, DOI 10.17487/RFC8446, , <https://www.rfc-editor.org/info/rfc8446>.
[RSA-FDH]
"Random Oracles are Practical: A Paradigm for Designing Efficient Protocols", , <https://cseweb.ucsd.edu/~mihir/papers/ro.pdf>.
[Sch01]
Schnorr, C., "Security of Blind Discrete Log Signatures against Interactive Attacks", DOI 10.1007/3-540-45600-7_1, Information and Communications Security pp. 1-12, , <https://doi.org/10.1007/3-540-45600-7_1>.
[WM99]
"Unknown key-share attacks on the station-to-station (STS) protocol", n.d..

Appendix A. Test Vector

This section includes a test vector for the blind signature scheme defined in this document. The following parameters are specified:

p = e1f4d7a34802e27c7392a3cea32a262a34dc3691bd87f3f310dc756734889305
59c120fd0410194fb8a0da55bd0b81227e843fdca6692ae80e5a5d414116d4803fca
7d8c30eaaae57e44a1816ebb5c5b0606c536246c7f11985d731684150b63c9a3ad9e
41b04c0b5b27cb188a692c84696b742a80d3cd00ab891f2457443dadfeba6d6daf10
8602be26d7071803c67105a5426838e6889d77e8474b29244cefaf418e381b312048
b457d73419213063c60ee7b0d81820165864fef93523c9635c22210956e53a8d9632
2493ffc58d845368e2416e078e5bcb5d2fd68ae6acfa54f9627c42e84a9d3f277401
7e32ebca06308a12ecc290c7cd1156dcccfb2311
q = c601a9caea66dc3835827b539db9df6f6f5ae77244692780cd334a006ab353c8
06426b60718c05245650821d39445d3ab591ed10a7339f15d83fe13f6a3dfb20b945
2c6a9b42eaa62a68c970df3cadb2139f804ad8223d56108dfde30ba7d367e9b0a7a8
0c4fdba2fd9dde6661fc73fc2947569d2029f2870fc02d8325acf28c9afa19ecf962
daa7916e21afad09eb62fe9f1cf91b77dc879b7974b490d3ebd2e95426057f35d0a3
c9f45f79ac727ab81a519a8b9285932d9b2e5ccd347e59f3f32ad9ca359115e7da00
8ab7406707bd0e8e185a5ed8758b5ba266e8828f8d863ae133846304a2936ad7bc7c
9803879d2fc4a28e69291d73dbd799f8bc238385
n = aec4d69addc70b990ea66a5e70603b6fee27aafebd08f2d94cbe1250c556e047
a928d635c3f45ee9b66d1bc628a03bac9b7c3f416fe20dabea8f3d7b4bbf7f963be3
35d2328d67e6c13ee4a8f955e05a3283720d3e1f139c38e43e0338ad058a9495c533
77fc35be64d208f89b4aa721bf7f7d3fef837be2a80e0f8adf0bcd1eec5bb040443a
2b2792fdca522a7472aed74f31a1ebe1eebc1f408660a0543dfe2a850f106a617ec6
685573702eaaa21a5640a5dcaf9b74e397fa3af18a2f1b7c03ba91a6336158de420d
63188ee143866ee415735d155b7c2d854d795b7bc236cffd71542df34234221a0413
e142d8c61355cc44d45bda94204974557ac2704cd8b593f035a5724b1adf442e78c5
42cd4414fce6f1298182fb6d8e53cef1adfd2e90e1e4deec52999bdc6c29144e8d52
a125232c8c6d75c706ea3cc06841c7bda33568c63a6c03817f722b50fcf898237d78
8a4400869e44d90a3020923dc646388abcc914315215fcd1bae11b1c751fd52443aa
c8f601087d8d42737c18a3fa11ecd4131ecae017ae0a14acfc4ef85b83c19fed33cf
d1cd629da2c4c09e222b398e18d822f77bb378dea3cb360b605e5aa58b20edc29d00
0a66bd177c682a17e7eb12a63ef7c2e4183e0d898f3d6bf567ba8ae84f84f1d23bf8
b8e261c3729e2fa6d07b832e07cddd1d14f55325c6f924267957121902dc19b3b329
48bdead5
e = 010001
d = 0d43242aefe1fb2c13fbc66e20b678c4336d20b1808c558b6e62ad16a2870771
80b177e1f01b12f9c6cd6c52630257ccef26a45135a990928773f3bd2fc01a313f1d
ac97a51cec71cb1fd7efc7adffdeb05f1fb04812c924ed7f4a8269925dad88bd7dcf
bc4ef01020ebfc60cb3e04c54f981fdbd273e69a8a58b8ceb7c2d83fbcbd6f784d05
2201b88a9848186f2a45c0d2826870733e6fd9aa46983e0a6e82e35ca20a439c5ee7
b502a9062e1066493bdadf8b49eb30d9558ed85abc7afb29b3c9bc644199654a4676
681af4babcea4e6f71fe4565c9c1b85d9985b84ec1abf1a820a9bbebee0df1398aae
2c85ab580a9f13e7743afd3108eb32100b870648fa6bc17e8abac4d3c99246b1f0ea
9f7f93a5dd5458c56d9f3f81ff2216b3c3680a13591673c43194d8e6fc93fc1e37ce
2986bd628ac48088bc723d8fbe293861ca7a9f4a73e9fa63b1b6d0074f5dea2a624c
5249ff3ad811b6255b299d6bc5451ba7477f19c5a0db690c3e6476398b1483d10314
afd38bbaf6e2fbdbcd62c3ca9797a420ca6034ec0a83360a3ee2adf4b9d4ba29731d
131b099a38d6a23cc463db754603211260e99d19affc902c915d7854554aabf608e3
ac52c19b8aa26ae042249b17b2d29669b5c859103ee53ef9bdc73ba3c6b537d5c34b
6d8f034671d7f3a8a6966cc4543df223565343154140fd7391c7e7be03e241f4ecfe
b877a051
msg = 8f3dc6fb8c4a02f4d6352edf0907822c1210a9b32f9bdda4c45a698c80023a
a6b59f8cfec5fdbb36331372ebefedae7d
salt = 051722b35f458781397c3a671a7d3bd3096503940e4c4f1aaa269d60300ce
449555cd7340100df9d46944c5356825abf
inv = 80682c48982407b489d53d1261b19ec8627d02b8cda5336750b8cee332ae26
0de57b02d72609c1e0e9f28e2040fc65b6f02d56dbd6aa9af8fde656f70495dfb723
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encoded_message = 6e0c464d9c2f9fbc147b43570fc4f238e0d0b38870b3addcf7
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209d9283dc7b93fecc04f3f9e7f566829ac41568ef799480c733c09759aa9734e201
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blinded_msg = 10c166c6a711e81c46f45b18e5873cc4f494f003180dd7f115
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blind_sig = 364f6a40dbfbc3bbb257943337eeff791a0f290898a67912
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sig = 6fef8bf9bc182cd8cf7ce45c7dcf0e6f3e518ae48f06f3c670c649ac737a8b
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c706195d52

Authors' Addresses

Frank Denis
Fastly Inc.
Frederic Jacobs
Apple Inc.
Christopher A. Wood
Cloudflare