Network Working Group                                 P. M. Hallam-Baker
Internet-Draft                                        ThresholdShare.com
Intended status: Informational                              9 March 2020
Expires: 10 September 2020


                Threshold Signatures in Elliptic Curves
                  draft-hallambaker-threshold-sigs-01

Abstract

   A Threshold signature scheme is described.  The signatures created
   are computationally indistinguishable from those produced using the
   Ed25519 and Ed448 curves as specified in RFC8032 except in that they
   are non-deterministic.  Threshold signatures are a form of digital
   signature whose creation requires two or more parties to interact but
   does not disclose the number or identities of the parties involved.

   https://mailarchive.ietf.org/arch/browse/cfrg/
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   Copyright (c) 2020 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

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   license-info) in effect on the date of publication of this document.




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   Please review these documents carefully, as they describe your rights
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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Applications  . . . . . . . . . . . . . . . . . . . . . .   4
       1.1.1.  HSM Binding . . . . . . . . . . . . . . . . . . . . .   4
       1.1.2.  Code Signing  . . . . . . . . . . . . . . . . . . . .   4
       1.1.3.  Signing by Redundant Services . . . . . . . . . . . .   5
   2.  Definitions . . . . . . . . . . . . . . . . . . . . . . . . .   5
     2.1.  Requirements Language . . . . . . . . . . . . . . . . . .   5
     2.2.  Defined Terms . . . . . . . . . . . . . . . . . . . . . .   5
     2.3.  Related Specifications  . . . . . . . . . . . . . . . . .   5
     2.4.  Implementation Status . . . . . . . . . . . . . . . . . .   5
   3.  Principles  . . . . . . . . . . . . . . . . . . . . . . . . .   6
     3.1.  Direct shared threshold signature . . . . . . . . . . . .   7
     3.2.  Shamir shared threshold signature . . . . . . . . . . . .   9
     3.3.  Stateless computation of final share  . . . . . . . . . .  10
       3.3.1.  Side channel resistance . . . . . . . . . . . . . . .  11
     3.4.  Security Analysis . . . . . . . . . . . . . . . . . . . .  11
       3.4.1.  Calculation of r values . . . . . . . . . . . . . . .  12
       3.4.2.  Replay Attack . . . . . . . . . . . . . . . . . . . .  13
       3.4.3.  Malicious Contribution Attack . . . . . . . . . . . .  13
       3.4.4.  Rogue Key Attack  . . . . . . . . . . . . . . . . . .  13
   4.  Ed2519 Signature  . . . . . . . . . . . . . . . . . . . . . .  14
   5.  Ed448 Signature . . . . . . . . . . . . . . . . . . . . . . .  15
   6.  Test Vectors  . . . . . . . . . . . . . . . . . . . . . . . .  16
     6.1.  Direct Threshold Signature Ed25519  . . . . . . . . . . .  16
     6.2.  Direct Threshold Signature Ed448  . . . . . . . . . . . .  18
     6.3.  Shamir Threshold Signature Ed25519  . . . . . . . . . . .  21
     6.4.  Shamir Threshold Signature Ed448  . . . . . . . . . . . .  24
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .  27
     7.1.  Rogue Key attack  . . . . . . . . . . . . . . . . . . . .  27
     7.2.  Disclosure or reuse of the value r  . . . . . . . . . . .  27
     7.3.  Resource exhaustion attack  . . . . . . . . . . . . . . .  27
     7.4.  Signature Uniqueness  . . . . . . . . . . . . . . . . . .  27
   8.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  28
   9.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  28
   10. Normative References  . . . . . . . . . . . . . . . . . . . .  28
   11. Informative References  . . . . . . . . . . . . . . . . . . .  28

1.  Introduction

   Threshold encryption and key generation provide compelling advantages
   over single private key approaches because splitting the private key
   permits the use of that key to be divided between two or more roles.




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   All existing digital signatures allow the signer role to be divided
   between multiple parties by attaching multiple signatures to the
   signed document.  This approach, known as multi-signatures, is
   distinguished from a threshold signature scheme in that the identity
   and roles of the individual signers is exposed.  In a threshold
   signature scheme, the creation of a single signature requires the
   participation of multiple signers and the signature itself does not
   reveal the means by which it was constructed.

   Rather than considering multi-signatures or threshold signatures to
   be inherently superior, it is more useful to regard both as two
   points on a continuum of choices:

   Multi-signatures  Multiple digital signatures on the same document.
      Multi-signatures are simple to create and provide the verifier
      with more information but require the acceptance criteria to be
      specified independently of the signature itself.  This requires
      that the application logic or PKI provide some means of describing
      the criteria to be applied.

   Multi-party key release  A single signature created using a single
      private key stored in an encrypted form whose use requires
      participation of multiple key decryption shares.

   Threshold signatures  A single signature created using multiple
      signature key shares.  Signature creation may be subject to
      complex criteria such as requiring an (n,t) quorum of signers but
      these criteria are fixed at the time the signature is created

   Aggregate Signatures  A single signature created using multiple
      signature key shares such that validation of the aggregate
      signature serves to validate the participation of each of the
      individual signers.

   This document builds on the approach described in
   [draft-hallambaker-threshold] to define a scheme that creates
   threshold signatures that are computationally indistinguishable from
   those produced according to the algorithm specified in [RFC8032].
   The scheme does not support the creation of aggregate signatures.

   The approach used is based on that developed in FROST [Komlo].  This
   document describes the signature scheme itself.  The techniques used
   to generate keys are described separately in
   [draft-hallambaker-threshold].

   As in the base document, we first describe signature generation for
   the case that _n_ = _t_ using 'direct' coefficients, that is the
   secret scalar is the sum of the secret shares.  We then show how the



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   scheme is modified using Shamir secret sharing [Shamir79] and
   Lagrange coefficients for the case that _n_ > _t_.

1.1.  Applications

   Threshold signatures have application in any situation where it is
   desired to have finer grain control of signing operations without
   this control structure being visible to external applications.  It is
   of particular interest in situations where legacy applications do not
   support multi-signatures.

1.1.1.  HSM Binding

   Hardware Security Modules (HSMs) prevent accidental disclosures of
   signature keys by binding private keys to a hardware device from
   which it cannot be extracted without substantial effort.  This
   provides effective mitigation of the chief causes of key disclosure
   but requires the signer to rely on the trustworthiness of a device
   that represents a black box they have no means of auditing.

   Threshold signatures allow the signer to take advantage of the key
   binding control provided by an HSM without trusting it.  The HSM only
   contributes one of the key shares used to create the signature.  The
   other is provided by the application code (or possibly an additional
   HSM).

1.1.2.  Code Signing

   Code signing is an important security control used to enable rapid
   detection of malware by demonstrating the source of authorized code
   distributions but places a critical reliance on the security of the
   signer's private key.  Inadvertent disclosure of code signing keys is
   commonplace as they are typically stored in a form that allows them
   to be used in automatic build processes.  Popular source code
   repositories are regularly scanned by attackers seeking to discover
   private signature keys and passwords embedded in scripts.

   Threshold signatures allow the code signing operation to be divided
   between a developer key and an HSM held locally or by a signature
   service.  The threshold shares required to create the signature can
   be mapped onto the process roles and personnel responsible for
   authorizing code release.  This last concern might be of particular
   advantage in open source projects where the concentration of control
   embodied in a single code signing key has proved to be difficult to
   reconcile with community principles.






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1.1.3.  Signing by Redundant Services

   Redundancy is as desirable for trusted services as for any other
   service.  But in the case that multiple hosts are tasked with
   compiling a data set and signing the result, there is a risk of
   different hosts obtaining a different view of the data set due to
   timing or other concerns.  This presents the risk of the hosts
   signing inconsistent views of the data set.

   Use of threshold signatures allows the criteria for agreeing on the
   data set to be signed to be mapped directly onto the requirement for
   creating a signature.  So if there are three hosts and two must agree
   to create a signature, three signature shares are created and with a
   threshold of two.

2.  Definitions

   This section presents the related specifications and standard, the
   terms that are used as terms of art within the documents and the
   terms used as requirements language.

2.1.  Requirements Language

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in [RFC2119].

2.2.  Defined Terms

   See [draft-hallambaker-threshold].

2.3.  Related Specifications

   This document extends the approach described in
   [draft-hallambaker-threshold] to support threshold signatures.  The
   deterministic mechanism described in specification
   [draft-hallambaker-mesh-udf] is used to generate the private keys
   used in the test vectors.

2.4.  Implementation Status

   The implementation status of the reference code base is described in
   the companion document [draft-hallambaker-mesh-developer].








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3.  Principles

   The threshold signatures created according to the algorithms
   described in this document are compatible with but not identical to
   the signatures created according to the scheme described in
   [RFC8032].  In particular:

   *  The signature verification algorithm is unchanged.

   *  The unanimous threshold scheme produces values of _R_ and _S_ that
      are deterministic but different from the values that would be
      obtained by using the aggregate private key to sign the same
      document.

   *  The deterministic quorate threshold scheme produces values of _R_
      and _S_ that are deterministic for a given set of signers but will
      change for a different set of signers or if the aggregate private
      key was used to sign the same document.

   *  ?The non-deterministic quorate threshold scheme produces values of
      _R_ and _S_ that will be different each time the document is
      signed.

   Recall that a digital signature as specified by [RFC8032] consists of
   a pair of values _S_, _R_ calculated as follows:

   _R_ = _r.B_

   S = _r_ + _k.s_ mod _L_

   Where  _B_ is the base point of the elliptic curve.

      _r_ is an unique, unpredictable integer value such that 0 r L

      _k_ is the result of applying a message digest function determined
      by the curve (Ed25519, Ed448) to a set of parameters known to the
      verifier which include the values _R_, _A_ and PH(_M_).

      _A_ is the public key of the signer, _A_ = _s.B_

      PH(_M_) is the prehash function of the message value.

      _s_ is the secret scalar value

      _L_ is the order of the elliptic curve group.

   To verify the signature, the verifier checks that:




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   _S.B_ = _R_ + _k.A_

   This equality must hold for a valid signature since:

   _S.B_  = (_r_ + _k.s_)._B_

      = _r.B_ +_k_.(_s.B_)

      = _R_ + _k.A_

   The value _r_ plays a critical role in the signature scheme as it
   serves to prevent disclosure of the secret scalar.  If the value _r_
   is known, _s_ can be calculated as _s_ = (_S-r_)._k_^(-1) mod _L_. It
   is therefore essential that the value _r_ be unguessable.

   Furthermore, if the same value of _r_ is used to sign two different
   documents, this results two signatures with the same value _R_ and
   different values of _k_ and _S_. Thus

   _S_(1)_ = _r_ + _k_(1)_._s_ mod _L_

   S_(2) = _r_ + _k_(2).s mod L_

   s = (_S_(1)_ - _S_(2)_)(_k_(1)_ - _k_(2)_)^(-1) mod _L_

   The method of constructing _r_ MUST ensure that it is unique and
   unguessable.

3.1.  Direct shared threshold signature

   A threshold signature R, S is constructed by summing a set of
   signature contributions from two or more signers.  For the case that
   the composite private key is the sum of the key shares (_n_ = _t_),
   each signer _i_ provides a contribution as follows:

   A_(i) = s_(i).B

   R_(i) = r_(i).B

   S_(i) = r_(i) + k.s_(i) mod L

   Where s_(i) and r_(i) are the secret scalar and unguessable value for
   the individual signer.

   The contributions of signers {1, 2, ... n} are then combined as
   follows:

   R = R_(1) + R_(2) + ... + R_(n)



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   S = S_(1) + S_(2) + ... + S_(n)

   A = s.B

   Where s = (s_(1) + s_(2) + ... + s_(n)) mod L

   The threshold signature is verified in the same manner as before:

   S.B = R + k.A

   Substituting for S.B we get:

   = (S_(1) + S_(2) + ... + S_(n)).B

   = S_(1).B + S_(2).B + ... + S_(n).B

   = (r_(1) + k.s_(1)).B + (r_(2) + k.s_(2)).B + ... + (r_(n) +
   k.s_(n)).B

   = (r_(1).B + k.s_(1).B) + (r_(2).B + k.s_(2).B) + ... + (r_(n).B +
   k.s_(n).B)

   = (R1 + k.A1) + (R1 + k.A1) + ... + (Rn + k.An)

   Substituting for R + k.A we get:

   = R_(1) + R_(2) + ... + R_(n) + k.(A_(1) + A_(2) + ... + A_(n))

   = R_(1) + R_(2) + ... + R_(n) + k.A_(1) + k.A_(2) + ... + k.A_(n)

   = (R_(1) + k.A_(1)) + (R_(1) + k.A_(1)) + ... + (R_(n) + k.A_(n))

   As expected, the operation of threshold signature makes use of the
   same approach as threshold key generation and threshold decryption as
   described in [draft-hallambaker-threshold].  As with threshold
   decryption it is not necessary for each key share holder to have a
   public key corresponding to their key share.  All that is required is
   that the sum of the secret scalar values used in calculation of the
   signature modulo the group order be the value of the aggregate secret
   scalar corresponding to the aggregate secret key.

   While verification of [RFC8032] signatures is unchanged, the use of
   threshold signatures requires a different approach to signing.  In
   particular, the fact that the value k is bound to the value R means
   that the participants in the threshold signature scheme must agree on
   the value R before the value k can be calculated.  Since k is
   required to calculate the signature contributions S_(i) can be
   calculated, it is thus necessary to calculate the values R_(i) and



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   S_(i) in separate phases.  The process of using a threshold signature
   to sign a document thus has the following stages orchestrated by a
   dealer as follows:

   0.  The dealer determines the values F, C and PH(M) as specified in
       [RFC8032] and transmits them to the signers {1, 2, ... n}.

   1.  Each signer generates a random value r_(i) such that 1 r_(i) L,
       calculates the value R_(i) = r_(i).B and returns R to the dealer
       .

   2.  The dealer calculates the value R = R_(1) + R_(2) + ... + R_(n)
       and transmits R and A to the signers {1, 2, ... n}.

   3.  Each signer uses the suppled data to determine the value k and
       hence S_(i) = r_(i) + k.s_(i) mod L and transmits it to the
       dealer .

   4.  The dealer calculates the value S = S_(1) + S_(2) + ... + S_(n)
       and verifies that the resulting signature R, S verifies according
       to the mechanism specified in [RFC8032].  If the signature is
       correct, the dealer publishes it.  Otherwise, the dealer MAY
       identify the signer(s) that provided incorrect contributions by
       verifying the values R_(i) and S_(i) for each.

   For clarity, the dealer role is presented here as being implemented
   by a single party.

3.2.  Shamir shared threshold signature

   To construct a threshold signature using shares created using Shamir
   Secret Sharing, each private key value _s_(i)_ is multiplied by the
   Lagrange coefficient _l_(i)_ corresponding to the set of shares used
   to construct the signature:

   A_(i) = s_(i)l_(i).B

   R_(i) = r_(i).B

   _S_(i) = ri + klisi mod L_

   It is convenient to combine the derivation of _S_(i)_ for the
   additive and Shamir shared threshold signatures by introducing a key
   multiplier coefficient _c_(i)_:

   _S_(i) = ri + kcisi mod L_





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   Where  _c_(i)_ = 1 for the additive shared threshold signature

      _c_(i)_ = _l_(i)_ for the Shamir shared threshold signature

3.3.  Stateless computation of final share

   One of the chief drawbacks to the algorithm described above is that
   it requires signers to perform two steps with state carried over from
   the first to the second to avoid reuse of the value _r_(i)_. This
   raises particular concern for implementations such as signature
   services or HSMs where maintaining state imposes a significant cost.

   Fortunately, it is possible to modify the algorithm so that the final
   signer does not need to maintain state between steps:

   0.  All the signers except the final signer _F_ generate their value
       _r_(i)_ and submit the corresponding value _R_(i)_ to the dealer

   1.  Dealer calculates the value _R_ - _R_(F)_ and sends it to the
       final signer together with the all the other parameters required
       to calculate _k_ and the final signer's key multiplier
       coefficient _c_(F)_.

   2.  The final signer generates its value _r_(F)_

   3.  The final signer calculates the value _R_(F)_ from which the
       values _R_ and _k_ can now be determined.

   4.  The final signer calculates its key share contribution _S_(F) =
       rF + kcFsF mod L._

   5.  The final signer returns the values _S_(F)_ and _R_ to the
       dealer.

   6.  The dealer reports the value R to the other signers and continues
       the signature process as before.

   While this approach to stateless computation of the signature
   contributions is limited to the final share, this is sufficient to
   cover the overwhelming majority of real-world applications where _n_
   = _t_ = 2.

   Note that the final signer MAY calculate its value _r_(F)_
   deterministically provided that the parameters _R_ - _R_(F)_ and
   _c_(F)_ are used in its determination.  Other signers MUST NOT use a
   deterministic means of generating their value _r_(i)_ since the
   information known to them at the time this parameter is generated is
   not sufficient to fix the value of _R_.



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3.3.1.  Side channel resistance

   The use of Kocher side channel resistance as described in
   [draft-hallambaker-threshold] entails randomly splitting the private
   key into two shares and performing the private key operation
   separately on each share to avoid repeated operations using the same
   private key value at the cost of performing each operation twice.

   This additional overhead MAY be eliminated when threshold approaches
   are used by applying blinding factors whose sum is zero to each of
   the threshold shares.

   For example, if generation of the threshold signature is divided
   between an application program A and an HSM B using the final share
   approach to avoid maintaining state in the HSM, we might generate a
   blinding factor thus:

   0.  A generates a random nonce _n_(A)_ and sends it to B with the
       other parameters required to generate the signature.

   1.  B generates a random nonce _n_(B)_

   2.  B calculates the blinding factor _x_ by calculating
       _H_(_n_(A,)nB) where H is a strong cryptographic digest function
       and converting the result to an integer in the range 1 x L._

   3.  B calculates the signature parameters as before except that the
       threshold signature contribution is now _S_(B) = rB + k(cBsB + x)
       mod L._

   4.  B returns the nonce _n_(B)_ to A with the other parameters.

   5.  A calculates the blinding factor _x_ using the same approach as B

   6.  A calculates the signature parameters as before except that the
       threshold signature contribution is now _S_(A) = rA + k(cAsA - x)
       mod L._

   This approach MAY be extended to the case that _t_ > 2 by
   substituting a Key Derivation Function (e.g.  [RFC5860]) for the
   digest function.

3.4.  Security Analysis

   We consider a successful breach of the threshold signature scheme to
   be any attack that allows the attacker to create a valid signature
   for any message without the participation of the required threshold
   of signers.



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   Potential breaches include:

   *  Disclosure of the signature key or signature key share.

   *  Modification of signature data relating to message M to allow
      creation of a signature for message M'.

   *  Ability of one of the signers to choose the value of the aggregate
      public key.

   *  Access control attacks inducing a signer to create a signature
      contribution that was not properly authenticated or authorized.

   We regard attacks on the access control channel to be out of scope
   for the threshold signature algorithm, though they are certainly a
   concern for any system in which a threshold signature algorithm is
   employed.

   We do not consider the ability of a signer to cause creation of an
   invalid signature to represent a breach.

3.4.1.  Calculation of r values

   The method of constructing the values _r_(i)_ MUST ensure that each
   is unique and unguessable both to external parties, the signers and
   the dealer.  The deterministic method specified in [RFC8032] cannot
   be applied to generation of the values r_(i) as it allows the dealer
   to cause signers to reveal their key shares by requesting multiple
   signature contributions for the same message but with different
   values of _k_. In particular, requesting signature contributions for
   the same message:

   With different Lagrange coefficients.

   With a false value of _R_

   To avoid these attacks, the value r_(i) is generated using a secure
   random number generator.  This approach requires the signer to ensure
   that values are never reused requiring that the signing API maintain
   state between the first and second rounds of the algorithm.

   While there are many approaches to deterministic generation of r_(i)
   that appear to be sound, closer inspection has demonstrated these to
   be vulnerable to rogue key and rogue contribution attacks.







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3.4.2.  Replay Attack

   The most serious concern in the implementation of any Schnorr type
   signature scheme is the need to ensure that the value r_(i) is never
   revealed to any other party and is never used to create signatures
   for two different values of k.s_(i).

   Ensuring this does not occur imposes significant design constraints
   as creating a correct signature contribution requires that the signer
   use the same value of r_(i) to construct its value or R_(i) and
   S_(i).

   For example, a HSM device may be required to perform multiple
   signature operations simultaneously.  Since the storage capabilities
   of an HSM device are typically constrained, it is tempting to attempt
   to avoid the need to track the value of r_(i) within the device
   itself using an appropriately authenticated and encrypted opaque
   state token.  Such mechanisms provide the HSM with the value of r_(i)
   but do not and cannot provide protection against a replay attack in
   which the same state token is presented with a request to sign
   different values of k.

3.4.3.  Malicious Contribution Attack

   In a malicious contribution attack, one or more parties present a
   signature contribution that does not meet the criteria R_(i) =
   r_(i).B and S_(i) = r_(i) + ks_(i).

   Such an attack is not considered to be a breach as it merely causes
   the signature process to fail.

3.4.4.  Rogue Key Attack

   A threshold signature scheme that allows the participants to 'bring
   their own key' may be vulnerable to a rogue key attack in which a
   signer is able to select the value of the aggregate public signature
   key by selecting a malicious public signature key value.

   The scheme described in this document is a threshold signature scheme
   and does not support this feature.  Consequently, this attack is not
   relevant.  It is described here for illustrative purposes only.

   This particular attack only applies when the individual signers
   create their own signature shares.  It is not a concern when the
   signature shares are created by splitting a master signature private
   key.





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   Consider the case where the aggregate public key signature is
   calculated from the sum of public signature key share values
   presented by the signers:

   A = A_(1) + A_(2) + ... + A_(n)

   If the public key values are presented in turn, the last signer
   presenting their key share can force the selection of any value of A
   that they choose by selecting A_(n) = A_(m) - (A_(1) + A_(2) + ... +
   A_(n-1))

   The attacker can thus gain control of the aggregate signature key by
   choosing A_(m) = s_(m).B where s_(m) is a secret scalar known only to
   the attacker.  But does so at the cost of not knowing the value s_(n)
   and so the signer cannot participate in the signature protocol.

   This attack allows the attacker and the attacker alone to create
   signatures which are validated under the aggregate signature key.

   The attack is a consequence of the mistaken assumption that a
   signature created under the signature key A_(1) + A_(2) + ... + A_(n)
   provides evidence of the individual participation of the
   corresponding key holders without separate validation of the
   aggregate key.

   Enabling the use of threshold signature techniques by ad-hoc groups
   of signers using their existing signature keys as signature key
   shares presents serious technical challenges that are outside the
   scope of this specification.

4.  Ed2519 Signature

   The means by which threshold shares are created is described in
   [draft-hallambaker-threshold].

   The dealer selects the signers who are to construct the signature.
   Each signer then computes the value R_(i):

   0.  Randomly generate an integer r_(i) such that 1 r_(i) L.

   1.  Compute the point R_(i) = r_(i)B.  For efficiency, do this by
       first reducing r_(i) modulo L, the group order of B.  Let the
       string R_(i) be the encoding of this point.

   2.  Transmit the value R_(i) to the dealer

   3.  At some later point, the dealer MAY complete the signature by
       returning the values F, C, A and R as specified in [RFC8032]



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       together with the key multiplier coefficient c_(i).  The signers
       MAY then complete their signature contributions:

   4.  Compute SHA512(dom2(F, C) || R || A || PH(M)), and interpret the
       64-octet digest as a little-endian integer k.

   5.  Compute S_(i) = (r_(i) + kc_(i)s_(i)) mod L.  For efficiency,
       again reduce k modulo L first.

   6.  Return the values R_(i), S_(i) to the dealer .

   The dealer then completes the signature by:

   0.  Computing the composite value S = S_(1) + S_(2) + ... + S_(n)

   1.  Verifying that the signature R, S is valid.

   2.  Publishing the signature.

5.  Ed448 Signature

   The means by which threshold shares are created is described in
   [draft-hallambaker-threshold].

   The dealer selects the signers who are to construct the signature.
   Each signer then computes the value R_(i):

   0.  Randomly generate an integer r_(i) such that 1 r_(i) L.

   1.  Compute the point R_(i) = r_(i)B.  For efficiency, do this by
       first reducing r_(i) modulo L, the group order of B.  Let the
       string R_(i) be the encoding of this point.

   Transmit the value R_(i) to the dealer

   0.  At some later point, the dealer MAY complete the signature by
       returning the values F, C, A and R as specified in [RFC8032]
       together with the key multiplier coefficient c_(i).  The signers
       MAY then complete the signature contributions:

   1.  Compute SHAKE256(dom4(F, C) || R || A || PH(M), 114), and
       interpret the 114-octet digest as a little-endian integer k.

   2.  Compute S_(i) = (r_(i) + kc_(i)s_(i)) mod L.  For efficiency,
       again reduce k modulo L first.

   3.  Return the values R_(i), S_(i) to the dealer.




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   The dealer then completes the signature by:

   0.  Computing the composite value S = S_(1) + S_(2) + ... + S_(n)

   1.  Verifying that the signature R, S is valid.

   2.  Publishing the signature.

6.  Test Vectors

6.1.  Direct Threshold Signature Ed25519

   The signers are Alice and Bob's Threshold Signature Service 'Bob'.
   Each creates a key pair:

   ED25519Alice's Key (ED25519)
       UDF:        ZAAA-GTSI-GXED-255X-XALI-CEXS-XKEY
       Scalar:     56271244081186130980636545017945156580516101894352492
           459594967614223399428880
       Encoded Private
     33 40 0E 22  D8 67 17 F4  8A 9F 6A 46  61 B4 0E AD
     8C D0 DD C3  79 CD 85 BD  95 5C 90 B9  6C CB 8C 23
       X: 11116793672970427161790264469280294507189044728140547954071022
           7976454124042406369344932655633664630560242213431409139866940
           284702002648469365756492647970
       Y: 61655404171611396573021808119108664749574235125343680206454285
           6299141386615046548323087409388548650272224487089895079970526
           0143544115364878870129761259200
       Encoded Public
     E2 AB 8F 37  62 C8 7B F9  E9 BC 59 0C  2E 99 A5 58
     0C C3 19 D5  CD DA 53 DF  3E C1 F0 C0  FE D3 55 5E
   ED25519Bob's Key (ED25519)
       UDF:        ZAAA-GTSI-G2ED-255X-XBOB-XSXK-EY
       Scalar:     54940772670153459146152925564198105262971485730889818
           986727608573229799020168
       Encoded Private
     68 9A 68 92  8A 06 17 84  35 3C B7 08  F8 56 00 3F
     BA 31 8C 42  B0 42 FE 2D  18 F2 7F AB  CD 10 49 F1
       X: 14271495069349838216379540196263140964032393512903842206168182
           5518850827098876289800868735522232908519794251130907125878675
           6343411484065706313568410880015
       Y: 28094328948004112428189466223757440886388684291254605355859923
           6240968229706795825282419594219442074647093851302547452470435
           9438513477629978601366725015573
       Encoded Public
     32 E5 8D 5E  66 B2 F9 E9  14 79 08 71  96 3B 9A 75
     A2 31 59 4B  8E ED 18 EF  BD FF 11 D4  47 2A 8C F4




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   The composite Signature Key A = A_(a) + A_(b)

   Aggregate Key = Alice + Bob ()
       UDF:        TBS
       Scalar:     26569330913556569171916721364983482306308422345436973
           56293312113171384684213
       Encoded Private
     B5 CE 0E B3  9C CF 18 99  CF 8D 4C BB  AE 81 79 1F
     CE 13 AA 3E  63 59 5B AC  8D 2C EB A4  55 C5 DF 05
       X: 67872685043898469012456949773240814121645904736114813455820339
           8688906486811443744733724675994181258029547346985079901494367
           752381127781166234556148580090
       Y: 36481740058369645484420180976004932062085375941522344052907594
           0118552792158551197107484892204562290802810655253510302448455
           4128548992118101415797909250954
       Encoded Public
     29 65 63 86  4F FB 10 8D  BA 7A 0A 68  04 6D 00 DA
     9B 1D C3 A4  AF BA 95 B4  5D 27 B4 35  00 2F DF 32

   To sign the text "This is a test", Alice first generates her value r
   and multiplies it by the base point to obtain the value R_(a):

   Alice:
   r_a =  26964569597283588823958971235480505062310410788972392836560088
       34751424110238
   R_a =
     00 76 86 F6  47 15 21 49  5D BC 40 61  C8 64 96 7B
     7A 77 B3 10  60 20 1D 50  C7 61 CF 27  CF 2F D3 E0

   Alice passes her value R_(A) to Bob along with the other parameters
   required to calculate i.  Bob then calculates his value R_(A) and
   multiplies it by the base point to obtain the value R_(b):

   Bob:
   r_b =  11540386535681085531005638949280260705343094809283082001566243
       3246392456462
   R_b =
     94 41 83 3D  D2 B1 FE 27  AD 28 EF 95  73 B5 5E E9
     4A D2 25 91  18 70 03 9C  C2 9B E6 F0  12 09 A9 5E

   Bob can now calculate the composite value R = R_(a) + R_(b) and thus
   the value k.

   R =
     C2 80 BF DE  1C 51 7D 11  34 74 7E A6  21 B8 65 B7
     1D 61 95 86  5E 21 7E 0A  FF 70 73 ED  1F AA 00 F0
   k =  1701029122539816017242429513798138718318684555383488776740740931
       828674348023



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   Bob calculates his signature scalar contribution and returns the
   value to Alice:

   Bob:
   S_b =  22552083956839743946415586440665231296779177689804281404209352
       33590198456755

   Alice can now calculate her signature scalar contribution and thus
   the signature scalar S.

   Alice:
   S_a =  64049035901867992939624616239692593314477498130612910883267027
       32679305425610
   S =  1423106408538511474630833704992788220268551222661811622745687027
       984049631376

   Alice checks to see that the signature verifies:

   S.B = R + kA =
       X: 87070195024672107970135683323709734641492765460361546859769679
           39147985870824
       Y: 51837226000842215744816324309390118726778447446024784069358107
           69430908552186

6.2.  Direct Threshold Signature Ed448

   The signers are Alice and Bob's Threshold Signature Service 'Bob'.
   Each creates a key pair:























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   ED448Alice's Key (ED448)
       UDF:        ZAAA-ITSI-GXED-44XA-LICE-XSXK-EY
       Scalar:     63495803583658817688110446314786076976347236361354035
           5597788771064742993095132758589292255654895141583596922516472
           738879360490167934280
       Encoded Private
     A0 53 4C 93  3C 34 00 76  AE 5D B5 4A  C2 71 5F 43
     E1 D6 63 2C  5C 56 53 C8  98 A0 8F 03  FF F5 22 96
     91 45 8C 2B  CF E3 FD 7E  2A 9E 0B D6  F4 CC 66 61
     43 62 72 7B  34 B4 79 92
       X: 24743197509267833262111449556527285120868167712209919570838426
           3466168536901525943558973091346360088759980994772668117646359
           614426660579
       Y: 21342899120576770537664462049685258390853729788303428349051130
           8752175233505795318243164692156369495328007220135137156078814
           081547431302
       Encoded Public
     0A 3B F3 27  E7 E1 67 63  2C 59 E2 1C  D1 84 C7 83
     E8 1E D1 68  9F 32 A1 16  99 00 5C DA  29 B9 6C 08
     E4 15 57 7E  E5 63 C2 32  08 23 41 68  5F 49 1F FF
     BC 4D CD 3A  4E A6 85 49  00
   ED448Bob's Key (ED448)
       UDF:        ZAAA-ITSI-G2ED-44XB-OBXS-XKEY
       Scalar:     72649803773199751564998543891898904839718409312910780
           0262041941160989643727331987658132182181970054245587322070535
           846720571414845714224
       Encoded Private
     BC 53 B4 74  3E A7 A7 FA  9F 05 9A BC  8C 22 26 15
     A1 4E BB 10  0E B5 59 6B  DE 9C 1B E9  F2 3C 65 42
     E7 B4 47 18  60 AC 18 A6  D2 78 B8 BC  CE F5 F4 28
     B2 3A FF 08  61 EF 6B 7C
       X: 58235851934808640621920816872959059172692411187640950432203039
           8116748997750134460231406698091317008063030408798536634284207
           667468558264
       Y: 34390767697909283892495761259186538632120422458392131201372282
           6056455656591826216381185634080685718154852726725624178995827
           091591132128
       Encoded Public
     93 63 5A 45  2D 4C 94 32  45 23 CD E2  A8 46 E4 78
     A0 80 59 DA  36 CB 6B 0C  06 64 6F BE  51 AB C0 BF
     1E DB A8 3F  2B 3B 80 0F  BF 00 E6 78  DD E0 83 E9
     AC 20 02 55  87 07 39 38  00

   The composite Signature Key A = A_(a) + A_(b)







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   Aggregate Key = Alice + Bob ()
       UDF:        TBS
       Scalar:     89488306051273634069773238262841883041784075539841550
           3672228636597106090916876462340541507950185640860121886233669
           49466515613996100051
       Encoded Private
     D3 29 DD AB  F6 0D 99 8B  75 65 B8 06  36 C9 3A 2C
     D4 08 C3 9B  7C F9 77 8C  68 29 0E 3D  5D C7 3E 00
     92 8B DC AE  26 FB 16 39  CD 25 1B 23  4A 5A 05 61
     1D 5C C4 70  0A C9 84 1F
       X: 17985659098670117617173315763082238685735647626871251468000984
           2080317111091696183607307614171726960576308774975742249260532
           199160570999
       Y: 31506323224859159594386181995639405170623657273945727288760063
           1624406694682617334725040181287905351066763414658543828623841
           509161975864
       Encoded Public
     9B 3E DF 49  55 40 9F 7B  EA 0B AA 40  B7 3D 15 82
     60 9F 7C 40  CF 67 DE 56  56 0D 03 87  63 3B 15 F2
     45 33 FE 48  BD 2D A0 A2  8B CC 74 DA  94 0F 39 00
     AC 39 CB 0A  9F A4 EB B0  00

   To sign the text "This is a test", Alice first generates her value r
   and multiplies it by the base point to obtain the value R_(a):

   Alice:
   r_a =  62565527493713919857865193519602365931012101673562577311838141
       54054766232567180678736888903730646941353640437421091869275000749
       6255278
   R_a =
     45 C6 4E FC  B3 CA 83 D6  12 9E 6C 9C  15 6A 63 E7
     50 B1 64 8A  F7 4C 45 BC  60 00 89 07  04 D3 6E ED
     55 15 E4 5D  BC 4F 61 A2  02 BD 37 FA  30 57 41 90
     CE 13 95 49  35 50 77 2D  00

   Alice passes her value R_(A) to Bob along with the other parameters
   required to calculate i.  Bob then calculates his value R_(A) and
   multiplies it by the base point to obtain the value R_(b):

   Bob:
   r_b =  68553581692021263576545409846872534945088226466517045820074454
       53563720130025310411907388118205768001112185289263525403420218781
       0825957
   R_b =
     21 27 22 F0  00 88 E8 D0  97 06 50 01  03 C8 37 24
     A2 59 35 40  2B 3A 6D F9  90 4F 05 22  0C 43 71 D2
     2C 0B 7D B0  4D 1D 95 7C  8C 0B 4D 19  19 6F 41 FF
     A4 6F BF 9C  D8 47 58 9C  00



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   Bob can now calculate the composite value R = R_(a) + R_(b) and thus
   the value k.

   R =
     44 E0 13 83  9E 90 CE 1B  8B F2 71 04  6C E9 AE 16
     F8 BE 72 22  3E 29 BE C4  FB 41 DF 47  31 CC E3 9C
     93 2A 71 6B  B1 32 3E EE  8D 00 98 B5  DE E9 FF 33
     1B 61 A9 F8  13 1E E2 5F  80
   k =  8559359322526978017385346909893058429591408900356440243822910260
       09124689192398260008095709124990846658648602608610131291879967363
       36428

   Bob calculates his signature scalar contribution and returns the
   value to Alice:

   Bob:
   S_b =  23225308473068317952317969434132903731627228506308768772207178
       49149309441840709888307940275398741726281603449230792143179257893
       4073681

   Alice can now calculate her signature scalar contribution and thus
   the signature scalar S.

   Alice:
   S_a =  15822521095547467691876148428778484160254716836077071285757716
       71810485457446614778358574022875083455217422316918648036906450706
       92022407
   S =  1814505194285429948710794537219177453341743968670794816297843456
       72541640163068576718936805041495762784558266184172725122437649626
       096088

   Alice checks to see that the signature verifies:

   S.B = R + kA =
       X: 20237958476218137983776478141950097846872887083363948281310043
           652612307164298
       Y: 42024667872160579723195748306782776970117818268757042395167033
           071351675224163

6.3.  Shamir Threshold Signature Ed25519

   The administrator creates the composite key pair









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   ED25519Aggregate Key (ED25519)
       UDF:        ZAAA-GTSI-GQED-255X-XAGG-REGA-TEXK-EY
       Scalar:     39348647608109113656999806950437958090469802387424444
           589375066079861075223816
       Encoded Private
     37 39 5E 7A  8B A5 A0 19  46 4B 58 22  EA 24 A5 71
     45 2C 2A AC  7A 3E FB CA  CE 3F D4 12  9A BA EB 70
       X: 14198837758377867455716504277518729070915183249890461230792115
           9904969716778427995951234766002164511738587575257530388758374
           7824906047250057721855068523970
       Y: 20211025649802071998810413948266748565975140520947927724517956
           2067625505077751598018629551746824533726709810990193455662385
           6152736116303441031851305458040
       Encoded Public
     6E 13 79 B4  39 DA 97 9C  5A 34 CE 79  CD 1B 50 DF
     A0 76 AD 49  81 6D 52 59  A4 2C DB CE  44 FF 3E F5

   Three key shares are required for Alice, Bob and Carol with a
   threshold of two.  The parameters of the Shamir Secret Sharing
   polynomial are:

   a0 = 3934864760810911365699980695043795809046980238742444458937506607
       9861075223816
   a1 = 2784115633880431304931825595251063942290923891174972818466623907
       0091375688

   The key share values for the participants are

   xa = 1
   ya = 3191460877786606900183192391175497525607129829436656287549977627
       503895344559

   xb = 2
   yb = 3219302034125411213232510647128008165030039068348406015734643866
       573986720247

   xc = 3
   yc = 3247143190464215526281828903080518804452948307260155743919310105
       644078095935

   Alice and Carol are selected to sign the message "This is another
   test"

   The Lagrange coefficients are:







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   la = 3618502788666131106986593281521497120428558179689953803000975469
       142727125496
   lc = 3618502788666131106986593281521497120428558179689953803000975469
       142727125494

   Alice and Carol select their values ra, rc

   ra = 5436932162843462293266539006078878918830367554401620912648914278
       387841243984
   Ra =
     8D 87 BC 1E  A9 84 92 05  56 93 31 35  B4 76 C0 98
     EA F4 60 12  A5 5F D6 9B  0A 69 41 1A  36 45 7A 83

   rc = 1538800080663244406394301816847739559173060560330113907829931117
       643929735504
   Rc =
     4B 65 A9 E9  87 7B 90 37  04 1F A1 64  16 69 F0 C3
     0E 45 1B F4  5A 55 5C F1  99 A4 71 2C  B0 00 EB 15

   The composite value R = R_(a) + R_(c)

   R =
     3B C4 76 D4  33 AC 43 21  D7 7E 4F 2D  58 E9 4B 92
     70 F2 36 37  2C D9 28 EA  4E 91 57 80  2F D4 21 E3

   The value k is

   k = 65022205971444234659067578861063910015740009952224679471431141734
       79939649783

   The values R and k (or the document to be signed) and the Lagrange
   coefficients are passed to Alice and Carol who use them to calculate
   their secret scalar values:

   sa = 1168688528013779243288195305241749167982136564465030628323990972
       113115891344;
   sc = 1994931193434023343845678829981237718202084026059875931041320416
       320688077527

   The signature contributions can now be calulated:

   Sa = 1642088151518325934823467054868266591252355171010295386239413063
       0650672438
   Sc = 2567122318027391708469410793879025926580216565704722275843007694
       783454379696

   The dealer calculates the composite value S = S_(a) + S_(b)




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   S =  2583543199542574967817645464427708592492740117414825229705401825
       414105052134

   The dealer checks to see that the signature verifies:

   S.B = R + kA =
       X: 38305755396533183244339161726284183266009090657176834266005039
           016134810784785
       Y: 29407699810490002191056006520525329213760985707980252137477485
           352643976612632

6.4.  Shamir Threshold Signature Ed448

   The administrator creates the composite key pair

   ED448Aggregate Key (ED448)
       UDF:        ZAAA-ITSI-GQED-44XA-GGRE-GATE-XKEY
       Scalar:     50890460656419721531273587958284096015810982760541575
           4207268050539683337837216003977228732536078674802149039736292
           653681850024283019712
       Encoded Private
     78 22 7E 3B  89 95 80 5D  04 19 DC 27  F1 7F 9B E4
     86 2B 0B DD  55 64 EE 04  19 49 4D DE  B9 04 3B 9E
     8B 7D DC EC  EC 8F DD 1D  E7 88 86 FD  11 FD 78 EF
     1A 8B 84 8F  77 00 73 65
       X: 44109173355278142669484438370724914685176368933547176239809629
           7503768465595321590690311221269514682222687386378631457535068
           446135118173
       Y: 53219402718535721212460981200104434180077825188675868294070079
           5084662920552823356888138706016038637934794839496624474125511
           419755284720
       Encoded Public
     43 61 20 A0  B1 DF AA BD  6B 55 00 97  A3 BE CB B8
     09 57 20 88  16 69 E4 B9  E1 7E 9C 13  C0 41 5B CB
     4D 3E E4 99  2E 2D 48 89  1C C0 FB 26  58 C2 DD 5C
     C1 DC 17 82  D7 A0 43 EE  80

   Three key shares are required for Alice, Bob and Carol with a
   threshold of two.  The parameters of the Shamir Secret Sharing
   polynomial are:

   a0 = 5089046065641972153127358795828409601581098276054157542072680505
       39683337837216003977228732536078674802149039736292653681850024283
       019712
   a1 = 1602922392607822292317783703384491077419282575745896489241768437
       65938863574713864292893351873520049039965264366659426161520599832
       630033




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   The key share values for the participants are

   xa = 1
   ya = 1240678026032742766325213940052866671348070646645168579203272449
       20184189527311111121534818534521595952183422975677092929049356536
       700408

   xb = 2
   yb = 1026503607901547832269688123717346412883249820672769917741315388
       90977049140485389698232415116349269028838393633244856785796200509
       680662

   xc = 3
   yc = 8123291897703528982141623073818261544184289947003712562793583286
       17699087536596682749300116981769421054933642908126206425430444826
       60916

   Alice and Carol are selected to sign the message "This is another
   test"

   The Lagrange coefficients are:

   la = 9085484053695086131866547598600056679420517008591475753518627489
       75730019807697928580978776458461879816551468545458311523868779298
       24891
   lc = 9085484053695086131866547598600056679420517008591475753518627489
       75730019807697928580978776458461879816551468545458311523868779298
       24889

   Alice and Carol select their values ra, rc

   ra = 2719648808372152695303324813004886411509189522037119771348270732
       01317204344949815636374307412539793080438754521149038747587752804
       50241
   Ra =
     5F 55 7D 9B  FF 93 7E 29  29 17 86 F3  5B 86 2F 7E
     00 2A 10 A4  C4 1B 5C 62  62 D8 C8 CD  50 AD 5D E0
     F6 6A 79 63  A4 E3 59 E7  3B 62 83 5B  6B 39 94 42
     2A C5 F1 3D  8C A2 EC 91  00

   rc = 9050006003211425294899204903626951978904214154147272814801034120
       83634683685763499954420274680944008271220208805334510963648274014
       41912
   Rc =
     57 94 25 6D  A6 B2 E3 AB  59 5D 6F 3D  F8 2E C6 9E
     3D 7C F8 04  C2 D6 98 5D  CF 3E 15 60  08 AB 8B 66
     19 AB 0A 6D  1A E6 B1 45  D1 A6 0B 21  23 CA 59 A7
     F3 A3 DE 35  8E 04 1D 02  80



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   The composite value R = R_(a) + R_(c)

   R =
     94 87 5A 3A  CE 75 7C 97  C0 24 93 5D  00 4C 0A AD
     8D 5B 74 9A  2F 1A 16 EE  85 8C EC EE  C3 17 91 68
     B2 80 4B 73  63 AC A1 04  65 82 16 80  B0 E1 48 3A
     02 63 18 A4  ED 88 15 3F  00

   The value k is

   k = 28597319766372023288919874253606353528116314247955304554628908699
       14741698440791352191812187685479498956177113463735604193977889389
       5471

   The values R and k (or the document to be signed) and the Lagrange
   coefficients are passed to Alice and Carol who use them to calculate
   their secret scalar values:

   sa = 4392022831009692311451139035928867113800256824945771810118317585
       13028032942708096610647251009001796496484075442397708880027894540
       0833;
   sc = 1410932215853840777266228366029098258674888904368109522564046333
       64261049584709751578730749442603904910563611563685351983502233618
       319321

   The signature contributions can now be calulated:

   Sa = 1237911193328598461098720576842938619833847324963757947381135045
       60734496037707134982632828188157194256327767616978405671604912622
       997176
   Sc = 4352294610170602596603144511079540327609274906403841682361237464
       38757029065808909094759113169492383225515142920572639508132292294
       64825

   The dealer calculates the composite value S = S_(a) + S_(b)

   S =  1673140654345658720759035027950892652594774815604142115617258792
       04610198944288025892108739505106432578879281909035669622418141852
       462001

   The dealer checks to see that the signature verifies:

   S.B = R + kA =
       X: 31370116705528265987661207265970557523081230091184475728092100
           801345054140398
       Y: 14983649831297656181541916312764508612330840642761296389133763
           572240307037592




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7.  Security Considerations

   All the security considerations of [RFC7748], [RFC8032] and
   [draft-hallambaker-threshold] apply and are hereby incorporated by
   reference.

7.1.  Rogue Key attack

   The rogue key attack described in [draft-hallambaker-threshold] is of
   particular concern to generation of threshold signatures.

   If _A_ and _B_ are public keys, the intrinsic degree of trust in the
   composite keypair _A_ + _B_ is that of the lesser of _A_ and _B_.

7.2.  Disclosure or reuse of the value r

   As in any Schnorr signature scheme, compromise of the value _r_
   results in compromise of the private key.  The base signature
   specification [RFC8032] describes a deterministic construction of _r_
   that ensures confidentiality and uniqueness for a given value of _k_.

   As described above, this approach is not applicable to the generation
   of values of _r_(i)_ to compute threshold signature contributions.
   Accordingly the requirements of [RFC4086] regarding requirements for
   randomness MUST be observed.

   Implementations MUST NOT use a deterministic generation of the value
   _r_(i)_ for any threshold contribution except for calculating the
   final contribution when all the other parameters required to
   calculate _k_ are known.

7.3.  Resource exhaustion attack

   Implementation of the general two stage signing algorithm requires
   that signers track generation and use of the values _r_(i)_ to avoid
   reuse for different values of _R_(i)_. Implementations MUST ensure
   that exhaustion of this resource by one party does not cause other
   parties to be denied service.

7.4.  Signature Uniqueness

   Signatures generated in strict conformance with [RFC8032] are
   guaranteed to be unique such that signing the same document with the
   same key will always result in the same signature value.

   The signature modes described in this document are computationally
   indistinguishable from those created in accordance with [RFC8032] but
   are not unique.



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   Implementations MUST not use threshold signatures in applications
   where signature values are used in place of cryptographic digests as
   unique content identifiers.

8.  IANA Considerations

   This document requires no IANA actions.

9.  Acknowledgements

   [TBS]

10.  Normative References

   [draft-hallambaker-mesh-udf]
              Hallam-Baker, P., "Mathematical Mesh 3.0 Part II: Uniform
              Data Fingerprint.", Work in Progress, Internet-Draft,
              draft-hallambaker-mesh-udf-08, 6 January 2020,
              <https://tools.ietf.org/html/draft-hallambaker-mesh-udf-
              08>.

   [draft-hallambaker-threshold]
              Hallam-Baker, P., "Threshold Key Generation and Decryption
              in Ed25519 and Ed448", Work in Progress, Internet-Draft,
              draft-hallambaker-threshold-00, 5 January 2020,
              <https://tools.ietf.org/html/draft-hallambaker-threshold-
              00>.

   [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/rfc/rfc2119>.

   [RFC4086]  Eastlake 3rd, D., Schiller, J., and S. Crocker,
              "Randomness Requirements for Security", BCP 106, RFC 4086,
              DOI 10.17487/RFC4086, June 2005,
              <https://www.rfc-editor.org/rfc/rfc4086>.

   [RFC7748]  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
              for Security", RFC 7748, DOI 10.17487/RFC7748, January
              2016, <https://www.rfc-editor.org/rfc/rfc7748>.

   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017,
              <https://www.rfc-editor.org/rfc/rfc8032>.

11.  Informative References



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   [draft-hallambaker-mesh-developer]
              Hallam-Baker, P., "Mathematical Mesh: Reference
              Implementation", Work in Progress, Internet-Draft, draft-
              hallambaker-mesh-developer-09, 23 October 2019,
              <https://tools.ietf.org/html/draft-hallambaker-mesh-
              developer-09>.

   [Komlo]    Komlo, C. and I. Goldberg, "FROST: Flexible Round-
              Optimized Schnorr Threshold Signatures", 2020.

   [RFC5860]  Vigoureux, M., Ward, D., and M. Betts, "Requirements for
              Operations, Administration, and Maintenance (OAM) in MPLS
              Transport Networks", RFC 5860, DOI 10.17487/RFC5860, May
              2010, <https://www.rfc-editor.org/rfc/rfc5860>.

   [Shamir79] Shamir, A., "How to share a secret.", 1979.



































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