Network Working Group N. Bindel
Internet-Draft SandboxAQ
Intended status: Informational B. Hale
Expires: 22 December 2025 Naval Postgraduate School
D. Connolly
SandboxAQ
F. Driscoll
UK National Cyber Security Centre
20 June 2025
Hybrid signature spectrums
draft-ietf-pquip-hybrid-signature-spectrums-07
Abstract
This document describes classification of design goals and security
considerations for hybrid digital signature schemes, including proof
composability, non-separability of the component signatures given a
hybrid signature, backwards/forwards compatibility, hybrid
generality, and simultaneous verification.
Discussion Venues
This note is to be removed before publishing as an RFC.
Discussion of this document takes place on the Post-Quantum Use In
Protocols Working Group mailing list (pqc@ietf.org), which is
archived at https://mailarchive.ietf.org/arch/browse/pqc/.
Source for this draft and an issue tracker can be found at
https://github.com/dconnolly/draft-connolly-pquip-hybrid-signature-
spectrums.
Status of This Memo
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Terminology . . . . . . . . . . . . . . . . . . . . . . . 4
1.2. Motivation for Use of Hybrid Signature Schemes . . . . . 7
1.2.1. Complexity . . . . . . . . . . . . . . . . . . . . . 7
1.2.2. Time . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3. Goals . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1. Hybrid Authentication . . . . . . . . . . . . . . . . 9
1.3.2. Proof Composability . . . . . . . . . . . . . . . . . 10
1.3.3. Weak Non-Separability . . . . . . . . . . . . . . . . 10
1.3.4. Strong Non-Separability . . . . . . . . . . . . . . . 11
1.3.5. Backwards/Forwards Compatibility . . . . . . . . . . 11
1.3.6. Simultaneous Verification . . . . . . . . . . . . . . 12
1.3.7. Hybrid Generality . . . . . . . . . . . . . . . . . . 13
1.3.8. High Performance . . . . . . . . . . . . . . . . . . 13
1.3.9. High Space Efficiency . . . . . . . . . . . . . . . . 13
1.3.10. Minimal Duplicate Information . . . . . . . . . . . . 13
2. Non-Separability Spectrum . . . . . . . . . . . . . . . . . . 13
3. Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1. Artifacts vs. Separability . . . . . . . . . . . . . . . 16
3.2. Artifact Locations . . . . . . . . . . . . . . . . . . . 16
3.3. Artifact Location Comparison Example . . . . . . . . . . 17
4. Need For Approval Spectrum . . . . . . . . . . . . . . . . . 21
5. EUF-CMA Challenges . . . . . . . . . . . . . . . . . . . . . 23
6. Discussion of Advantages/Disadvantages . . . . . . . . . . . 25
6.1. Backwards Compatibility vs. SNS . . . . . . . . . . . . . 25
6.2. Backwards Compatibility vs. Hybrid Unforgeability . . . . 25
6.3. Simultaneous Verification vs. Low Need for Approval . . . 25
7. Security Considerations . . . . . . . . . . . . . . . . . . . 26
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 26
9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 26
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10. References . . . . . . . . . . . . . . . . . . . . . . . . . 26
10.1. Normative References . . . . . . . . . . . . . . . . . . 26
10.2. Informative References . . . . . . . . . . . . . . . . . 26
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 28
1. Introduction
Plans to transition protocols to post-quantum cryptography sometimes
focus 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
sufficiently powerful quantum computer, also known as a
Cryptographically-Relevant Quantum Computer (CRQC).
It is important to also consider transitions to post-quantum
authentication; delaying such transitions creates risks. For
example, attackers may be able to carry out quantum attacks against
RSA-2048 [RSA] years before the public is aware of these
capabilities. Furthermore, there are applications where algorithm
turn-over is complex or takes a long time. For example, root
certificates are often valid for about 20 years or longer. There are
also applications where future checks on past authenticity play a
role, such as long-lived digital signatures on legal documents.
Still, there have been successful attacks against algorithms using
post-quantum cryptography, as well as implementations of such
algorithms. Sometimes an attack exploits implementation issues, such
as [KYBERSLASH], which exploits timing variations, or [HQC_CVE] which
exploits implementation bugs. Sometimes an attack works for all
implementations of the specified algorithm. Research has indicated
that implementation-independent attacks published in 2023 or earlier
had broken 48% of the proposals in Round 1 of the NIST Post-Quantum
Cryptography Standardization Project, 25% of the proposals not broken
in Round 1, and 36% of the proposals selected by NIST for Round 2
[QRCSP].
Such cryptanalysis and security concerns are one reason to consider
'hybrid' cryptographic algorithms, which combine both traditional and
post-quantum (or more generally a combination of two or more)
algorithms. A core objective of hybrid algorithms is to protect
against quantum computers while at the same time making clear that
the change is not reducing security. A premise of security of these
algorithms being that if at least one of the two component algorithms
of the hybrid scheme holds, the confidentiality or authenticity
offered by that scheme is maintained. It should be noted that the
word 'hybrid' has many uses, but this document uses 'hybrid' only in
this algorithm sense.
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Whether or not hybridization is desired depends on the use case and
security threat model. Users may recognize a need to start post-
quantum transition, even while issues such as those described above
are a concern. For this, hybridization can support transition. It
should be noted that hybridization is not necessary for all systems;
recommendations on system types or analysis methods for such
determination are out of scope of this document. For cases where
hybridization is determined to be advantageous, a decision on how to
hybridize needs to be made. With many options available, this
document is intended to provide context on some of the trade-offs and
nuances to consider.
Hybridization of digital signatures, where the hybrid signature may
be expected to attest to both standard and post-quantum components,
is subtle to design and implement due to the potential separability
of the hybrid/dual signatures and the risk of downgrade/stripping
attacks. There are also a range of requirements and properties that
may be required from hybrid signatures, which will be discussed in
this document. Some of these are mutually exclusive, which
highlights the importance of considering use-case specific
requirements.
This document focuses on explaining a spectrum of different hybrid
signature scheme design categories 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 use case. In scope limitations, it
does not attempt to give concrete recommendations for any use case.
It also intentionally does not propose concrete hybrid signature
combiners or instantiations thereof. As with the data authenticity
guarantees provided by any digital signature, the security guarantees
discussed in this document are reliant on correct provisioning and
management of the keys involved, e.g. entity authentication, key
revocation etc. This document only considers scenarios with a single
signer and a single verifier, constructions with multiple signers or
verifiers are out of scope.
1.1. Terminology
We follow existing Internet documents on hybrid terminology [RFC9794]
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].
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* 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. In this draft, the secret signing
key sk is assumed to be implicit for notational simplicity, and
the following notation is used: Sign(m) -> (sig). If the
message m is comprised of multiple fields, m1, m2, ..., mN,
this is notated Sign(m) = Sign (m1, m2, ... mN) -> (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 [RFC9794], we define a hybrid
signature scheme to be "a multi-algorithm digital signature scheme
made up of two or more component digital signature algorithms".
While it often makes sense for security purposes to require that
the security of the component schemes is based on the hardness of
different cryptographic assumptions, in other cases hybrid schemes
might be motivated, e.g., by interoperability of variants on the
same scheme and as such both component schemes are based on the
same hardness assumption (e.g., both post-quantum assumptions or
even both the same concrete assumption such as Ring LWE). We
allow this explicitly. This means in particular that in contrast
to [RFC9794], 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' has sometimes been used as a synonym for
'hybrid scheme'. This is different from [RFC9794] where the term
is used as a specific instantiation of hybrid schemes such that
"where multiple cryptographic algorithms are combined to form a
single key or signature such that they can be treated as a single
atomic object at the protocol level." To avoid confusing we will
avoid the term 'composite scheme'.
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* Hybrid signature: A hybrid signature is the output of the hybrid
signature scheme's signature generation. As synonyms we might use
'dual signature'. For example, NIST defines a dual signature as
"two or more signatures on a common message" [NIST_PQC_FAQ]. For
the same reason as above we will avoid using the term 'composite
signature' although it sometimes appears as a synonym for 'hybrid/
dual signature'.
* Component (signature) scheme: Component signature schemes are the
cryptographic algorithms contributing to the hybrid signature
scheme. This has a similar purpose as in [RFC9794]. 'Ingredient
(signature) scheme' may be used as a synonym.
* Next-generation algorithms: Following
[I-D.ietf-tls-hybrid-design], we define next-generation algorithms
to be "algorithms which are not yet widely deployed but which may
eventually be widely deployed". Hybrid signatures are mostly
motivated by preparation for post-quantum transition or use in
long-term post-quantum deployment, hence the reference to post-
quantum algorithms through this document. However, the majority
of the discussion in this document applies equally well to future
transitions to other next-generation algorithms.
* Policy: Throughout this document we refer to 'policy' in the
context of e.g., a system policy requiring verification of two
signatures, an RFC description, guidance documentation, etc.
Similar terminology may include 'security configuration settings',
or related phrasing. We treat these terms as equivalent for the
purposes of this document.
* Artifact: An artifact is evidence of the sender's intent to
hybridize a signature that remains even if a component signature
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.
* Stripping attack: A stripping attack refers to a case where an
adversary takes a message and hybrid signature pair and attempts
to submit (a potential modification of) the pair to a component
algorithm verifier, meaning that the security is based only on the
remaining component scheme. A common example of a stripping
attack includes a message and hybrid signature, comprised of
concatenated post-quantum and traditional signatures, where an
adversary with a quantum computer simply removes the post-quantum
component signature and submits the (potentially changed) message
and traditional component signature to a traditional verifier.
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This could include an authentic traditional certificate authority
signature on a certificate that was originaly hybrid-signed. An
attribute of this is that the an honest signer would attest to
generating the signature. Stripping attacks should not be
confused with component message forgery attacks.
* Component message forgery attacks: A forgery attack refers to a
case where an adversary attempts to forge a (non-hybrid) signature
on a message using the public key associated with a component
algorithm. An common example of such an attack would be a quantum
attacker compromising the key associated with a traditional
component algorithm and forging a message and signature pair.
Message forgery attacks may be formalized with experiments such as
existential unforgeability under chosen-message attack (EUF-CMA)
[EUFCMA], while the difference introduced in component message
forgery attacks is that the key is accepted for both hybrid and
single algorithm use. Further discussions on this appear under
Section 5.
1.2. Motivation for Use of Hybrid Signature Schemes
Before diving into the design goals for hybrid digital signatures, it
is worth taking a look at motivations for them. 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.
1.2.1. Complexity
Next-generation algorithms and their underlying hardness assumptions
are often more complex than traditional algorithms. For example, the
signature scheme ML-DSA [MLDSA] (also known as CRYSTALS-DILITHIUM)
follows the well-known Fiat-Shamir transform [FS] to construct the
signature scheme, but also relies on rejection sampling that is known
to give cache side channel information (although this does not lead
to a known attack). Likewise, the signature scheme Falcon [FALCON]
uses complex sampling during signature generation. Furthermore,
attacks against the next-generation multivariate schemes Rainbow
[RAINBOW] and GeMSS [GEMSS] might raise concerns for conservative
adopters of other algorithms, which could hinder adoption.
As such, some next-generation algorithms 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 (e.g., component algorithm
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forgeries). 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, some next-generation algorithms have received considerable
analysis, for example, following attention gathered during the NIST
Post-Quantum Cryptography Standardization Process [NIST_PQC_FAQ].
However, if and when further information on caveats and
implementation issues come to light, it is quite possible that
vulnerabilities will represent a weakening of security rather than a
full "break". Such weakening may also be offset if a hybrid approach
has been used.
1.2.2. Time
In large systems, including national systems, space systems, large
healthcare support systems, and critical infrastructure, acquisition
and procurement time can be measured in years and algorithm
replacement may be difficult or even practically impossible. Long-
term commitment creates further urgency for immediate post-quantum
algorithm selection, for example when generating root certificates
with their long validity windows. Additionally, for some sectors,
future checks on past authenticity plays a role (e.g., many legal,
financial, auditing, and governmental systems). This means there is
a need to transition some systems to post-quantum signature
algorithms imminently. However, as described above, there is a need
to remain aware of potential, hidden subtleties in next-generation
algorithms' resistance to standard attacks, particularly in cases
where it is difficult to replace algorithms. This combination of
time pressure and complexity drives some transition designs towards
hybridization.
1.3. Goals
There are various security and usability goals that can be achieved
through hybridization. The following provides a summary of these
goals, while also noting where goals are in conflict, i.e., that
achievement of one goal precludes another, such as backwards
compatibility.
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1.3.1. Hybrid Authentication
One goal of hybrid signature schemes is security. As defined in
[RFC9794], ideally a hybrid signature scheme can achieve 'hybrid
authentication' which is the property that (cryptographic)
authentication is achieved by the hybrid signature scheme provided
that a least one component signature algorithm remains 'secure'.
There might be, however, other goals in competition with this one,
such as backward-compatibility - referring to the property where a
hybrid signature may be verified by only verifying one component
signature (see description below). Hybrid authentication is an
umbrella term that encompasses more specific concepts of hybrid
signature security, such as 'hybrid unforgeability' described next.
1.3.1.1. Hybrid Unforgeability
Hybrid unforgeability is a specific type of hybrid authentication,
where the security assumption for the scheme, e.g. EUF-CMA, is
maintained as long as at least one of the component schemes maintains
that security assumption. We call this notion 'hybrid
unforgeability'; it is a specific type of hybrid authentication. For
example, the concatenation combiner in [HYBRIDSIG] is 'hybrid
unforgeable'. As mentioned above, this might be incompatible with
backward-compatibility, where the EUF-CMA security of the hybrid
signature relies solely on the security of one of the component
schemes instead of relying on both, e.g., the dual message combiner
using nesting in [HYBRIDSIG]. For more details, we refer to our
discussion below.
Use cases where a hybrid scheme is used with, e.g., EUF-CMA security
assumed for only one component scheme generally use hybrid techniques
for their 'functional transition' pathway support. For example,
hybrid signatures may be used as a transition step for gradual post-
quantum adoption, while ensuring interoperability when a system
includes both verifiers that only support traditional signatures and
verifiers that have been upgraded to support post-quantum signatures.
In contrast, use cases where a hybrid scheme is used with e.g., EUF-
CMA security assumed for both component schemes without
prioritisation between them can use hybrid techniques for both
functional transition and security transition (i.e., a transition to
ensure security even if it may not be known which algorithm should be
relied upon).
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1.3.2. Proof Composability
Under proof composability, the component algorithms are combined in
such a way that it is possible to prove a security reduction from the
security properties of a hybrid signature scheme to the properties of
the respective component signature schemes and, potentially, other
building blocks such as hash functions, key derivation functions,
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 component
algorithms. The resulting hybrid signature would be, in effect, an
entirely new algorithm of its own. The more the component signature
schemes are entangled, the more likely it is that an entirely new
proof is required, thus not meeting proof composability.
1.3.3. 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
component 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
component digital signature from a hybrid one (a signature stripping
attack), but rather implies that such a digital signature will
contain artifacts of the separation. Thus, authentication that is
normally assured under correct verification of digital signature(s),
is now potentially also reliant on 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 component algorithms/the
hybrid signature scheme. If an adversary removes one component
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 have 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.
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1.3.4. 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
component 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 component
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, a new message m' = (hybridAlgID, m) along with
signature sig_1 and Sigma_1.pk, with the hybrid artifact embedded in
the message instead of the signature, could be correctly verified.
The separation would be 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.ietf-lamps-pq-composite-sigs] has looked at reliance
on the public key certificate chains to explicitly define hybrid use
of the public key. Namely, that Sigma_1.pk cannot be used without
Sigma_2.pk. 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 its 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
achievable based on dependencies at the system level.
1.3.5. Backwards/Forwards Compatibility
Backwards compatibility refers to the property where a hybrid
signature may be verified by only verifying one component signature,
allowing the scheme to be used by legacy receivers. In general, this
means verifying the traditional component signature scheme,
potentially ignoring the post-quantum signature entirely. This
provides an option to transition sender systems to post-quantum
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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
policy and/or implementation, to only verify one component signature.
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 post-quantum) 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 component 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 post-quantum 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.
1.3.6. Simultaneous Verification
Simultaneous Verification (SV) builds on SNS and was first introduced
in [HYBRIDSIGDESIGN]. SV requires that not only is the entire hybrid
signature (e.g., all component signature elements) needed to achieve
a successful verification present in the signature presented for
verification, but also that verification of both component algorithms
occurs roughly simultaneously. Namely, "missing" information needs
to be computed by the verifier so that a normally functioning
verification algorithm cannot "quit" the verification process before
the hybrid signature elements attesting for both component algorithms
are verified. This may additionally cover some error-injection and
similar attacks, where an adversary attempts to make an otherwise
honest verifier skip component algorithm verification. 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 otherwise honest verifier to
initiate termination of the hybrid verification upon successful
verification of one component algorithm without also knowing if the
other component succeeded. Note that SV does not prevent dishonest
verification, such as if a verifier maliciously implements a
customized verification algorithm that is designed with intention to
subvert the hybrid verification process or skips signature
verification altogether.
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1.3.7. 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 whereas
more tailored hybrid variants might be more efficient in terms of
sizes and performance.
1.3.8. 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.
1.3.9. 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 be smaller than the
concatenation of the two component signatures.
1.3.10. Minimal Duplicate Information
Duplicated information should be avoided when possible, as a general
point of efficiency. This might include repeated information in
hybrid certificates or in the communication of component certificates
in additional to hybrid certificates (for example, to achieve
backwards/forwards-compatibility) 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.
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|-----------------------------------------------------------------------------|
|**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
| ----------------------------------------------------------------------------|
â–¼
Figure 1: Spectrum of non-separability from weakest to strongest.
At one end of the spectrum are schemes in which one of the component
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 component 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 contents 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 is an artifact.
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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 can detect artifacts on an algorithmic
level during verification. For example, the signature itself may
encode 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 notion, Strong Non-
Separability with Simultaneous Verification, verification succeeds
only when both of the component signatures are present and the
verifier has verified both signatures. Moreover, no information is
leaked to the receiver during the verification process on the
possible validity of the component signatures until both verify (or
verification failure may or may not be attributable to a specific
component algorithm). 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 the full signature or a failure if the signature is
not valid. For fused hybrid signatures, a full signature implies the
fusion of both component algorithms, and therefore this type of
construction has the potential to achieve the strongest non-
separability notion which ensures an all-or-nothing approach to
verification, regardless of adversarial action. 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 intent 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 at the algorithmic level). Even commonly
discussed hybrid approaches, such as concatenation, are not
inherently tied to one type of security (e.g., WNS or SNS). This can
lead to ambiguities when comparing different approaches and
assumptions about security or lack thereof. Thus, in this section we
cover artifact locations and also walk through a high-level
comparison of a few hybrid 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.
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3.1. Artifacts vs. Separability
Note that non-separability is a security notion of the signature
scheme and not directly related to artifacts – artifacts may be used
for detection of separation, however. For instance, under strong
non-separability, the scheme would fail verification if separation
occurs, while for weak non-separability some artifacts exist if
separation occurs but verification would not necessarily fail. The
verifier could indeed ignore the artifact, hence the scheme achieving
only weak non-separability and not strong non-separability. It is
rather that an artifact exists that could be identified if an
investigation occurred, etc. Under weak non-separability, detection
of separation may depend on non-cryptographic configurations or other
dependencies. Also, strong non-separability and weak non-
separability are properties of the signature scheme – artifacts are
not necessarily in the signature and may appear in the signed
message, the certificate, the protocol, or policy (hence them not
necessarily being related to the strong non-separability and weak
non-separablity security notions). Artifacts may still be useful
(albeit dependent on system configurations) even if separable
signatures are used.
3.2. Artifact Locations
There are a variety of artifact locations possible, ranging from
within the message to the signature algorithm 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, it might be possible
to strip this away. For example, a quantum attacker could strip away
the post-quantum signature of a concatenated dual signature, and,
being able to forge the traditional signature, remove the label
artifact from the message as well. So, for many applications and
threat models, adding an artifact in the message might be
insufficient under stripping attacks. Another artifact location
could be in the public key certificates as described in
[I-D.ietf-lamps-pq-composite-sigs]. In such a case, the artifacts
are still present even if a stripping attack occurs. In yet another
case, artifacts may be present through the fused hybrid method, thus
making them part of the signature at the algorithmic level. Note
that in this latter case, it is not possible for an adversary to
strip one of the component signatures or use a component of the
hybrid to create a forgery for a component algorithm. Such
signatures provide SNS. This consequently also implies that the
artifacts of hybridization are absolute in that verification failure
would occur if an adversary tries to remove them.
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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. In
this case, it is necessary to consider the configuration of a
particular implementation or use to assess security, which could
increase the risk of unknown and unanticipated vulnerabilities,
regardless of the algorithms in use.
+=============================================+===========+
| Location of Artifacts of Hybrid Intent | Level |
+=============================================+===========+
| Signature | Algorithm |
+---------------------------------------------+-----------+
| Certificate | Protocol |
+---------------------------------------------+-----------+
| Algorithm agreement / negotiation | Protocol |
+---------------------------------------------+-----------+
| Message | Policy |
+---------------------------------------------+-----------+
Table 1: Artifact placement levels
3.3. Artifact Location Comparison Example
Here we provide a high-level example of how artifacts can appear in
different locations even within a single, common approach. We look
at the following categories of approaches: concatenation, nesting,
and fusion. This is to illustrate that a given approach does not
inherently imply a specific non-separability notion and that there
are subtleties to the selection decision, since hybrid artifacts are
related to non-separability guarantees. Additionally, this
comparison highlights how artifacts placement can be identical in two
different hybrid approaches.
We briefly summarize the hybrid approach categories (concatenation,
nesting, and fusion) for clarity in description, before showing how
each one may have artifacts in different locations in Table 2.
* Concatenation: 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).
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* Nesting: 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: variants of hybridization where for component
algorithms Sigma_1.Sign and Sigma_2.Sign, the hybrid signature is
calculated to generate a single hybrid signature sig_h that cannot
be cleanly separated to form one or more valid component
constructs. For example, if both signature schemes are signatures
schemes constructed through the Fiat-Shamir transform, the
component signatures would include responses r_1 and r_2 and
challenges c_1 and c_2, where c_1 and c_2 are hashes computed over
the respective commitments comm_1 and comm_2 (and the message). A
fused hybrid signature could consist of the component responses,
r_1 and r_2 and a challenge c that is computed as a hash over both
commitments, i.e., c = Hash((comm_1 || comm_2) || Hash2(message)).
As such, c does not belong to either of the component signatures
but rather both, meaning that the signatures are 'entangled'.
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+====+=======================+=============================+
| # | Location of artifacts | Category |
| | of hybrid intent | |
+====+=======================+=============================+
| | | *Concatenated* |
+----+-----------------------+-----------------------------+
| 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 |
+----+-----------------------+-----------------------------+
| | | *Nested* |
+----+-----------------------+-----------------------------+
| 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 |
+----+-----------------------+-----------------------------+
| | | *Fused* |
+----+-----------------------+-----------------------------+
| 8 | In signature | Public keys are in separate |
| | | certs |
+----+-----------------------+-----------------------------+
| 9 | In signature and | Label in message, public |
| | message | keys are in separate certs |
+----+-----------------------+-----------------------------+
| 10 | In signature and cert | Public keys are in combined |
| | | cert |
+----+-----------------------+-----------------------------+
| 11 | In signature and | Label in message, public |
| | message and cert | keys are in combined cert |
+----+-----------------------+-----------------------------+
Table 2: Artifact locations depending on the hybrid
signature type
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As can be seen, while concatenation may appear to refer to a single
type of combiner, there are in fact several possible artifact
locations depending on implementation choices. Artifacts help to
support detection in the case of stripping attacks, which means that
different artifact locations imply different overall system
implementation considerations to be able to achieve such detection.
Case 1 provides the weakest guarantees of hybrid identification, as
there are no prescribed artifacts and therefore non-separability is
not achieved. However, as can be seen, this does not imply that
every implementation using concatenation fails to achieve non-
separability. Thus, it is advisable for implementors to be
transparent about artifact locations.
In cases 2 and 5 the artifacts lie within the message. This is
notable as the authenticity of the message relies on the validity of
the signature, and the artifact location means that the signature in
turn relies on the authentic content of the message (the artifact
label). This creates a risk of circular dependency. Alternative
approaches such as cases 3, 4, 6 and 7 solve this circular dependency
by provisioning keys in a combined certificate.
Another observation from this comparison is that artifact locations
may be similar among some approaches. For instance, case 3 and case
6 both contain artifacts in the certificate. Naturally these
examples are high-level and further specification on concrete schemes
in the categories are needed before prescribing non-separability
guarantees to each, but this does indicate how there could be a
strong similarity between such guarantees. Such comparisons allow
for a systematic decision process, where security is compared and
identified and, if schemes are similar in the desired security goal,
then decisions between schemes can be based on performance and
implementation ease.
A final observation that this type of comparison provides is how
various combiners may change the security analysis assumptions in a
system. For instance, cases 3, 4, 6, and 7 all push artifacts - and
therefore the signature validity - into the certificate chain.
Naturally the entire chain must then also use a similar combiner if a
straightforward security argument is to be made. Other cases, such
as 8, 9, 10, and 11 put artifacts within the signature itself,
meaning that these bear the closest resemblance to traditional
schemes where message authenticity is dependent on signature
validity.
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4. Need For Approval Spectrum
In practice, use of hybrid digital signatures relies on standards
where applicable. This is particularly relevant in the cases where
use of FIPS (Federal Information Processing Standard) approved
software modules is required, but applies equally to any guidance or
policy direction that specifies that at least one component algorithm
of the hybrid has passed some certification type while not specifying
requirements on the other component. NIST provides the following
guidance (emphasis added),
Assume that in a [hybrid] signature, _one signature is generated
with a NIST-approved signature scheme as specified in FIPS 186,
while another signature(s) can be generated using different
schemes_, e.g., ones that are not currently specified in NIST
standards..._hybrid signatures can be accommodated by current
standards in FIPS mode, as defined in FIPS 140, provided at least
one of the component methods is a properly implemented, NIST-
approved signature algorithm_. For the purposes of FIPS 140
validation, any signature that is generated by a non-approved
component scheme would not be considered a security function,
since the NIST-approved component is regarded as assuring the
validity of the hybrid signature. [NIST_PQC_FAQ]
This draft does not define a formal interpretation of the NIST
statement; however, we use it as motivation to highlight some points
that implementors of hybrids may wish to consider when following any
guidance documents that specify that 1) the signature scheme for one
of the component algorithms must be approved and 2) the said
algorithm must be a well implemented or a certified implementation.
This type of need for approval (i.e., a requirement that an
implementor is looking to follow regarding approval or certification
of the software module implementation of hybrid or its component
algorithms) can drive some logistical decisios on what types of
hybrids an implementor should consider.
In this respect, there is a scale of approval that developers may
consider as to whether they are using at least one approved component
algorithm implementation (1-out-of-n approved software module), or
whether every component algorithm implementation is individually
approved (all approved software module).
We provide a scale for the different nuances of approval of the
hybrid combiners, where "approval" means that a software
implementation of a component algorithm can be used unmodified for
creation of the hybrid signature. This may be related to whether a
hybrid combiner is likely to need dedicated certification.
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| ---------------------------------------------------------------------------------|
| **New Algorithm**
| New signature scheme based on a selection of hardness assumptions
| Separate approval needed
| ---------------------------------------------------------------------------------|
| **No Approved Software Module**
| Hybrid combiner supports security analysis that can be reduced to
| approved component algorithms, potentially changing the component implementations
| Uncertainty about whether separate approval is needed
| ---------------------------------------------------------------------------------|
| **1-out-of-n Approved Software Module**
| Combiner supports one component algorithm and implementation in a black-box way
| but potentially changes the other component algorithm implementation(s)
| No new approval needed if the black-box component (implementation) is approved
| ---------------------------------------------------------------------------------|
| **All Approved Software Modules**
| Hybrid combiner acts as a wrapper, fully independent of the component
| signature scheme implementations
| No new approval needed if at least one component implementation is approved
| ---------------------------------------------------------------------------------|
â–¼
Figure 2: Generality / Need for Approval spectrum
The first listed "combiner" would be a new construction with a
security reduction to different hardness assumptions but not
necessarily to approved (or even existing) signature schemes. Such a
new, singular algorithm relies on both traditional and next-gen
principles.
Next, is a combiner that might take inspiration from existing/
approved signature schemes such that its security can be reduced to
the security of the approved algorithms. The combiner may, however,
alter the implementations. As such it is uncertain whether new
approval would be needed as it might depend on the combiner and
changes. Such a case may potentially imply a distinction between a
need for fresh approval of the algorithm(s) and approval of the
implementation(s).
The 1-out-of-n combiner uses at least one approved algorithm
implementation in a black-box way (i.e., without modification to the
software module implementaton for that algorithm). It may
potentially change the specifics of the other component algorithm
implementations. If the premise is that no new approval is needed so
long as at least one component is approved, then this is likely
considered sufficient.
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In an all-approved combiner, every algorithm implementation is used
in a black-box way. A concatenation combiner is a simple example
(where a signature is valid if all component signatures are valid).
Thus, as all algorithm implementations are approved, a requirement
that at least one of hybrid component algorithms is approved would be
satisfied.
5. EUF-CMA Challenges
Unforgeability properties for hybrid signature schemes are more
nuanced than for single-algorithm schemes.
Under the traditional EUF-CMA security assumption, an adversary can
request signatures for messages of their choosing and succeeds if
they are able to produce a valid signature for a message that was not
part of an earlier request. EUF-CMA can be seen as applying to the
hybrid signature scheme in the same way as single-algorithm schemes.
Namely, the most straightforward extension of the traditional EUF-CMA
security game would be that an adversary can request hybrid
signatures for messages of their choosing and succeeds if they are
able to produce a valid hybrid signature for a message that was not
part of an earlier request. However, this has several layers of
nuance under a hybrid construct.
Consider, for example, a simplistic hybrid approach using
concatenated component algorithms. If the hybrid signature is
stripped, such that a single component signature is submitted to a
verification algorithm for that component along with the message that
was signed by the hybrid, the result would be an EUF-CMA forgery for
the component signature. This is because as the component signing
algorithm was not previously called for the message - the hybrid
signing algorithm was used to generate the signature. This is an
example of a component algorithm forgery, a.k.a. a case of cross-
algorithm attack or cross-protocol attack.
The component algorithm forgery verifier target does not need to be
the intended recipient of the hybrid-signed message and may even be
in an entirely different system. This vulnerability is particularly
an issue among concatenated or nested hybrid signature schemes where
individual component verification could be possible. It should be
noted that policy enforcement of a hybrid verification does not
mitigate the issue on the intended message recipient: the component
forgery could occur on any system that accepts the component keys.
Thus, if EUF-CMA security for hybrids is considered to be informally
defined in the straightfoward way as that an adversary can request
hybrid signatures for messages of their choosing and succeeds if they
are able to produce a valid hybrid signature for a message that was
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not part of an earlier request, implicit requirements must hold in
order to avoid real-world implications. Namely, either component
algorithm forgeries, a.k.a. cross-protocol attacks, must be out of
scope for the use case or the hybrid signature choice must be
strongly non-separable. Otherwise, component algorithm forgeries,
which can be seen as a type of cross-protocol attack, affect the type
of EUF-CMA properties offered and are a practical consideration that
system designers and managers should be aware of when selecting among
hybrid approaches for their use case.
There are a couple approaches to alleviating this issue, as noted
above. One is on restricting key reuse. As described in
[I-D.ietf-lamps-pq-composite-sigs], prohibiting hybrid algorithm and
component algorithm signers and verifiers from using the same keys
can help ensure that a component verifier cannot be tricked into
verifying the hybrid signature. This would effectively put component
forgeries out of scope for a use case. One means for restricting key
reuse is through allowed key use descriptions in certificates. While
prohibiting key reuse reduces the risk of such component forgeries,
and is the mitigation described in
[I-D.ietf-lamps-pq-composite-sigs], it is still a policy requirement
and not a cryptographic assurance. Component forgery attacks may be
possible if the policy is not followed or is followed inconsistently
across all entities that might verify signatures using those keys.
This needs to be accounted for in any security analysis. Since
cryptographic provable security modeling has not historically
accounted for key reuse in this way, it should not be assumed that
systems with existing analyses are robust to this issue.
The other approach noted for alleviating the component forgery risk
is through hybrid signature selection of a scheme that provides
strong non-separability. Under this approach, the hybrid signature
cannot be separated into component algorithm signatures that will
verify correctly, thereby preventing the signature separation
required for the component forgery attack to be successful.
It should be noted that weak non-separability is insufficient for
mitigating risks of component forgeries. As noted in
[I-D.ietf-lamps-pq-composite-sigs], Sect. 11.3, in cases of hybrid
algorithm selection that provide only weak non-separability, key
reuse should be avoided, as mentioned above, to mitigate risks of
introducing EUF-CMA vulnerabilities for component algorithms.
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6. Discussion of Advantages/Disadvantages
The design (and hence, security guarantees) of hybrid signature
schemes depend heavily on the properties needed for the application
or protocol using hybrid signatures. It seems that not all goals can
be achieved simultaneously as exemplified below.
6.1. Backwards Compatibility vs. SNS
There is an inherent mutual exclusion between backwards compatibility
and SNS. While WNS allows for a valid separation under leftover
artifacts, SNS will ensure verification failure if a receiver
attempts separation.
6.2. Backwards Compatibility vs. Hybrid Unforgeability
Similarly, there is an inherent mutual exclusion between backwards
compatibility, when acted upon, and hybrid unforgeability as briefly
mentioned above. Since the goal of backwards compatibility is
usually to allow legacy systems without any software change to be
able to process hybrid signatures, all differences between the legacy
signature format and the hybrid signature format must be allowed to
be ignored, including skipping verification of signatures additional
to the classical signature. As such, if a system does skip a
component signature, security does not rely on the security of all
component signatures. Note that this mutual exclusion occurs at the
verification stage, as a hybrid signature that is verified by a
system that can process both component schemes can provide hybrid
unforgeability even if another (legacy) system, processing the same
hybrid signature, loses that property.
6.3. Simultaneous Verification vs. Low Need for Approval
Hybrid algorithms that achieve simultaneous verification tend to fuse
(or 'entangle') the verification of component algorithms such that
verification operations from the different component schemes depend
on each other in some way. Consequently, there may be a natural
connection between achieving simultaneous verification and a higher
need-for-approval. As a contrasting example, NIST accommodate
concatenation of a FIPS approved signature and another (potentially
non-FIPS approved) signature without any artifacts in FIPS 140
validation [NIST_PQC_FAQ], however as the component signatures are
verified separately it is not possible to enforce 'simultaneous
verification'.
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7. 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.
8. IANA Considerations
This document has no IANA actions.
9. Acknowledgements
This document 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 document forward, offered
useful insights and perspectives, and/or stimulated work in the area:
D.J. Bernstein, Scott Fluhrer, Felix Günther, John Gray, Serge
Mister, Max Pala, Mike Ounsworth, Douglas Stebila, Falko Strenzke,
Brendan Zember
10. References
10.1. Normative References
[I-D.ietf-tls-hybrid-design]
Stebila, D., Fluhrer, S., and S. Gueron, "Hybrid key
exchange in TLS 1.3", Work in Progress, Internet-Draft,
draft-ietf-tls-hybrid-design-13, 17 June 2025,
<https://datatracker.ietf.org/doc/html/draft-ietf-tls-
hybrid-design-13>.
[RFC4949] Shirey, R., "Internet Security Glossary, Version 2",
FYI 36, RFC 4949, DOI 10.17487/RFC4949, August 2007,
<https://www.rfc-editor.org/rfc/rfc4949>.
[RFC9794] Driscoll, F., Parsons, M., and B. Hale, "Terminology for
Post-Quantum Traditional Hybrid Schemes", RFC 9794,
DOI 10.17487/RFC9794, June 2025,
<https://www.rfc-editor.org/rfc/rfc9794>.
10.2. Informative References
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[EUFCMA] Green, M., "EUF-CMA and SUF-CMA", n.d.,
<https://blog.cryptographyengineering.com/euf-cma-and-suf-
cma/>.
[FALCON] "FALCON: Fast-Fourier Lattice-based Compact Signatures
over NTRU", n.d., <https://falcon-sign.info/falcon.pdf>.
[FS] Fiat, A. and A. Shamir, "How To Prove Yourself: Practical
Solutions to Identification and Signature Problems", 1986,
<https://doi.org/10.1007%2F3-540-47721-7_12>.
[GEMSS] "GeMSS: A Great Multivariate Short Signature", 15 April
2020, <https://www-polsys.lip6.fr/Links/NIST/
GeMSS_specification_round2_V2.pdf>.
[HQC_CVE] "Correctness error in HQC decapsulation", 6 December 2024,
<https://nvd.nist.gov/vuln/detail/CVE-2024-54137>.
[HYBRIDSIG]
Bindel, N., Herath, U., McKague, M., and D. Stebila,
"Transitioning to a Quantum-Resistant Public Key
Infrastructure", May 2017,
<https://eprint.iacr.org/2017/460>.
[HYBRIDSIGDESIGN]
Bindel, N. and B. Hale, "A Note on Hybrid Signature
Schemes", March 2023, <https://eprint.iacr.org/2023/423>.
[I-D.becker-guthrie-noncomposite-hybrid-auth]
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, 22 March 2022,
<https://datatracker.ietf.org/doc/html/draft-becker-
guthrie-noncomposite-hybrid-auth-00>.
[I-D.ietf-lamps-pq-composite-sigs]
Ounsworth, M., Gray, J., Pala, M., Klaußner, J., and S.
Fluhrer, "Composite ML-DSA for use in X.509 Public Key
Infrastructure", Work in Progress, Internet-Draft, draft-
ietf-lamps-pq-composite-sigs-06, 18 June 2025,
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pq-composite-sigs-06>.
[KYBERSLASH]
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in Kyber implementations", 30 June 2024,
<https://eprint.iacr.org/2024/1049>.
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[MLDSA] National Institute of Standards and Technology (NIST),
"Module-Lattice-Based Digital Signature Standard", 13
August 2024, <https://doi.org/10.6028/NIST.FIPS.204>.
[NIST_PQC_FAQ]
National Institute of Standards and Technology (NIST),
"Post-Quantum Cryptography FAQs", 5 July 2022,
<https://csrc.nist.gov/Projects/post-quantum-cryptography/
faqs>.
[QRCSP] Bernstein, D., "Quantifying risks in cryptographic
selection processes", 24 November 2023,
<https://cr.yp.to/papers/qrcsp-20231124.pdf>.
[RAINBOW] "PQC Rainbow", n.d., <https://www.pqcrainbow.org/>.
[RSA] Rivest, R. L., Shamir, A., and L. Adleman, "A Method for
Obtaining Digital Signatures and Public-Key
Cryptosystems", n.d.,
<https://people.csail.mit.edu/rivest/Rsapaper.pdf>.
Authors' Addresses
Nina Bindel
SandboxAQ
Email: nina.bindel@sandboxaq.com
Britta Hale
Naval Postgraduate School
Email: britta.hale@nps.edu
Deirdre Connolly
SandboxAQ
Email: durumcrustulum@gmail.com
Florence Driscoll
UK National Cyber Security Centre
Email: flo.d@ncsc.gov.uk
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