Internet-Draft hale-pquip-hybrid-spectrums October 2023
Bindel, et al. Expires 25 April 2024 [Page]
Network Working Group
Intended Status:
N. Bindel
B. Hale
Naval Postgraduate School
D. Connolly
F. Driscoll
UK National Cyber Security Centre

Hybrid signature spectrums


This document describes classification of design goals and security considerations for hybrid digital signature schemes, including proof composability, non-separability of the ingredient signatures given a hybrid signature, backwards/forwards compatiblity, hybrid generality, and simultaneous verification.

Discussion of this work is encouraged to happen on the IETF PQUIP mailing list or on the GitHub repository which contains the draft:

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Table of Contents

1. Introduction

Initial focus on the transition to use of post-quantum algorithms in protocols has largely been on confidentiality, given the potential risk of store and decrypt attacks, where data encrypted today using traditional algorithms could be decrypted in the future by an attacker with a Cryptographically-Relevant Quantum Computer (CRQC). While traditional authentication is only at risk once a CRQC exists, it is important to consider the transition to post-quantum authentication before this point. This is particularly relevant for systems where algorithm turn-over is complex or takes a long time (e.g., long-lived systems with hardware roots of trust), or where future checks on past authenticity play a role (e.g., digital signatures on legal documents).

One approach to designing quantum-resistant protocols, particularly during the transition period from traditional to post-quantum algorithms, is incorporating PQ/T Hybrid schemes, which combine both traditional and post-quantum algorithms in one cryptographic scheme. Hybridization has been looked at for key encapsulation [HYBRIDKEM], and in an initial sense for digital signatures [HYBRIDSIG]. Compared to key encapsulation, hybridization of digital signatures, where the verification tag may be expected to attest to both standard and post-quantum components, is subtler to design and implement due to the potential separability of the composite signatures and the risk of downgrade attacks. There are also a range of requirements and properties that may be required from PQ/T signatures, not all of which can be achieved at once.

This document focuses on explaining advantages and disadvantages of different hybrid signature scheme designs and different security goals for them. It is intended as a resource for designers and implementers of hybrid signature schemes to help them decide what properties they do and do not require from their scheme. It intentionally does not propose concrete hybrid signature combiners or instantiations thereof.

1.1. Revision history

  • RFC Editor's Note: Please remove this section prior to publication of a final version of this document.

  • TBD-00: Initial version.

1.2. Terminology

We follow existing Internet drafts on hybrid terminology [I-D.ietf-pquip-pqt-hybrid-terminology] and hybrid key encapsulation mechanisms (KEM) [I-D.ietf-tls-hybrid-design] to enable settling on a consistent language. We will make clear when this is not possible. In particular, we follow the definition of 'post-quantum algorithm', 'traditional algorithms', and 'combiner'. Moreover, we use the definition of 'certificate' to mean 'public-key certificate' as defined in [RFC4949].

  • 'Signature scheme': A signature scheme is defined via the following three algorithms:

    • KeyGen() -> (pk, sk): A probabilistic key generation algorithm, which generates a public verifying key pk and a secret signing key sk.

    • Sign(sk, m) -> (sig): A probabilistic signature generation, which takes as input a secret signing key sk and a message m, and outputs a signature sig.

    • Verify(pk, sig, m) -> b: A verification algorithm, which takes as input a public verifying key pk, a signature sig and a message m, and outputs a bit b indicating accept ($b=1$) or reject ($b=0$) of the signature for message m.

  • 'Hybrid signature scheme': Following [I-D.ietf-pquip-pqt-hybrid-terminology], we define a hybrid signature scheme to be "a multi-algorithm digital signature scheme made up of two or more component digital signature algorithms ...". We require that the security of the component schemes is based on the hardness of different cryptographic assumptions. In contrast to [I-D.ietf-pquip-pqt-hybrid-terminology], we will use the more general term 'hybrid signature scheme' instead of requiring one post-quantum and one traditional algorithm (i.e., PQ/T hybrid signature schemes) to allow also the combination of several post-quantum algorithms. The term 'composite' scheme is often used as a synonym for 'hybrid scheme'. This is different from [I-D.ietf-pquip-pqt-hybrid-terminology] where the term is used at the protocol level.

  • 'Hybrid signature': A hybrid signature is the output of the hybrid signature scheme's signature generation. As synonyms we might use 'composite signature' or 'dual signature'. For example, NIST define a dual signature as "two or more signatures on a common message" [NIST_PQC_FAQ].

  • 'Component (signature) scheme': Component signature schemes are the cryptographic algorithms contributing to the hybrid signature scheme. This has a similar purpose as in [I-D.ietf-pquip-pqt-hybrid-terminology]. In this draft, we will use 'ingredient signature scheme' as a synonym.

  • 'Next-generation algorithms': Similarly to the case of hybrid KEMs [I-D.ietf-tls-hybrid-design], hybrid signatures are mostly motiviated as preparation for the post-quantum migration. Following [I-D.ietf-tls-hybrid-design], we opt to use the more generic term "next-generation" and "traditional" algorithm instead of "post-quantum" and "classical" algorithms.

  • 'Artifact': An artifact is evidence of the sender's intent to hybridize a signature that remains even if a component algorithm tag is removed. Artifacts can be e.g., at the algorithmic level (e.g., within the digital signature), or at the protocol level (e.g., within the certificate), or on the system policy level (e.g., within the message). Artifacts should be easily identifiable by the receiver in the case of signature stripping.

1.3. Motivation for use of hybrid signature schemes

Before diving into the design goals for hybrid digital signatures, it is worth taking a look at why hybrid digital signatures are desirable for some applications. As many of the arguments hold in general for hybrid algorithms, we again refer to [I-D.ietf-tls-hybrid-design] that summarizes these well. In addition, we explicate the motivation for hybrid signatures here.

Complexity. Next-generation algorithms and their underlying hardness assumptions are often more complex than traditional algorithms and as such carry a higher risk of implementation mistakes and revision of parameters compared to traditional algorithms, such as RSA. RSA is a relatively simple algorithm to understand and explain, yet during its existence and use there have been multiple attacks and refinements, such as adding requirements to how padding and keys are chosen, and implementation issues such as cross-protocol attacks. Thus, even in a relatively simple algorithm subtleties and caveats on implementation and use can arise over time. Given the complexity of next generation algorithms, the chance of such discoveries and caveats needs to be taken into account.

Of note, next generation algorithms have been heavily vetted. Thus, if and when further information on caveats and implementation issues come to light, it is less likely that a "break" will be catastrophic. Instead, such vulnerabilities and issues may represent a weakening of security - which may in turn be offset if a hybrid approach has been used.

The complexity of next-generation algorithms needs to be balanced against the fact that hybridization itself adds more complexity to a protocol and introduces the risk of implementation mistakes in the hybridization process.

One example of a next generation algorithm is the signature scheme ML-DSA (a.k.a. Kyber-Dilithium) that has been selected for standardization by NIST. While the scheme follows the know Fiat-Shamir transform to construct the signature scheme, it also relies on rejection sampling that is known to give cache side channel information (not necessarily leading to a known attack though). Furthermore, recent attacks again some next-generation signature schemes such as MULTI-VARIATE schemes might call into question the asymptotic and concrete security for conservative adopters and therefore might hinder adoption.

Time. The need to transition to quantum-resistant algorithms now while simultaneously being aware of potential, hidden subtleties in their resistance to standard attacks drives transition designs towards hybridization. Mosca’s equation [MOSCA] very simply illustrates the risk of post-quantum transition delay: $l + d > q$, where l is the information life-span, d is the time for system transition to post-quantum algorithms, and q is the time before a quantum computer is ready to execute cryptanalysis. As opposed to key exchange and KEMs, it may not be obvious why there is urgency for an adoption of next-generation signatures; namely, while encryption is subject to store-now-decrypt-later attacks, there may not seem to be a parallel notion for authenticity, i.e., 'store-now-modify-later attacks'. However, in larger systems, including national systems, space systems, large healthcare support systems, and critical infrastructure, where acquisition and procurement time can be measured in years and algorithm replacement may be difficult or even practically impossible, this equation can have drastic implications. In such systems, algorithm turn-over can be complex and difficult and can take considerable time (such as in long-lived systems with hardware deployment), meaning that an algorithm may be committed to long-term, with no option for replacement. Long-term committment creates further urgency for immediate next-generation algorithm selection. Additionally, for some sectors future checks on past authenticity plays a role (e.g., many legal, financial, auditing, and governmental systems). The 'store-now-modify-later' analogy would present challenges in such sectors, where future analysis of past authentication may be more critical than in e.g., internet connection use cases. As such there is an eagerness to use next-generation signatures algorithms for some applications.

1.4. Goals

There are various security goals that can be achieved through hybridization. The following provides a summary of these goals, while also noting where security goals are in conflict, i.e., that achievement of one goal precludes another, such as backwards compatibility.

1.4.1. Unforgeability

One goal is security of hybrid signature schemes, in particular that EUF-CMA security is maintained as long as at least one of the ingredient schemes is EUF-CMA secure. There might be, however, other goals in competition with this one, such as backward-compatibility, where the EUF-CMA seucurity of the hybrid signature relies solely on the security of one of the ingredient schemes instead of relying on both.

1.4.2. Proof Composability.

Under proof composability, the ingredient algorithms are combined in such a way that it is possible to prove a security reduction from the security properties of hybrid signature scheme to the properties of the respective ingredient signature schemes and, potentially, other building blocks such as hash functions, KDF, etc. Otherwise an entirely new proof of security is required, and there is a lack of assurance that the combination builds on the standardization processes and analysis performed to date on ingredient algorithms. The resulting hybrid signature would be, in effect, an entirely new algorithm of its own. The more two signature schemes are entangled, the more likely it is that an entirely new proof is required, thus not meeting proof composability.

1.4.3. Weak Non-Separability

Non-Separability was one of the earliest properties of hybrid digital signatures to be discussed [HYBRIDSIG]. It was defined as the guarantee that an adversary cannot simply “remove” one of the ingredient signatures without evidence left behind. For example there are artifacts that a carefully designed verifier may be able to identify, or that are identifiable in later audits. This was later termed Weak Non-Separability (WNS) [HYBRIDSIGDESIGN]. Note that WNS does not restrict an adversary from potentially creating a valid ingredient digital signatures from a hybrid one (a signature stripping attack), but rather implies that such a digital signature will contain artifacts of the separation. Thus authentication is not simply provided by the sender to the receiver through correct verification of the digital signature(s), but potentially through further investigation on the receiver side that may extend well beyond traditional signature verification behavior. For instance, this can intuitively be seen in cases of a message containing a context note on hybrid authentication, that is then signed by all ingredient algorithms/the hybrid signature scheme. If an adversary removes one ingredient signature but not the other, then artifacts in the message itself point to the possible existence of hybrid signature such as a label stating “this message must be hybrid signed”. This might be a counter measure against stripping attacks if the verifier expects a hybrid signature scheme to ensure this property. However, it places the responsibility of signature validity not only on the correct format of the message, as in a traditional signature security guarantee, but the precise content thereof.

1.4.4. Strong Non-Separability

Strong Non-Separability (SNS) is a stronger notion of WNS, introduced in [HYBRIDSIGDESIGN]. SNS guarantees that an adversary cannot take as input a hybrid signature (and message) and output a valid ingredient signature (and potentially different message) that will verify correctly. In other words, separation of the hybrid signature into component signatures implies that the component signature will fail verification (of the component signature scheme) entirely. Therefore, authentication is provided by the sender to the receiver through correct verification of the digital signature(s), as in traditional signature security experiments. It is not dependent on other components, such as message content checking, or protocol level aspects, such as public key provenance. As an illustrative example distinguishing WNS from SNS, consider the case of ingredient algorithms $\Sigma_1.Sign$ and $\Sigma_2.Sign$ where the hybrid signature is computed as a concatenation $(sig_1, sig_2)$, where $sig_1 = \Sigma_1.Sign(hybridAlgID,m)$ and $sig_2 = \Sigma_2.Sign(hybridAlgID,m)$. In this case, separation and delivery of a new message $m^* = (hybridAlgID,m)$ along with signature $sig_1$ and $\$ could allow for correct verification and the hybrid artifact is embedded in the message instead of the signature (identifiable through further investigation but the signature verification itself would not fail). Thus, this case shows WNS (assuming the verification algorithm is defined accordingly) but not SNS.

Some work [I-D.ounsworth-pq-composite-sigs] has looked at reliance on the public key certificate chains to explicitly define hybrid use of the public key. Namely, that $\$ cannot be used without $\$. This implies pushing the hybrid artifacts into the protocol and system level and a dependency on the security of other verification algorithms (namely those in the certificate chain). This further requires that security analysis of a hybrid digital signature requires analysis of the key provenance, i.e., not simply that a valid public key is used but how it hybridization and hybrid artifacts have been managed throughout the entire chain. External dependencies such as this may imply hybrid artifacts lie outside the scope of the signature algorithm itself. SNS may potentially be achieveable based on dependencies at the system level.

1.4.5. Backwards/Forwards Compatibility

Backwards compatibility refers to the property where a hybrid algorithm may also be used for legacy receivers that may take a hybrid signature and verify it as being valid using only the verification algorithm of one component scheme (e.g., the verification algorithm of the traditional signature scheme used in the hybrid scheme), potentially ignoring the next-generation signature entirely. This provides an option to transition various sender system attributes to next-generation algorithms while still supporting select legacy receivers. Notably, this is a verification property; the sender has provided a hybrid digital signature, but the verifier is allowed, due to internal restrictions and/or implementation, to only verify one component signature. Backwards compatibility may be further decomposed to subcategories where ingredient key provenance is either separate or hybrid so as to support implementations that cannot recognize (and/or process) hybrid signatures.

Forwards compatibility has also been a consideration in hybrid proposals [I-D.becker-guthrie-noncomposite-hybrid-auth]. Forward compatibility assumes that hybrid signature schemes will be used for some time, but that eventually all systems will transition to use only one (particularly, only one next-generation) algorithm. As this is very similar to backwards compatibility, it also may imply separability of a hybrid algorithm; however, it could also simply imply capability to support separate ingredient signatures. Thus the key distinction between backwards and forwards compatibility is that backwards compatibility may be needed for legacy systems that cannot use and/or process hybrid or next-generation signatures, whereas in forwards compatibility the system has those capabilities and can choose what to support (e.g., for efficiency reasons).

As noted in [I-D.ietf-tls-hybrid-design], ideally, forward/backward compatibility is achieved using redundant information as little as possible.

1.4.6. Simultaneous Verification

Simultaneous Verification (SV) builds on SNS and was first introduced in [HYBRIDSIGDESIGN]. SV requires that not only are all ingredient signatures needed to achieve a successful verification present in the hybrid signature, but also that verification of both component algorithms occurs simultaneously. Namely, "missing" information needs to be computed by the verifier so they cannot “quit” the verification process before both component signatures are verified. SV mimics traditional digital signatures guarantees, essentially ensuring that the hybrid digital signature behaves as a single algorithm vs. two separate component stages. Alternatively phrased, under an SV guarantee it is not possible for an unerring verifier to initiate termination of the hybrid verification upon successful verification of one component algorithm without also knowing if the other component succeeded or failed.

1.4.7. Hybrid Generality

Hybrid generality means that a general signature combiner is defined, based on inherent and common structures of component digital signatures "categories." For instance, since multiple signature schemes use a Fiat-Shamir Transform, a hybrid scheme based on the transform can be made that is generalizable to all such signatures. Such generality can also result in simplified constructions where as more tailored hybrid variants might be more efficient in terms of sizes and performance.

1.4.8. High performance

Similarly to performance goals noted for hybridization of other cryptographic components [I-D.ietf-tls-hybrid-design] hybrid signature constructions are expected to be as performant as possible. For most hybrid signatures this means that the computation time should only minimally exceed the sum of the component signature computation time. It is noted that performance of any variety may come at the cost of other properties, such as hybrid generality.

1.4.9. High space efficiency

Similarly to space considerations in [I-D.ietf-tls-hybrid-design], hybrid signature constructions are expected to be as space performant as possible. This includes messages (as they might increase if artifacts are used), public keys, and the hybrid signature. For the hybrid signature, size should no more than minimally exceed the signature size of the two component signatures. In some cases, it may be possible for a hybrid signature to even be smaller than two component signatures.

1.4.10. Minimal duplicate information

Similarly to [I-D.ietf-tls-hybrid-design], duplicated information should be avoided when possible. This might concern repeated information in hybrid certificates or in the communication of component certificates in additional to hybrid certificates (for example to acheive backwards/forwards-comptability), or sending multiple public keys or signatures of the same component algorithm.

2. Non-separability spectrum

Non-separability is not a singular definition but rather is a scale, representing degrees of separability hardness, visualized in Figure 1.

|**No Non-Separability**
| no artifacts exist
|**Weak Non-Separability**
| artifacts exist in the message, signature, system, application, or protocol
| ----------------------------------------------------------------------------|
|**Strong Non-Separability**
| artifacts exist in hybrid signature
| ----------------------------------------------------------------------------|
|**Strong Non-Separability w/ Simultaneous Verification**
| artifacts exist in hybrid signature and verification or failure of both
| components occurs simultaneously
| ----------------------------------------------------------------------------|
Figure 1: Spectrum of non-separability from weakest to strongest.

At one end of the spectrum are schemes in which one of the ingredient signatures can be stripped away with the verifier not being able to detect the change during verification.An example of this includes simple concatenation of signatures without any artifacts used. Nested signatures (where a message is signed by one component algorithm and then the message-signature combination is signed by the second component algorithm) may also fall into this category, dependent on whether the inner or outer signature is stripped off without any artifacts remaining.

Next on the spectrum are weakly non-separable signatures. Under Weak Non-Separability, if one of the composite signatures of a hybrid is removed artifacts of the hybrid will remain (in the message, signature, or at the protocol level, etc.). This may enable the verifier to detect if a component signature is stripped away from a hybrid signature, but that detectability depends highly on the type of artifact and permissions. For instance, if a message contains a label artifact "This message must be signed with a hybrid signature" then the system must be allowed to analyze the message components for possible artifacts. Whether a hybrid signature offers (Weak/Strong) Non-Separability might also depend on the implementation and policy of the protocol or application the hybrid signature is used in on the verifier side. Such policies may be further ambiguous to the sender, meaning that the type of authenticity offered to the receiver is unclear. In another example, under nested signatures the verifier could be tricked into interpreting a new message as the message/inner signature combination and verify only the outer signature. In this case, the inner signature-tag is an artifact.

Third on the scale is the Strong Non-Separability notion, in which separability detection is dependent on artifacts in the signature itself. Unlike in Weak Non-Separability, where artifacts may be in the actual message, the certificate, or in other non-signature components, this notion more closely ties to traditional algorithm security notions (such as EUF-CMA) where security is dependent on the internal construct of the signature algorithm and its verification. In this type, the verifier is enabled to detect artifacts on an algorithmic level during verification. For example, the signature itself encodes the information that a hybrid signature scheme is used. Examples of this type may be found in [HYBRIDSIGDESIGN].

For schemes achieving the most demanding security notions, i.e., Strong Non-Separability with Simultaneous Verification, verification succeeds not only when both of the component signatures are present but also only when the verifier has verified both signatures. Moreover, no information is leaked to the receiver during the verification process on the possibile validity/invalidity of the component signatures until both verify. This construct most closely mirrors traditional digital signatures where, assuming that the verifier does verify a signature at all, the result is either a positive verification of a the full signature or a failure if the signature is not valid. For hybrid signatures, a full signature implies the hybridization of both component algorithms, and therefore the strongest non-separability notion enforces an all-or-nothing approach to verification. Examples of algorithms providing this type of security can be found in [HYBRIDSIGDESIGN].

3. Artifacts

Hybridization benefits from the presence of artifacts as evidence of the sender's intend to decrease the risk of successful stripping attacks. This, however, depends strongly on where such evidence resides (e.g., in the message, the signature, or somewhere on the protocol level instead of the algorithmic level). Even commonly discussed hybrid approaches, such as concatenation, are not inherently tied one type of security (e.g., WNS, SNS, etc.). This can lead to ambiguities when comparing different approaches and assumptions about approach security or lack thereof. Thus in this section we cover artifact locations and also walk through a highlevel comparison of a few hybrid approach categories to show how artifact location can differ within a given approach. Artifact location is tied to non-separability notions above; thus the selection of a given security guarantee and general hybrid approach must also include finer grained selection of artifact placement.

3.1. Artifact locations

There are a variety of artifact locations possible, ranging from within the signature algorithm itself to the protocol level and even into policy, as shown in Table 1 For example, one artifact location could be in the message-to-be-signed, e.g., containing a label artifact. Depending on the hybrid type, this might be stripped away though. For example, a quantum attacker could strip away the quantum-secure signature of a concatenated dual signature, and (being able to forge, e.g., ECDSA signatures) remove the label artifact from the message as well. So, for many applications and threat models, adding an artificat in the message might not prevent stripping attacks. Another artifact location could be in the public key certificates as described in [I-D.ounsworth-pq-composite-sigs]. In still yet another case, artifacts may be present through the fused hybrid method, thus making them part of the signature at the algorithmic level.

Eventual security analysis may be a consideration in choosing between levels. For example, if the security of the hybrid scheme is dependent on system policy, then cryptographic analysis must necessarily be reliant on specific policies and it may not be possible to describe a scheme's security in a standalone sense.

Table 1: Artifact placement levels
Location of artifacts of hybrid intent     Level
Signature                                                      Algorithm
Certificate Protocol
Algorithm agreement / negotiation  
Message                                                       Policy

3.2. Artifact Location Comparison Example

Decisions on artifact locations present further considerations for implementors, namely that if artifact placement is identical in two hybrid schemes, for example that hybridization artifacts reside in the certificate in both cases, and if the above non-separability guarantees are also the same for each scheme, then implementing the most performant scheme maybe preferred. Thus it is advisable to not assume performance or lack thereof is inherent to a particular security goal selection.

We briefly summarize our example hybrid approach categories (concatenation, nesting, and fusion) before showing how each one may have artifacts in different locations in Table 2.

  • Concatenation refers to variants of hybridization where, for component algorithms $\Sigma_1.Sign$ and $\Sigma_2.Sign$, the hybrid signature is calculated as a concatenation $(sig_1, sig_2)$ such that $sig_1 = \Sigma_1.Sign(hybridAlgID,m)$ and $sig_2 = \Sigma_2.Sign(hybridAlgID,m)$.

  • Nesting refers to variants of hybridization where for component algorithms $\Sigma_1.Sign$ and $\Sigma_2.Sign$, the hybrid signature is calculated in a layered approach as $(sig_1, sig_2)$ such that, e.g., $sig_1 = \Sigma_1.Sign(hybridAlgID,m)$ and $sig_2 = \Sigma_2.Sign(hybridAlgID,(m, sig_1))$.

  • Fused hybrid refers to variants of hybridization where for component algorithms $\Sigma_1.Sign$ and $\Sigma_2.Sign$, the hybrid signature is calculated with entaglement to produce a single hybrid signature $sig_h$ without clear component constructs.

Table 2: Artifact locations depending on the hybrid signature type
Ref Location of artifacts of hybrid intent Category
1 None No label in message, public keys are in separate certs
2 In message Label in message, public keys are in separate certs
3 In cert No label in message, public keys are in combined cert
4 In message and cert Label in message, public keys are in combined cert
5 In message Label in message, public keys are in separate certs
6 In cert No label in message, public keys are in combined cert
7 In message and cert Label in message, public keys are in combined cert
8 In signature Public keys are in separate certs
9 In signature and message Label in message, public keys are in separate certs
10 In signature and cert Public keys are in combined cert
11 In signature and message and cert Label in message, public keys are in combined cert

Under a concatenation combiner in option case 2, the artifacts lie within the message, and therefore validity of the message >NB: what's validity of a message? do you mean signautre? >BH: No, I mean message. I.e. that pointing out that it is circular reasoning to base the validity of the message on the message... > I added more discussion

depends on the message itself. Option cases 3 and 4 solve this circular dependancy by provisioning keys in a combined certificate. Option case 1 provides the weakest guarantees of hybrid identification, as there are no prescribed artifacts.

The artifact guarantees provided by a nesting combiner are similar to those provided by concatenation option cases 2, 3, and 4. Namely, if $sig_2 = \Sigma_2.Sign(hybridAlgID,(m, sig_1))$, then the "message" $(m, sig_1)$ input into $\Sigma_2.Sign$ actually contains the artifact and acts as a label. Unless an additional label is provided within $m$ itself, $sig_1$ does not therefore contain an artifact. This presents an implementation challenge as it is necessary to guess which algorithm is more at risk for a stripping attack and choose the order of nesting accordingly.

Under a fused combiner, SNS also implies that artifacts of hybridization are present within the signature. This can be coupled with artifacts in the message, such as through use of a label, and/or artifacts in the certificate if keys are also provisioned in a combined certificate.

The above comparison shows similarities among some hybrid scheme choices, for insance option case 3 and option case 6 both contain artifacts in the certificate. However, performance and correct implementation of option case 3 and option case 6 may not be the same, illustrating possible decision points.

4. Need-For-Approval Spectrum

In practice, use of hybrid digital signatures relies on standards specifications where applicable. This is particularly relevant in the case of FIPS approval considerations as well as NIST, which has provided basic guidance on hybrid signature use. NIST provides the following guidance (emphasis added),

We provide a scale for the different nuances of generality of the hybrid combiners. This is related to whether the resulting combiner needs a new approval process or falls under already approved specifications. together with respective example constructions.

| --------------------------------------------------------------------|
| **New Algorithm**
| New signature scheme base on different hard assumptions
| Separate approval needed
| --------------------------------------------------------------------|
| **No generality**
| Combiner supporting algorithms that can be reduced to (at least one)
| approved algorithm, potentially changing the component algorithms
| Uncertainty about whether separate approval is needed
| --------------------------------------------------------------------|
| **1-out-of-n generality**
| Combiner supports one component algorithm in a black-box way but
| potentially changes the other component algorithms
| No new approval needed if the black-box component is approved
| --------------------------------------------------------------------|
| **Full generality**
| Hybrid combiner acts as a wrapper, fully independet of the component
| signature schemes
| No new approval needed if at least one component is approved
| --------------------------------------------------------------------|
Figure 2: Generality / Need-for-approval spectrum

The least general"combiner" would be a new construction with a security reduction to different hardness assumptions but not necessarily to existing/approved signature schemes.

Next, is a combiner that might take inspiration from existing/approved signature schemes but changes the specifics but such that its security can be reduced to the security of an approved algorithm. As such it is uncertain whether a new approval would be needed as it might depend on the combiner and changes.

The second-most general construction, would be a combiner that uses at least one approved algorithm in a black-box way while it might change the specifics of the other component algorithms.

The most or 'fully' general combiner is using all algorithms in a blackbox way, for example the concatenation combiner (with a signature being valid if all componentn signatures are valid). As long as at least one component is approved, no new approval is needed.

5. EUF-CMA Challenges

Under traditional signature scheme security assumptions such as EUF-CMA, the adversary 'wins' the security experiment if it can produce a new message such that a message-signature pair (m, sig) with it correctly verifies. This traditional security notion is challenged under a hybrid construct.

The most straightforward comparison would be for the adversary to attempt to produce a new message m' that a message-hybrid signature pair (m', sig_h) correctly verifies. However, such a guarantee depends on the signature being strongly non-separable. Otherwise, in practical terms a security experiment must capture the case that an existing or new message m could be verified with a component signature, e.g., to produce (m', sig_1) that correctly verifies under Sigma_1.Sign. Such considerations are beyond the scope of traditional security analysis and represent considerations that would need to be accounted for depending on the signature combiner method chosen.

6. Security Considerations

This document discusses digital signature constructions that may be used in security protocols. It is an informational document and does not directly affect any other Internet draft. The security considerations for any specific implementation or incorporation of a hybrid scheme should be discussed in the relevant specification documents.

7. Discussion of Advantages/Disadvantages

8. Acknowledgements

This draft is based on the template of [I-D.ietf-tls-hybrid-design].

We would like to acknowledge the following people in alphabetical order who have contributed to pushing this draft forward, offered insights and perspectives, and/or stimulated work in the area:

Scott Fluhrer Felix Günther John Gray Serge Mister Max Pala Mike Ounsworth Douglas Stebila

9. Informative References

Bindel, N., Brendel, J., Fischlin, M., Goncalves, B., and D. Stebila, "Hybrid Key Encapsulation Mechanisms and Authenticated Key Exchange", Post-Quantum Cryptography pp.206-226, DOI 10.1007/978-3-030-25510-7_12, , <>.
Bindel, N., Herath, U., McKague, M., and D. Stebila, "Transitioning to a Quantum-Resistant Public Key Infrastructure", , <>.
Bindel, N. and B. Hale, "A Note on Hybrid Signature Schemes", , <>.
Becker, A., Guthrie, R., and M. J. Jenkins, "Non-Composite Hybrid Authentication in PKIX and Applications to Internet Protocols", Work in Progress, Internet-Draft, draft-becker-guthrie-noncomposite-hybrid-auth-00, , <>.
D, F., "Terminology for Post-Quantum Traditional Hybrid Schemes", Work in Progress, Internet-Draft, draft-ietf-pquip-pqt-hybrid-terminology-01, , <>.
Stebila, D., Fluhrer, S., and S. Gueron, "Hybrid key exchange in TLS 1.3", Work in Progress, Internet-Draft, draft-ietf-tls-hybrid-design-09, , <>.
Ounsworth, M., Gray, J., Pala, M., and J. Klaußner, "Composite Signatures For Use In Internet PKI", Work in Progress, Internet-Draft, draft-ounsworth-pq-composite-sigs-10, , <>.
Kaye, P., Laflamme, R., and M. Mosca, "An Introduction to Quantum Computing, Oxford University Press", .
National Institute of Standards and Technology (NIST), "Post-Quantum Cryptography FAQs", , <>.
Shirey, R., "Internet Security Glossary, Version 2", FYI 36, RFC 4949, DOI 10.17487/RFC4949, , <>.

Authors' Addresses

Nina Bindel
Britta Hale
Naval Postgraduate School
Deirdre Connolly
Florence Driscoll
UK National Cyber Security Centre