RSA Blind Signatures
draft-irtf-cfrg-rsa-blind-signatures-01

Document Type Active Internet-Draft (cfrg RG)
Authors Frank Denis  , Frederic Jacobs  , Christopher Wood 
Last updated 2021-07-12
Replaces draft-wood-cfrg-rsa-blind-signatures
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Network Working Group                                           F. Denis
Internet-Draft                                               Fastly Inc.
Intended status: Informational                                 F. Jacobs
Expires: 13 January 2022                                      Apple Inc.
                                                               C.A. Wood
                                                              Cloudflare
                                                            12 July 2021

                          RSA Blind Signatures
                draft-irtf-cfrg-rsa-blind-signatures-01

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
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   This Internet-Draft will expire on 13 January 2022.

Copyright Notice

   Copyright (c) 2021 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

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   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.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Requirements Notation . . . . . . . . . . . . . . . . . . . .   3
   3.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   3
   4.  Blind Signature Protocol Overview . . . . . . . . . . . . . .   4
   5.  RSABSSA Signature Instantiation . . . . . . . . . . . . . . .   4
     5.1.  Signature Generation  . . . . . . . . . . . . . . . . . .   4
       5.1.1.  Blind . . . . . . . . . . . . . . . . . . . . . . . .   5
       5.1.2.  BlindSign . . . . . . . . . . . . . . . . . . . . . .   6
       5.1.3.  Finalize  . . . . . . . . . . . . . . . . . . . . . .   6
     5.2.  Encoding Options  . . . . . . . . . . . . . . . . . . . .   7
   6.  Public Key Certification  . . . . . . . . . . . . . . . . . .   8
   7.  API Considerations  . . . . . . . . . . . . . . . . . . . . .   8
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .   8
     8.1.  Timing Side Channels  . . . . . . . . . . . . . . . . . .   9
     8.2.  Message Robustness  . . . . . . . . . . . . . . . . . . .   9
     8.3.  Randomized and Deterministic Signatures . . . . . . . . .   9
     8.4.  Key Substitution Attacks  . . . . . . . . . . . . . . . .  10
     8.5.  Alternative RSA Encoding Functions  . . . . . . . . . . .  10
     8.6.  Alternative Blind Signature Schemes . . . . . . . . . . .  10
     8.7.  Post-Quantum Readiness  . . . . . . . . . . . . . . . . .  11
   9.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  12
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  12
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  12
     10.2.  Informative References . . . . . . . . . . . . . . . . .  12
   Appendix A.  Test Vectors . . . . . . . . . . . . . . . . . . . .  15
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  19

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,

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   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:

   *  I2OSP and OS2IP: Convert a byte string to and from a non-negative
      integer as described in [RFC8017].  Note that these functions
      operate on byte strings in big-endian byte order.

   *  random_integer_uniform(M, N): Generate a random, uniformly
      distributed integer R such that M <= R < N.

   *  inverse_mod(x, n): Compute the multiplicative inverse of x mod n.

   *  len(s): The length of a byte string, in octets.

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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.

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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_msg = 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_msg)
   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

   The blinding factor r must be randomly chosen from a uniform
   distribution.  This is typically done via rejection sampling.

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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.

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   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, defined as in [RFC8017], Section 9.1.1,
   are as follows:

   *  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

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   Implementations MUST support PS384-encoding, using SHA-384 as Hash
   and MGF functions and sLen = 48, as described in [RFC8230],
   Section 2.  It is RECOMMENDED that implementations also support
   encoding using SHA-384 as Hash and MGF functions and sLen = 0.  Note
   that setting sLen = 0 has the result of making the signature scheme
   deterministic.

   The blinded functions in Section 5.1 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.  API Considerations

   It is NOT RECOMMENDED that APIs allow clients to specify RSA-PSS
   parameters directly, e.g., to set the PSS salt length to 0 as a way
   of producing deterministic signatures.  Instead, implementations
   should offer separate abstractions for randommized or deterministic
   signature generation.  See Section 8.3 for more information about
   deterministic signature considerations.

8.  Security Considerations

   Bellare et al.  [BNPS03] proved the following properties of Chaum's
   original blind signature scheme based on RSA-FDH:

   *  One-more-forgery polynomial security.  This means the adversary,
      interacting with the server (signer) as a client, cannot output
      n+1 valid message and signature tuples after only interacting with
      the server n times, for some n which is polynomial in the scheme's
      security parameter.

   *  Concurrent polynomial security.  This means that servers can
      engage in polynomially many invocations of the protocol without
      compromising security.

   Both results rely upon the RSA Known Target Inversion Problem being
   hard.

   The design in this document differs from the analysis in [BNPS03]
   only in message encoding, i.e., using PSS instead of FDH.  Note,
   importantly, that an empty salt effectively reduces PSS to FDH, so
   the same results apply.

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8.1.  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.

8.2.  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.

8.3.  Randomized and Deterministic Signatures

   When sLen > 0, the PSS salt is a randomly generated string chosen
   when a message is encoded.  This means the resulting signature is
   non-deterministic, meaning that two signatures over the same message
   will be different.  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.  As a result, APIs SHOULD NOT allow clients to provide
   the salt directly; see Section 7 for API considerations.

   When sLen = 0, the PSS salt is empty and the resulting signature is
   deterministic.  Such signatures may be useful for applications
   wherein the only desired source of entropy is the input message.

   Applications that use deterministic signatures SHOULD carefully
   analyze the security implications.  When the required signature
   scheme is not clear, applications SHOULD default to randomized
   signatures.

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8.4.  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.

8.5.  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.

8.6.  Alternative Blind Signature Schemes

   The protocol in this document is not without shortcomings, including:

   *  RSA key and signature sizes are larger than those of alternative
      blind signature protocols;

   *  No evaluation batching support, which means that the cost of the
      protocol scales linearly with the number of invocations; and

   *  Extensions for features such as threshold signing are more complex
      to instantiate compared to other protocols based on, for example,
      Schnorr signatures.

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

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   *  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.

8.7.  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.)

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9.  IANA Considerations

   This document makes no IANA requests.

10.  References

10.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <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, February
              2003, <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, January 2010,
              <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, November 2016,
              <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,
              May 2017, <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, September 2017,
              <https://www.rfc-editor.org/info/rfc8230>.

10.2.  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, 2001,
              <https://doi.org/10.1007/3-540-44987-6_9>.

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   [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, June 2003,
              <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, 2020,
              <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,
              13 July 2020, <https://www.ietf.org/archive/id/draft-
              davidson-pp-protocol-01.txt>.

   [I-D.irtf-cfrg-bls-signature]
              Boneh, D., Gorbunov, S., Wahby, R. S., Wee, H., and Z.
              Zhang, "BLS Signatures", Work in Progress, Internet-Draft,
              draft-irtf-cfrg-bls-signature-04, 10 September 2020,
              <https://www.ietf.org/archive/id/draft-irtf-cfrg-bls-
              signature-04.txt>.

   [I-D.irtf-cfrg-voprf]
              Davidson, A., Faz-Hernandez, A., Sullivan, N., and C. A.
              Wood, "Oblivious Pseudorandom Functions (OPRFs) using
              Prime-Order Groups", Work in Progress, Internet-Draft,
              draft-irtf-cfrg-voprf-07, 6 July 2021,
              <https://www.ietf.org/archive/id/draft-irtf-cfrg-voprf-
              07.txt>.

   [JKK14]    "Round-Optimal Password-Protected Secret Sharing and
              T-PAKE in the Password-Only model",
              <https://eprint.iacr.org/2014/650>.

Denis, et al.            Expires 13 January 2022               [Page 13]
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   [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,
              January 2018, <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, April 2017,
              <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", 2003,
              <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, June 2005,
              <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, May 2008,
              <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, August 2018,
              <https://www.rfc-editor.org/info/rfc8446>.

   [RSA-FDH]  "Random Oracles are Practical: A Paradigm for Designing
              Efficient Protocols", October 1995,
              <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, 2001,
              <https://doi.org/10.1007/3-540-45600-7_1>.

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   [WM99]     "Unknown key-share attacks on the station-to-station (STS)
              protocol", n.d..

Appendix A.  Test Vectors

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

   *  p, q, n, e, d: RSA private and public key parameters, each encoded
      as a hexadecimal string.

   *  msg: Messsage being signed, encoded as a hexadecimal string.  The
      hash is computed using SHA-384.

   *  salt: Randomly-generated salt used when computing the signature.
      The length (sLen) is either 48 or 0 bytes.

   *  inv: The message blinding inverse, encoded as a hexadecimal
      string.

   *  encoded_msg: EMSA-PSS encoded message.  The mask generation
      function is MGF1 with SHA-384.

   *  blinded_msg, blind_sig: The protocol values exchanged during the
      computation, encoded as hexadecimal strings.

   *  sig: The output message signature.

   Test vector for probabilistic signatures (sLen=48):

   p = e1f4d7a34802e27c7392a3cea32a262a34dc3691bd87f3f310dc756734889305
   59c120fd0410194fb8a0da55bd0b81227e843fdca6692ae80e5a5d414116d4803fca
   7d8c30eaaae57e44a1816ebb5c5b0606c536246c7f11985d731684150b63c9a3ad9e
   41b04c0b5b27cb188a692c84696b742a80d3cd00ab891f2457443dadfeba6d6daf10
   8602be26d7071803c67105a5426838e6889d77e8474b29244cefaf418e381b312048
   b457d73419213063c60ee7b0d81820165864fef93523c9635c22210956e53a8d9632
   2493ffc58d845368e2416e078e5bcb5d2fd68ae6acfa54f9627c42e84a9d3f277401
   7e32ebca06308a12ecc290c7cd1156dcccfb2311
   q = c601a9caea66dc3835827b539db9df6f6f5ae77244692780cd334a006ab353c8
   06426b60718c05245650821d39445d3ab591ed10a7339f15d83fe13f6a3dfb20b945
   2c6a9b42eaa62a68c970df3cadb2139f804ad8223d56108dfde30ba7d367e9b0a7a8
   0c4fdba2fd9dde6661fc73fc2947569d2029f2870fc02d8325acf28c9afa19ecf962
   daa7916e21afad09eb62fe9f1cf91b77dc879b7974b490d3ebd2e95426057f35d0a3
   c9f45f79ac727ab81a519a8b9285932d9b2e5ccd347e59f3f32ad9ca359115e7da00
   8ab7406707bd0e8e185a5ed8758b5ba266e8828f8d863ae133846304a2936ad7bc7c
   9803879d2fc4a28e69291d73dbd799f8bc238385
   n = aec4d69addc70b990ea66a5e70603b6fee27aafebd08f2d94cbe1250c556e047
   a928d635c3f45ee9b66d1bc628a03bac9b7c3f416fe20dabea8f3d7b4bbf7f963be3

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   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
   ba01173d4707a12fddac628ca29f3e32340bd8f7ddb557cf819f6b01e445ad96f874
   ba235584ee71f6581f62d4f43bf03f910f6510deb85e8ef06c7f09d9794a008be7ff
   2529f0ebb69decef646387dc767b74939265fec0223aa6d84d2a8a1cc912d5ca25b4
   e144ab8f6ba054b54910176d5737a2cff011da431bd5f2a0d2d66b9e70b39f4b050e
   45c0d9c16f02deda9ddf2d00f3e4b01037d7029cd49c2d46a8e1fc2c0c17520af1f4
   b5e25ba396afc4cd60c494a4c426448b35b49635b337cfb08e7c22a39b256dd032c0
   0adddafb51a627f99a0e1704170ac1f1912e49d9db10ec04c19c58f420212973e0cb
   329524223a6aa56c7937c5dffdb5d966b6cd4cbc26f3201dd25c80960a1a111b3294
   7bb78973d269fac7f5186530930ed19f68507540eed9e1bab8b00f00d8ca09b3f099
   aae46180e04e3584bd7ca054df18a1504b89d1d1675d0966c4ae1407be325cdf623c
   f13ff13e4a28b594d59e3eadbadf6136eee7a59d6a444c9eb4e2198e8a974f27a39e

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   b63af2c9af3870488b8adaad444674f512133ad80b9220e09158521614f1faadfe85
   05ef57b7df6813048603f0dd04f4280177a11380fbfc861dbcbd7418d62155248dad
   5fdec0991f
   encoded_msg = 6e0c464d9c2f9fbc147b43570fc4f238e0d0b38870b3addcf7
   a4217df912ccef17a7f629aa850f63a063925f312d61d6437be954b45025e8282f9c
   0b1131bc8ff19a8a928d859b37113db1064f92a27f64761c181c1e1f9b251ae5a2f8
   a4047573b67a270584e089beadcb13e7c82337797119712e9b849ff56e04385d144d
   3ca9d8d92bf78adb20b5bbeb3685f17038ec6afade3ef354429c51c687b45a7018ee
   3a6966b3af15c9ba8f40e6461ba0a17ef5a799672ad882bab02b518f9da7c1a96294
   5c2e9b0f02f29b31b9cdf3e633f9d9d2a22e96e1de28e25241ca7dd04147112f5789
   73403e0f4fd80865965475d22294f065e17a1c4a201de93bd14223e6b1b999fd548f
   2f759f52db71964528b6f15b9c2d7811f2a0a35d534b8216301c47f4f04f412cae14
   2b48c4cdff78bc54df690fd43142d750c671dd8e2e938e6a440b2f825b6dbb3e19f1
   d7a3c0150428a47948037c322365b7fe6fe57ac88d8f80889e9ff38177bad8c8d8d9
   8db42908b389cb59692a58ce275aa15acb032ca951b3e0a3404b7f33f655b7c7d83a
   2f8d1b6bbff49d5fcedf2e030e80881aa436db27a5c0dea13f32e7d460dbf01240c2
   320c2bb5b3225b17145c72d61d47c8f84d1e19417ebd8ce3638a82d395cc6f7050b6
   209d9283dc7b93fecc04f3f9e7f566829ac41568ef799480c733c09759aa9734e201
   3d7640dc6151018ea902bc
   blinded_msg = 10c166c6a711e81c46f45b18e5873cc4f494f003180dd7f115
   585d871a28930259654fe28a54dab319cc5011204c8373b50a57b0fdc7a678bd74c5
   23259dfe4fd5ea9f52f170e19dfa332930ad1609fc8a00902d725cfe50685c95e5b2
   968c9a2828a21207fcf393d15f849769e2af34ac4259d91dfd98c3a707c509e1af55
   647efaa31290ddf48e0133b798562af5eabd327270ac2fb6c594734ce339a14ea4fe
   1b9a2f81c0bc230ca523bda17ff42a377266bc2778a274c0ae5ec5a8cbbe364fcf0d
   2403f7ee178d77ff28b67a20c7ceec009182dbcaa9bc99b51ebbf13b7d542be33717
   2c6474f2cd3561219fe0dfa3fb207cff89632091ab841cf38d8aa88af6891539f263
   adb8eac6402c41b6ebd72984e43666e537f5f5fe27b2b5aa114957e9a580730308a5
   f5a9c63a1eb599f093ab401d0c6003a451931b6d124180305705845060ebba6b0036
   154fcef3e5e9f9e4b87e8f084542fd1dd67e7782a5585150181c01eb6d90cb958838
   37384a5b91dbb606f266059ecc51b5acbaa280e45cfd2eec8cc1cdb1b7211c8e1480
   5ba683f9b78824b2eb005bc8a7d7179a36c152cb87c8219e5569bba911bb32a1b923
   ca83de0e03fb10fba75d85c55907dda5a2606bf918b056c3808ba496a4d955322120
   40a5f44f37e1097f26dc27b98a51837daa78f23e532156296b64352669c94a8a855a
   cf30533d8e0594ace7c442
   blind_sig = 364f6a40dbfbc3bbb257943337eeff791a0f290898a67912
   83bba581d9eac90a6376a837241f5f73a78a5c6746e1306ba3adab6067c32ff69115
   734ce014d354e2f259d4cbfb890244fd451a497fe6ecf9aa90d19a2d441162f7eaa7
   ce3fc4e89fd4e76b7ae585be2a2c0fd6fb246b8ac8d58bcb585634e30c9168a43478
   6fe5e0b74bfe8187b47ac091aa571ffea0a864cb906d0e28c77a00e8cd8f6aba4317
   a8cc7bf32ce566bd1ef80c64de041728abe087bee6cadd0b7062bde5ceef308a23bd
   1ccc154fd0c3a26110df6193464fc0d24ee189aea8979d722170ba945fdcce9b1b4b
   63349980f3a92dc2e5418c54d38a862916926b3f9ca270a8cf40dfb9772bfbdd9a3e
   0e0892369c18249211ba857f35963d0e05d8da98f1aa0c6bba58f47487b8f663e395
   091275f82941830b050b260e4767ce2fa903e75ff8970c98bfb3a08d6db91ab1746c
   86420ee2e909bf681cac173697135983c3594b2def673736220452fde4ddec867d40
   ff42dd3da36c84e3e52508b891a00f50b4f62d112edb3b6b6cc3dbd546ba10f36b03
   f06c0d82aeec3b25e127af545fac28e1613a0517a6095ad18a98ab79f68801e05c17

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   5e15bae21f821e80c80ab4fdec6fb34ca315e194502b8f3dcf7892b511aee45060e3
   994cd15e003861bc7220a2babd7b40eda03382548a34a7110f9b1779bf3ef6011361
   611e6bc5c0dc851e1509de1a
   sig = 6fef8bf9bc182cd8cf7ce45c7dcf0e6f3e518ae48f06f3c670c649ac737a8b
   8119a34d51641785be151a697ed7825fdfece82865123445eab03eb4bb91cecf4d69
   51738495f8481151b62de869658573df4e50a95c17c31b52e154ae26a04067d5ecdc
   1592c287550bb982a5bb9c30fd53a768cee6baabb3d483e9f1e2da954c7f4cf492fe
   3944d2fe456c1ecaf0840369e33fb4010e6b44bb1d721840513524d8e9a3519f40d1
   b81ae34fb7a31ee6b7ed641cb16c2ac999004c2191de0201457523f5a4700dd64926
   7d9286f5c1d193f1454c9f868a57816bf5ff76c838a2eeb616a3fc9976f65d4371de
   ecfbab29362caebdff69c635fe5a2113da4d4d8c24f0b16a0584fa05e80e607c5d9a
   2f765f1f069f8d4da21f27c2a3b5c984b4ab24899bef46c6d9323df4862fe51ce300
   fca40fb539c3bb7fe2dcc9409e425f2d3b95e70e9c49c5feb6ecc9d43442c33d5000
   3ee936845892fb8be475647da9a080f5bc7f8a716590b3745c2209fe05b17992830c
   e15f32c7b22cde755c8a2fe50bd814a0434130b807dc1b7218d4e85342d70695a5d7
   f29306f25623ad1e8aa08ef71b54b8ee447b5f64e73d09bdd6c3b7ca224058d7c67c
   c7551e9241688ada12d859cb7646fbd3ed8b34312f3b49d69802f0eaa11bc4211c2f
   7a29cd5c01ed01a39001c5856fab36228f5ee2f2e1110811872fe7c865c42ed59029
   c706195d52

   Test vector for deterministic signatures (sLen=0):

   p = ca9d82e9059fa3b145da850e0c451ff31093d819644ba29a3409393de2adfa1b
   cd65e8669a5c5140142c1404204edbc380d4e7a5c866c06bb2427c76b9e3d16bbfc1
   b1668dec219b8c59fee90b7baf557fc2feb13f2f4b30d8606d20b9928f4f588a3b34
   baa659b3bd1dd590c83e90e6251b5239fbbb73b12e90534a375e3f71
   q = c075694f69db6a07456e19eeace01b430f2d6cc6cd5495d569e242b6f5e8ded7
   df27e6aeea4db4e307554fb519b68279a58d9e2d25cee4b37668554eec2f2feb7924
   6955a07bd526f02a6afedc7a3aff2b8953287fef2c4a02207ccb9f14e4612e9af344
   7dd3401728a8957871b759b6bbf22aa0e8271b82f32dd5a2d2550197
   n = 98530f850dcc894d84ecfce9dec3a475bf30ec3ce4606f677ac4a6ef63f763ff
   64a162ef1c991d8094b5652d0d78c126b3e97d1d77eba2f833b5be9a124e003065ec
   2a3ea4fbc31bc283de1c7cd8a971eb57aa7284b082562ccde572b73702068a6143e6
   dabf886538ff419874c300a85f3d9d50f0731fc6b9c92a121fefb7911f5ea92d25b1
   7a4f3b2883eff34a221b5c28c488e35067a8460d8fab1c405704ebfa1ca165d69cd4
   e425995a03a447f6cbba5d20d459707ab4a2c537a5dbd02801d7b19a03aaa9aec21d
   1c363996c6b9fee2cab370d501c9b67e7dc4a20eb0cdc3b24be242093b5a66119b96
   da0fb0ec0b1b0da0bd0b92236ece47d5c95bdca7
   e = 010001
   d = 6b15d18e4f8220709fe75f7226ca517ef9b7320d28dc66d54fa89a5727670f24
   c7a0f1857a0c6682338946a4a298e6e90788390e137553afbbe2a4297a7edd8128d6
   1b68c8e1b96b7596f0fa0406e9308e2ba64735e344edc237c97b993411b7796721ae
   54d05bda1574d5af913e59e30479b373e86676cb6566f7ada0480d3ae21d50ac94c0
   b41c476e566d6bcdef88eeab3042ef1016527558e794b6029cff1120596fe2104fac
   928a66ad2fb1094d1ae1231abf95206cae7cd4e7aad388199d7ac1fe17e3f9174362
   32cffe70e12056e02cfb9604e73cc34984bb83f7112ed197bf3a4d9f6d0c0e3c4dd8
   f2d9cbe17185f1e63561b08f7d14bd36112f3ea1
   msg = 5465737420766563746f7220776974682064657465726d696e697374696320

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   70616464696e67
   encoded_msg = 4021ac68705782fb7587bf24ac0528853025aa4a998db7b1a503af
   b5b266cbd1876710a2b0aa6e37b70fca538d42285beddd61d965c02b2162c8644587
   3bdaf687a29bf6b2ab10fa22013cae53ff1c78969ef6c3eb069bfef339a5df788044
   d159678e571e50fc3fa40a30fe183348453542f258c663dc9c4b372895120ad12ff8
   b8ec1d37d766b2604fbf50bf9d84432a59593d21d7f379d6bf9198ea2fa90ee5abad
   b27eada5d6f40a2ec45aa4bb8710042beab5c6afb4381fc57012e61b3a815800e53e
   69fe2fdccb3b4ee51968c1ef6281d7e8fe08c4337bad73d99e947df834e5402378a6
   6142bf032dfade7e6e2d43ae90b145055861e06eff189b63bc
   inv = 6e69972553327ee6240ce0de7146aea2243927cf9f7f52c0103367df79e3ba
   febfa61c2ffdc41ea397a38523654a1a806f4eebcd5fe9a2592a463f1faa26c3601f
   83f29141eda488f14f7c0aa82faa025e37adbe77e02e575f72f7b9d095882923476f
   2328dfaeb23b607d2f706c6c8ef6c2aee50ddb14e6d27e043e7dec8e5dede6844aa8
   0b2206b6019350d37925bb8819653aa7a13bfb9cc3c95b53378f278903b5c06a10c0
   b3ce0aa028e9600f7b2733f0278565f9b88e9d92e039db78300170d7bbd32ce2b89a
   d8944167839880e3a2aeba05bf00edc8032a63e6279bf42a131ccc9bb95b8693764b
   27665274fb673bdfb7d69b7957ee8b64a99efbeed9
   blinded_msg = 5a631b41e7759a634cef04359436e358143ee2892fbebd072d1e5c
   c45475ff55b6b492e13c59979f4b968994ddca3cc068084d3b176a6132039c584707
   acbb9862c009fa5b63cfb7b6f6d577825c1e81ad11059cb87a524083230f906ea0a4
   d9db3434d49cf9f0ea52b2425db4d319f51540e5de6cfb30b86d5e5d810a284f3478
   f6259f054407c854303ec23c2e0989dd57aa002e56ab6287594c25154a1646060cb4
   f6479b07f627991f7089ac0491d5841d6461166b324b3d448b2a8071de68505503fe
   adf7d8182d18d8b0d3b91d77b627a5ffae68f913efbbb2fc082437f845880f94f07d
   873bc0c0688f60033235bcc1701dcba83dca944b05227884e3
   blind_sig = 817596a0b568088b60c29482c0178d34e0d54dc34a9375152701e4e6
   d5ef76c92f4281a377d8b2f438f6af4ef9c26dd38ad2cc932f90fe45d4c0a1ba10e6
   95a1c8e797aa5023f84385904e5f378df5677b8eb7312f835f9e3a097b1b7e55fece
   0d00ec3f52ba26b39c91322b6404eef4e567d909195bfc0f72690805ea3f71736d7e
   b51e84556c5241786f5f37bf9d2a0305bf36454d9ab8b5a9f6fe03fd4ab472b5379d
   7e8ab92e803c7c15bf3d0234653e1f6d80d23c7f127bed7fba3d297b62fee51b8e71
   b04d402cf291ac87460011fd222cfd27b5669d79d1e0dcc8d911c2dc6d0edcd205a9
   1278cc97019cfc709ce8a50297409e66f27b1299e386a6cd
   sig = 848fc8a032ea073280a7d9146ae55bb0199cd1941c10a03cce1dc38579c4e7
   7e87f259e250b16a9912ce2c085cb9489846f803fd6ed09bf8605c4aa8b0ebf2c938
   093e53ad025a48b97f7975255805118c33fa0f73ec204b9723acefacd8031ab3d9f7
   ebeaf996eee3678c788cea96932dd723b236355c0e6864fad2fc87b00e4eda476e90
   f000936b0d9fa65bf1112fc296e8aa5bb05ca7cb32dec01407e3d3ed94c1ebb0dc43
   0ea59588ccc0995a6e2f1423dbe06c6f27650b23b12eb343b9e461ba532825e5e265
   72fbe723b69753c178361e7a834a566ce950df55ff97d314b384b3fa8c0098d560d4
   c6ba519a9b6040f908adf34f6b2d5d30c265cd0fb1

Authors' Addresses

   Frank Denis
   Fastly Inc.

   Email: fd@00f.net

Denis, et al.            Expires 13 January 2022               [Page 19]
Internet-Draft            RSA Blind Signatures                 July 2021

   Frederic Jacobs
   Apple Inc.

   Email: frederic.jacobs@apple.com

   Christopher A. Wood
   Cloudflare

   Email: caw@heapingbits.net

Denis, et al.            Expires 13 January 2022               [Page 20]