Threshold Signatures in Elliptic Curves
ThresholdSecrets.com
phill@hallambaker.com
Threshold Cryptography
Elliptic Curve
Threshold Signature
Digital Signature
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 nondeterministic. 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/Discussion of this draft should take place on the CFRG mailing list (cfrg@irtf.org), which is archived at .
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.
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 multisignatures, 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 multisignatures or threshold signatures to be inherently superior, it is more useful to regard both as two points on a continuum of choices:
 Multisignatures

Multiple digital signatures on the same document. Multisignatures 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.
 Multiparty 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 to define a scheme that creates threshold signatures that are computationally indistinguishable from those produced according to the algorithm specified in . The scheme does not support the creation of aggregate signatures.
The approach used is based on that developed in FROST . This document describes the signature scheme itself. The techniques used to generate keys are described separately in .
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 scheme is modified using Shamir secret sharing and Lagrange coefficients for the case that n > t.
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 multisignatures.
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).
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.
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.
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.
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 .
This document extends the approach described in to support threshold signatures. The deterministic mechanism described in specification is used to generate the private keys used in the test vectors.
The implementation status of the reference code base is described in the companion document .
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 . 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 nondeterministic 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 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:
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 = (Sr).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.
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}
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 . 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 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 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:
 The dealer determines the values F, C and PH(M) as specified in and transmits them to the signers {1, 2, ... n}.
 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 .
 The dealer calculates the value R = R_{1} + R_{2} + ... + R_{n} and transmits R and A to the signers {1, 2, ... n}.
 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 .
 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 . 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.
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
 Where

c_{i} = 1 for the additive shared threshold signature
c_{i} = l_{i} for the Shamir shared threshold signature
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:
 All the signers except the final signer F generate their value r_{i} and submit the corresponding value R_{i} to the dealer
 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}.
 The final signer generates its value r_{F}
 The final signer calculates the value R_{F} from which the values R and k can now be determined.
 The final signer calculates its key share contribution S_{F} = rF + kcFsF mod L.
 The final signer returns the values S_{F} and R to the dealer.
 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 realworld 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.
The use of Kocher side channel resistance as described in 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:
 A generates a random nonce n_{A} and sends it to B with the other parameters required to generate the signature.
 B generates a random nonce n_{B}
 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.
 B calculates the signature parameters as before except that the threshold signature contribution is now S_{B} = rB + k(cBsB + x) mod L.
 B returns the nonce n_{B} to A with the other parameters.
 A calculates the blinding factor x using the same approach as B
 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. ) for the digest function.
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.
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.
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 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.
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.
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.
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.
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_{n1})
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 adhoc groups of signers using their existing signature keys as signature key shares presents serious technical challenges that are outside the scope of this specification.
The means by which threshold shares are created is described in .
The dealer selects the signers who are to construct the signature. Each signer then computes the value R_{i}:
 Randomly generate an integer r_{i} such that 1 r_{i} L.
 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
 At some later point, the dealer MAY complete the signature by returning the values F, C, A and R as specified in together with the key multiplier coefficient c_{i}. The signers MAY then complete their signature contributions:
 Compute SHA512(dom2(F, C)  R  A  PH(M)), and interpret the 64octet digest as a littleendian integer k.
 Compute S_{i} = (r_{i} + kc_{i}s_{i}) mod L. For efficiency, again reduce k modulo L first.
 Return the values R_{i}, S_{i} to the dealer .
The dealer then completes the signature by:
 Computing the composite value S = S_{1} + S_{2} + ... + S_{n}
 Verifying that the signature R, S is valid.
 Publishing the signature.
The means by which threshold shares are created is described in .
The dealer selects the signers who are to construct the signature. Each signer then computes the value R_{i}:
 Randomly generate an integer r_{i} such that 1 r_{i} L.
 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
 At some later point, the dealer MAY complete the signature by returning the values F, C, A and R as specified in together with the key multiplier coefficient c_{i}. The signers MAY then complete the signature contributions:
 Compute SHAKE256(dom4(F, C)  R  A  PH(M), 114), and interpret the 114octet digest as a littleendian integer k.
 Compute S_{i} = (r_{i} + kc_{i}s_{i}) mod L. For efficiency, again reduce k modulo L first.
 Return the values R_{i}, S_{i} to the dealer.
The dealer then completes the signature by:
 Computing the composite value S = S_{1} + S_{2} + ... + S_{n}
 Verifying that the signature R, S is valid.
 Publishing the signature.
The signers are Alice and Bob's Threshold Signature Service 'Bob'. Each creates a key pair:
ED25519Alice's Key (ED25519)
UDF: ZAAAGTSIGXED255XXALICEXSXKEY
Scalar: 312191303806394376947696888962276115420485359001
34467943432016761653342335248
Encoded Private
10 AE C0 C2 16 65 9B 4F 7C 9D DE 82 3E 49 7F D4
9B 14 BB F8 2D 9F 0C 11 24 D7 15 E3 43 79 57 20
X: 13697699435406080999251131063344049965140553452
752305353714819106646919347160064793506327635954342719144289
2305566686088586980395284289746495530409889930
Y: 278793875610616080844162800185864399625503938157
569374174700414845758479331294424147393776831767266487579098
7675375777043504113387553916769515911310193558
Encoded Public
45 16 53 7C 26 50 CF DA F1 A4 DF 4C 45 DC 3D 95
4E B6 8E EB A6 5A 27 D6 CD 5B 43 C5 F4 06 53 ED
ED25519Bob's Key (ED25519)
UDF: ZAAAGTSIG2ED255XXBOBXSXKEY
Scalar: 567212843891509414800308620158891720685508995620
72140666211075925337851277632
Encoded Private
E5 CD 34 01 FD 8C 0E 27 81 4B 11 DD 12 68 50 A1
4B 5A D5 E1 E1 41 D7 68 5F 51 ED B4 3A 84 58 5C
X: 13809282472298084436735987888897423507149580966
952791761446670884044433963975178482398144657564565223270588
5322459642470946347570575475534141406285323257
Y: 263684226342871984706317411760423095947068088366
393546798602378437432707482089806653755881399592963068751759
9645362525866308283171284327931970404321458677
Encoded Public
F1 5F C0 78 F8 32 49 2C D9 64 CC 2B CF 90 5C 4F
23 EA BB F8 38 99 C5 FE F3 AA 67 BE AB EC D2 5E
The composite Signature Key A = A_{a} + A_{b}
Aggregate Key = Alice + Bob ()
UDF: TBS
Scalar: 109634784180323260712231215560085272031403914964
7717337619681427565742601012
Encoded Private
34 33 AB 10 9A 09 A9 61 65 8B 3A EC 58 21 FB 2D
0D 45 74 49 45 BA E2 CF A8 98 C2 94 C9 82 6C 02
X: 83837675294300852842901121613445594296352372347
711317409367737761568353629718805151940195325485285476438422
923698718220652243749390297055882388709313280
Y: 160553422944358144751060009820735322036903773802
361117046457476895165059738086663330972263850675453249990301
0398473811263196653225446124160025082144761534
Encoded Public
48 1A 27 66 06 AF 4E 3C 20 A4 02 CD 8A 13 46 99
02 B7 75 F8 AC D4 7E 89 68 FB 68 EB D8 EF 4A C7
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: 304554767184319354570910632330847275245170925807
471005852065308719163006873
R_a =
4D D3 35 87 4D D2 A9 35 79 CF 33 11 4E 3C 4C 86
A3 46 83 F8 77 2B 69 B9 BE 8A D0 03 5B 07 1B 3D
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: 211453017009495036941255861540216441186480571963
4308458833747992420728606380
R_b =
9E 92 BB 50 CD DA 6A 97 70 83 54 82 4A C5 A2 5E
CF 0C 53 65 19 55 74 E0 66 F7 49 0B F1 9E 42 C1
Bob can now calculate the composite value R = R_{a} + R_{b} and thus the value k.
R =
6D 04 0A 15 D3 26 95 7E 60 4B 31 B9 40 03 34 24
C1 E2 94 CE 4C 2D 67 AC 23 F9 09 EB 01 35 27 BC
k: 141815370378674010309754096097287620917810351837
8705827473796209228380712742
Bob calculates his signature scalar contribution and returns the value to Alice:
Bob:
S_b: 536363782415883723339410938370364298390876250983
1717008575694722457748759583
Alice can now calculate her signature scalar contribution and thus the signature scalar S.
Alice:
S_a: 268700124091647466934677864874812484315783329013
4699034602004495702143596514
S: 813633487743049688767701469408773586209479440586
508437175748279874438105108
Alice checks to see that the signature verifies:
S.B = R + kA =
X: 303888032885010422619559054491601097433934565460
00461769461185182044799973523
Y: 242312166466765598632540958273647355363821360549
75658447998613064394185428840
The signers are Alice and Bob's Threshold Signature Service 'Bob'. Each creates a key pair:
ED448Alice's Key (ED448)
UDF: ZAAAITSIGXED44XALICEXSXKEY
Scalar: 672286477331130983513039743350616227864346753924
962787860729757222511999618443513569403793186398096717924945
854846544396984088344823264
Encoded Private
6F 85 B1 91 9A 37 06 A6 B2 15 79 AD 5B 69 16 6A
5A CD C8 17 D4 14 1F 68 DA 97 C5 B4 44 79 CE EA
3C 17 7B E1 29 44 70 DF 41 C8 98 38 1E 7C 9B 3B
03 63 6F 85 E8 39 31 91
X: 526046019655043632868470952286947529492283092344
122476077151423645243648974512182548405702873560533846673262
767064019365470830861106049
Y: 145374550785380850812934424757986866673485237047
938554544492694946608060986459495807055455048208713991919477
720250115717234689256856152
Encoded Public
59 55 F4 7A 66 08 91 35 F8 15 63 F4 90 91 7F 38
12 E3 49 22 51 F8 BC 4A 41 C9 44 59 5A 64 9B 40
0B C5 7E 53 48 0F 32 12 90 32 69 38 47 28 94 BB
99 D1 16 6F 2D D5 3D 4F 80
ED448Bob's Key (ED448)
UDF: ZAAAITSIG2ED44XBOBXSXKEY
Scalar: 455052626698262385397736547727159423941520792904
908612603542850909167215987713902322619933929404455741806848
064294945283113799683261212
Encoded Private
CA 15 22 BD F4 0F 9E 0A EC A7 61 79 BE 9E E3 38
BF 93 D3 5B B3 E6 FC F0 A7 5B 7C F0 E7 B5 89 F6
2E F6 D1 0E 72 49 4D DF 34 5E 2F 7C 9E 42 1D 85
AB AB 30 BD 68 C6 3E 35
X: 752024108200272710832187535557164455078689734595
171189993383259892607253027500878543439908750525763880661232
171322059854852522782265
Y: 619329873102159676791326142073166790594683111409
729383584199833441028484525583699421181422168190856074786324
020492214873796495570056511
Encoded Public
76 2B FC F8 AC 96 79 DE 1C 72 07 65 DD 49 5B 28
C7 04 CB A8 A5 96 3D D9 9E 23 FA 05 83 15 33 95
85 82 F8 CF A3 7A 2F 24 F8 EB D6 AE 20 0A 25 D0
44 1A F9 C0 86 D7 87 B7 00
The composite Signature Key A = A_{a} + A_{b}
Aggregate Key = Alice + Bob ()
UDF: TBS
Scalar: 370810175859830330867905792457688502754055057988
943100420373093608031918369199015948491953656482966798700316
64591515851455352870185802
Encoded Private
4A AB 7A BB 2D 95 72 75 B1 3A 1D 22 24 17 76 2D
A1 D5 55 94 67 35 8C E7 A1 A0 ED 0C E7 88 FF 9F
6E 2F 70 80 89 F5 01 2A C0 AD 4C 4E 7B 90 68 6C
F4 53 BA 32 9B 70 0F 0D
X: 583249553407699999284154112964835446252412293188
857058051552519639906663406776316984154017062023869075790536
30514579317017660114474427
Y: 518040437562811181169413740718290938351269168888
257124107164689245721852001077758864406412789756149699111633
051823234569886260996269341
Encoded Public
34 70 8D 08 DE 63 0B A6 49 2A 33 D8 B7 15 A9 84
A4 87 F6 B6 C7 4B 1C AE 5A 1F 7C 4B 12 70 FB CF
5A A9 3C 20 31 BA 9A 53 A0 FE 2A 43 24 97 06 F8
DA 40 0D 88 E3 D9 DE 2E 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: 177274411620382331885613075298112943809338383162
174753240420771790910425395072244220798681166866026098695507
899343208324484682831848544
R_a =
EC 4E A5 8E BE 3E DC C4 AA DC 67 65 D9 0D B1 24
82 97 7E 57 83 4C BC CA D0 68 98 72 B8 1A F2 61
E2 CC 2C AF AE F3 A9 8A 16 DF F1 1E 91 EC E2 FC
3C 09 DD 25 AB 6E 7D 02 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: 914384096261894497392108217856312041404007433725
737987383860198449871749775140845071471047786135423317371867
85115509488719418941845277
R_b =
7E 8C 1C B1 EA 5F 9E 58 F4 E1 4C AB 9D CF C9 4B
42 8E 01 F4 B4 27 AA 74 D8 66 E8 5B D5 8C 23 10
31 A7 99 FC 69 3D 4A 81 56 9B 31 07 A4 7C 3B 83
9D 79 6B 12 08 D0 2C CC 80
Bob can now calculate the composite value R = R_{a} + R_{b} and thus the value k.
R =
AE 82 1A 7D A9 B6 35 7A 3E CF 58 95 9E 97 3A 45
62 77 E5 DB F6 B5 B7 5F 5A 27 A0 29 0A 4D D4 2C
9F 39 97 BB A3 55 DC C8 BF 02 C1 8F CE 8E 92 D9
97 38 19 6B 73 07 19 FE 00
k: 120812996308912938173796275547209139305812356493
218326541177300961108122972500434047109753132978603955927000
049579948492518627324723657
Bob calculates his signature scalar contribution and returns the value to Alice:
Bob:
S_b: 131642872805495450312819108720996394951179966909
422047047106195752645599694599197550402234644818088495417690
338344590538474713569000455
Alice can now calculate her signature scalar contribution and thus the signature scalar S.
Alice:
S_a: 135660088492295551401234992018778138767930165450
263985669170229118586124164862063605687569508563259045492355
878328728103423148386029682
S: 855932802238892790767231487677734001306997921878
565176459038750760857198979216754398940488616889715775997525
07581656337124106095380358
Alice checks to see that the signature verifies:
S.B = R + kA =
X: 225696302738968030426017676274023389357161605713
77841610835599611770420843463
Y: 366869559370146495596698412318900310141515842791
80057095036770676314039224332
The administrator creates the composite key pair
ED25519Aggregate Key (ED25519)
UDF: ZAAAGTSIGQED255XXAGGREGATEXKEY
Scalar: 367238470592488326468789252109412889361910680229
03089760692844779165588879504
Encoded Private
FE 48 94 1F EB 3D 28 E1 61 81 E2 1E E1 CF F2 1E
1E 70 91 30 DF 98 9F 1C 34 EB BB 74 C5 C8 07 EB
X: 143576564277195758046684172284175869008525477709
640743490221115123376609940386394888392330104965579307772627
313244177612005636942740116142030215202393600
Y: 844838272625277895849027219595751726665225134917
547580682441821283235675507225396641352769322822815561632929
543097074319051436285787045255908364074589900
Encoded Public
DF E8 0A 2B E9 6C 53 C0 AB 9B BC BC 39 95 9A 61
9C 33 2E 22 24 A7 F7 F2 21 06 AC 6D 01 5D 0B E2
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 = 367238470592488326468789252109412889361910680229030897606928
44779165588879504
a1 = 338318986010852412366041291642398977925879116179901190258252
3043521639876163
The key share values for the participants are
xa = 1
ya = 392200903269604570067340531215030751116427738780256363326561
3131259957500722
xb = 2
yb = 681933154723076103606316655313030495659521902216679298461852
36496143125896
xc = 3
yc = 345138317558083173402104458195529282882474335202067983242870
8280017783002059
Alice and Carol are selected to sign the message "This is another test"
The Lagrange coefficients are:
la = 361850278866613110698659328152149712042855817968995380300097
5469142727125496
lc = 361850278866613110698659328152149712042855817968995380300097
5469142727125494
Alice and Carol select their values ra, rc
ra = 183324669043432475889177343290874841197580255869543278119963
6861783584446189
Ra =
54 9B F7 DF F9 4E FA 95 DE D8 27 4E 0D AD A4 81
4B D7 1F CA 21 B4 B2 7D D6 06 4F 59 28 41 87 92
rc = 161475617387612449161820030097871350703159449773365674965881
1083650872383824
Rc =
95 25 84 AA B3 F7 39 14 10 67 E8 45 21 68 67 76
11 73 88 28 D6 50 71 0F D4 21 67 12 98 CC 80 C7
The composite value R = R_{a} + R_{c}
R =
22 B8 67 CA 63 65 00 7D AD 85 96 83 FD 86 CB 92
88 E7 E7 73 F9 3E 48 8D AE 7E 43 E3 2D AC 5F 7F
The value k is
k = 300461162806835024067835386157211645931753120888211250552921
1847964183554048
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 = 588301354904406855101010796822546126674641608170384544989841
9696889936251083
sc = 189281120087571523997607099054385070601618650367961388678662
1329133835624465
The signature contributions can now be calulated:
Sa = 128238145780800590303153436041482532412837879827391619588852
5270458485017264
Sc = 425392829290723185011341307483608802636424146029130066221513
8481933637068796
The dealer calculates the composite value S = S_{a} + S_{b}
S = 553630975071523775314494743525091335049262025856521685810366
3752392122086060
The dealer checks to see that the signature verifies:
S.B = R + kA =
X: 386656916198500913898974718482142916932478347228
53747339220827400653000735042
X: 272627300420736046999889168909581852465293124122
69338237353604076047710443564
The administrator creates the composite key pair
ED448Aggregate Key (ED448)
UDF: ZAAAITSIGQED44XAGGREGATEXKEY
Scalar: 723088510822916843359337925516642493307623385482
113107480846498794254549074097051759295396782499503452909258
978468506553055366989547456
Encoded Private
59 DC 8A 5F 5E AF 8C FA 96 19 F8 EE 78 13 00 12
33 0E 12 80 2D 25 E6 EF E8 E2 56 B5 83 6A 0C CF
DC 11 96 A5 A5 D1 39 AA 34 25 0B 52 ED 9F 38 92
5D 9F 7B BC B9 BC 86 45
X: 600163199260212879671026282440221570752543874569
276531213297382365938924845597497264583528185273760383031589
25167107013312482098672476
Y: 568007995844826855892481230051783440873263817862
016100095069663100696528804467952219402043387612562057320585
561865068046226655443122582
Encoded Public
ED C3 90 99 38 0B 8F CD 60 29 24 04 6C DE 52 33
A2 07 3E 56 8D 27 B5 B9 21 60 CF E9 E7 9D D6 4A
11 47 20 E6 9D FE 75 C7 04 14 70 18 B4 52 10 83
D0 EC 98 BD F5 E6 E3 D5 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 = 723088510822916843359337925516642493307623385482113107480846
49879425454907409705175929539678249950345290925897846850
6553055366989547456
a1 = 253947292459661473537056525563228503335644586909535883819036
29593189848969312285942082195525418378983878506351793852
012228645407360825
The key share values for the participants are
xa = 1
ya = 216445157732761001637197701849608092875464834857486355812599
29206860382197250994836594571141148378583546590493895709
346188988958309165
xb = 2
yb = 470392450192422475174254227412836596211109421767022239631635
58800050231166563280778676766666566757567425096845689561
358417634365669990
xc = 3
yc = 724339742652083948711310752976065099546754008676558123450671
88393240080135875566720758962191985136551303603197483413
370646279773030815
Alice and Carol are selected to sign the message "This is another test"
The Lagrange coefficients are:
la = 908548405369508613186654759860005667942051700859147575351862
74897573001980769792858097877645846187981655146854545831
152386877929824891
lc = 908548405369508613186654759860005667942051700859147575351862
74897573001980769792858097877645846187981655146854545831
152386877929824889
Alice and Carol select their values ra, rc
ra = 597061760142011172791892795402895896258128262343360777519356
25449868368724094466154383697912358541824000038036669052
993742082445345445
Ra =
35 E5 61 55 78 E7 27 24 55 2D C9 76 4B 49 2B 46
16 E3 FA 97 2F 9A E6 47 1B 22 CD 2D 54 F5 1A 1C
7A A7 67 B0 CE 65 84 05 39 33 0E A6 33 5E 67 BD
58 CA ED 7E F3 EE DD 59 00
rc = 134782448747121628062621888359149856538463227176060814858572
40759552504564125328966600869858667833252205589918638910
3286110005426517374
Rc =
3C 7B 81 57 A9 C9 23 90 E2 1D 82 42 7B 0C D9 88
E7 E5 98 81 DB 52 76 E3 77 67 E8 34 DC D7 DC 81
20 B6 3A 6F EC E8 0C D9 0E BC C5 5A A0 E6 91 D5
D9 14 0B 60 C2 A4 D0 A8 80
The composite value R = R_{a} + R_{c}
R =
00 27 F8 F7 A7 39 F7 AE B6 8F E4 0F A8 4E 3A 71
45 87 6A 1E C4 C3 29 52 CC 8A C7 6C 36 47 4B A6
04 F8 55 03 7B 49 86 87 E4 91 AB 73 E0 AF 3F 12
AB 7F 09 2D 4E 06 4A 71 00
The value k is
k = 111997156629233344141251678175691985810414419806912701822160
03679477455027316936425566511876628316232490805931954430
9467825781335347270
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 = 123321614196865011564245131263441780725524895314537710907076
16870786357527664628511298973435756875585697503259538939
5171670361367288637
sc = 546378534043466638830999383371973118168674696520868513626526
80700952961912832009497718396549853619706003345255804124
467063738043309482
The signature contributions can now be calulated:
Sa = 458780069801903308724659679047126206540023727016859370284301
27448209487257982695731719827860225173821194895861101407
212569594681924953
Sc = 401983114222685811776093088236307108408769238787937252473320
58598217288865359359764076141664751937484813828720687591
206436379246811336
The dealer calculates the composite value S = S_{a} + S_{b}
S = 860763184024589120500752767283433314948792965804796622757621
86046426776123342055495795969524977111306008724581788998
419005973928736289
The dealer checks to see that the signature verifies:
S.B = R + kA =
X: 565863868058207644748379075787643375969163605886
69329263413068616160478393570
X: 150478297775841135131577263700053908124463098444
5517859016637728894764938481
All the security considerations of , and apply and are hereby incorporated by reference.
The rogue key attack described in 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.
As in any Schnorr signature scheme, compromise of the value r results in compromise of the private key. The base signature specification 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 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.
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.
Signatures generated in strict conformance with 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 but are not unique.
Implementations MUST not use threshold signatures in applications where signature values are used in place of cryptographic digests as unique content identifiers.
This document requires no IANA actions.
EdwardsCurve Digital Signature Algorithm (EdDSA)
Key words for use in RFCs to Indicate Requirement Levels
Elliptic Curves for Security
Randomness Requirements for Security
Threshold Modes in Elliptic Curves
Mathematical Mesh 3.0 Part II: Uniform Data Fingerprint.
FROST: Flexible RoundOptimized Schnorr Threshold Signatures
University of Waterloo, Zcash Foundation
University of Waterloo
How to share a secret.
Requirements for Operations, Administration, and Maintenance (OAM) in MPLS Transport Networks
Mathematical Mesh: Reference Implementation