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/ (http://whatever)Discussion of this draft should take place on the CFRG mailing list (cfrg@irtf.org), which is archived at . Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet- Drafts is at https://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." This Internet-Draft will expire on 10 September 2020. Copyright Notice Copyright (c) 2020 IETF Trust and the persons identified as the document authors. All rights reserved. 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. Hallam-Baker Expires 10 September 2020 [Page 1]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 Please review these documents carefully, as they describe your rights and restrictions with respect to this document. 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. Hallam-Baker Expires 10 September 2020 [Page 2]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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 Hallam-Baker Expires 10 September 2020 [Page 3]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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. Hallam-Baker Expires 10 September 2020 [Page 4]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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]. Hallam-Baker Expires 10 September 2020 [Page 5]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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: Hallam-Baker Expires 10 September 2020 [Page 6]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 _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) Hallam-Baker Expires 10 September 2020 [Page 7]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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 Hallam-Baker Expires 10 September 2020 [Page 8]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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_ Hallam-Baker Expires 10 September 2020 [Page 9]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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_. Hallam-Baker Expires 10 September 2020 [Page 10]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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. Hallam-Baker Expires 10 September 2020 [Page 11]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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. Hallam-Baker Expires 10 September 2020 [Page 12]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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. Hallam-Baker Expires 10 September 2020 [Page 13]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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] Hallam-Baker Expires 10 September 2020 [Page 14]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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. Hallam-Baker Expires 10 September 2020 [Page 15]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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 Hallam-Baker Expires 10 September 2020 [Page 16]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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 Hallam-Baker Expires 10 September 2020 [Page 17]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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: Hallam-Baker Expires 10 September 2020 [Page 18]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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) Hallam-Baker Expires 10 September 2020 [Page 19]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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 Hallam-Baker Expires 10 September 2020 [Page 20]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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 Hallam-Baker Expires 10 September 2020 [Page 21]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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: Hallam-Baker Expires 10 September 2020 [Page 22]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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) Hallam-Baker Expires 10 September 2020 [Page 23]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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 Hallam-Baker Expires 10 September 2020 [Page 24]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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 Hallam-Baker Expires 10 September 2020 [Page 25]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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 Hallam-Baker Expires 10 September 2020 [Page 26]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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. Hallam-Baker Expires 10 September 2020 [Page 27]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 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 Hallam-Baker Expires 10 September 2020 [Page 28]

Internet-Draft Threshold Signatures in Elliptic Curves March 2020 [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. Hallam-Baker Expires 10 September 2020 [Page 29]