lamps R. Struik
Internet-Draft Struik Security Consultancy
Intended status: Standards Track March 11, 2021
Expires: September 12, 2021
ECDSA Signatures in Verification-Friendly Format
draft-struik-lamps-verification-friendly-ecdsa-01
Abstract
This document specifies how to represent ECDSA signatures so as to
facilitate accelerated verification of single signatures and fast
batch verification. We demonstrate that this representation
technique can be applied retroactively by any device (rather than
only by the signer), thereby facilitating transitioning to always
generating ECDSA signatures in this way, without changing
standardized ECDSA specifications. This facilitates verifying
devices to reap the significant speed-up potential (ranging from
~1.3x to more than 2x) fast verification techniques afford.
Requirements Language
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in BCP
14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
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This Internet-Draft will expire on September 12, 2021.
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Copyright Notice
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Table of Contents
1. Fostering Fast Verification with ECDSA . . . . . . . . . . . 2
2. Review of ECDSA and ECDSA* . . . . . . . . . . . . . . . . . 3
3. Signature Verification with ECDSA and ECDSA* . . . . . . . . 4
4. Transitionary Considerations . . . . . . . . . . . . . . . . 5
5. Implementation Status . . . . . . . . . . . . . . . . . . . . 6
6. Informal Comparison with Speed-ups for EdDSA Signatures . . . 6
7. Security Considerations . . . . . . . . . . . . . . . . . . . 7
8. Privacy Considerations . . . . . . . . . . . . . . . . . . . 7
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 7
9.1. OIDs for Use with PKIX and CMS . . . . . . . . . . . . . 7
10. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 9
11. References . . . . . . . . . . . . . . . . . . . . . . . . . 9
11.1. Normative References . . . . . . . . . . . . . . . . . . 9
11.2. Informative References . . . . . . . . . . . . . . . . . 10
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 10
1. Fostering Fast Verification with ECDSA
ECDSA is one of the most widely used elliptic-curve digital signature
algorithms. It has been standardized in FIPS Pub 186-4, ANSI X9.62,
BSI, SECG, and IETF, and is widely deployed by a plethora of internet
protocols specified by the Internet Engineering Task Force (IETF),
with industry specifications in the areas of machine-to-machine
communication, such as ZigBee, ISA, and Thread, with wireless
communication protocols, such as IEEE 802.11, with payment protocols,
such as EMV, with vehicle-to-vehicle (V2V) specifications, as well as
with electronic travel documents and other specifications developed
under a more stringent regulatory oversight regime, such as, e.g.,
ICAO and PIV. ECDSA is the only elliptic-curve based signature
scheme endorsed by regulatory bodies in both the United States and
the European Union.
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While methods for accelerated verification of ECDSA signatures and
for combining this with key computations have been known for over 1
1/2 decade (see, e.g., [SAC2005] and [SAC2010]), these have been
commonly described in technical papers in terms of ECDSA*, a slightly
modified version of ECDSA, where their use with standardized ECDSA
seems less well known. It is the purpose of this document to bridge
this gap and describe how ECDSA signatures can be easily generated to
facilitate more efficient verification, without failing. We
emphasize that this does not require changes to standardized
specifications of ECDSA, thereby allowing reuse of existing standards
and easy integration with existing implementations. We exemplify
this for ECDSA certificates.
2. Review of ECDSA and ECDSA*
In this section, we summarize the properties of the signature scheme
ECDSA and of the modified signature scheme ECDSA* that are relevant
for our exposition. The signature schemes are defined in terms of a
suitable elliptic curve E, hash function H, and several
representation functions, where n is the (prime) order of the base
point G of this curve, and where E is an elliptic curve in short-
Weierstrass form. For full details, we refer to the relevant
standards.
With the ECDSA signature scheme, the signature over a message m
provided by a signing entity with static private key d is an ordered
pair (r,s) of integers in the interval [1,n-1], where the value r is
derived from a so-called ephemeral signing key R:=k*G generated by
the signer via a fixed public conversion function and where the value
s is a function of the ephemeral private key k, the static private
key d, the value r and the value e derived from message m via hash
function H and representation hereof in the interval [0,n-1]. (More
specifically, one has e=s*k-d*r (mod n), where r is a function of the
x-coordinate of R.) A signature (r,s) over message m purportedly
signed by an entity with public key Q:=d*G is accepted if Q is indeed
a valid public key, if both signature components r and s are integers
in the interval [1,n-1] and if the reconstructed value R' derived
from the purported signature, message, and public key yields r, via
the same fixed conversion function as used during the signing
operation. (More specifically, one computes R':=(1/s)*(e*G+r*Q) and
checks that r is the same function of the x-coordinate of R'.)
With the ECDSA* signature scheme, one follows the same signing
operation, except that one outputs as signature the ordered pair
(R,s), rather than the pair (r,s), where R is the ephemeral signing
key; one accepts a signature (R,s) over message m purportedly signed
by an entity with public key Q by first computing the value r derived
from signature component R via the conversion function, checking that
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Q is indeed a valid public key and that both r and s are integers in
the interval [1,n-1], computing R':=(1/s)*(e*G+r*Q) and checking
whether, indeed, R'=R.
It is known that ECDSA signatures and the corresponding ECDSA*
signatures have the same success/failure conditions (i.e., ECDSA and
ECDSA* are equally secure): if (r,s) is a valid ECDSA signature for
message m purportedly signed by an entity with public key Q, then
(R',s) is a valid corresponding ECDSA* signature, where R':=(1/
s)(e*G+r*Q) is a point for which the conversion function yields r.
Conversely, if (R,s) is a valid ECDSA* signature for message m
purportedly signed by an entity with public key Q, then (r,s) is a
valid corresponding ECDSA signature, where r is obtained from R via
the conversion function.
It is well-known that if an ECDSA signature (r,s) is valid for a
particular message m and public key Q, then so is (r,-s) -- the so-
called malleability -- and that, similarly, if an ECDSA* signature
(R,s) is valid, then so is (-R,-s), where this relies on the fact
that the conversion function only depends on the x-coordinate of R.
3. Signature Verification with ECDSA and ECDSA*
In this section, we more closely scrutinize ECDSA and ECDSA*
verification processes.
With ECDSA*, signature verification primarily involves checking an
elliptic curve equation, viz. checking whether R = (1/s)*(e*G+r*Q),
which lends itself to accelerated signature verification techniques
and the ability to use batch verification techniques, with
significant potential for accelerated verification (with ~1.3x and up
and more than 2x speed-up potential, respectively). Here, speed-ups
are due to the availability of the point R, which effectively allows
checking an equation of the form -s*R + (e*G+r*Q)=O instead (where O
is the identity element of the curve). Similarly to the case with
EdDSA [RFC8032] (which natively represents the ephemeral signing key
R as part of the signature), this offers the potential for batch
verification, by checking a randomized linear combination of this
equation instead (thereby sharing the so-called point doubling
operations amongst all individual verifications and, potentially,
sharing scalars for signers of more than one message). In the case
of single verifications, efficient tricks allow reducing the bit-size
of the scalars involved in evaluating this expression (thereby
effectively halving the required point doubling operations).
With ECDSA itself, these techniques are generally not available,
since one cannot uniquely (and efficiently) reconstruct R from r:
both R and -R yield the same r value. If the conversion function
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only has two pre-images, though, one can use malleability to remove
ambiguity altogether.
The modified ECDSA signing procedure is as follows:
a. Generate ECDSA signature (r,s) of message m;
b. If the ephemeral signing key R has odd parity of the
y-coordinate, change (r,s) to (r,-s).
Note that this modified signing procedure removes the ambiguity in
the reconstruction of R from r if the conversion function would
otherwise only have two preimages, since R and -R have different
parity of the y-coordinate. In practice, this is the case for all
prime-order curves, including the NIST prime curves P-256, P-384,
P-521, all standardized Brainpool curves, and, e.g., the "BitCoin"
curve secp256k1. (This follows from the observation that, for prime-
order curves, r generally uniquely represents the x-coordinate of R.)
NOTE: With ECDSA, any party (not just the signer) can recompute the
ephemeral signing key R' from a valid signature, since R':=(1/
s)(e*G+r*Q). In particular, any party can retroactively put the
ECDSA signature in the required form above, thereby allowing
subsequent unique reconstruction of the R value from r by verifying
entities that know this modified signing procedure was indeed
followed (again, subject to the assumption that r would only have two
preimages otherwise, as is generally the case with prime-order
curves).
One can extend this technique to also apply to curves that have a
small co-factor h, e.g., h=4 or h=8 (rather than h=1, as is the case
with prime-order curves). This extension is out of scope for the
current document.
4. Transitionary Considerations
The modified signing procedure described in Section 3 facilitates the
use of accelerated ECDSA verification techniques by devices that wish
to do so, provided these know that this modified signing procedure
was indeed followed. This can be realized explicitly via a new
"fast-verification-friendly" label (e.g., OID) indicating that this
was indeed the case. This has the following consequences:
a. New device: accept both old and new label and apply speed-ups
with new label if possible (and desired);
b. Old device: implement flimsy parser that replaces new label by
old label and proceed as with traditional ECDSA verification.
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Note that this parser "label replacement" step is a public operation,
so any interface can implement this step.
A label can also be realized implicitly (e.g., by stipulating the
modified signing procedure in protocol specifications that use ECDSA
signatures), where the benefit of not having to introduce a new label
explicitly should be weighed against potential disadvantages of
implicit labels, such as requiring extra care with specification work
to avoid confusion and the likely need to reintroduce an explicit
label if ECDSA signatures are processed outside the original context
(e.g., using a generic crypographic token).
As suggested before, any device can implement the modified ECDSA
signing procedure retroactively, so one could conceivably implement
this once for all existing ECDSA signatures and only use "new" labels
once this task has been completed (i.e., old labels could be
mothballed from then on).
NOTE: the above labeling procedures assume that old and new labels
are not part of the message to be signed. If they are, one may not
be able to mothball old labels. In this case, signing devices should
always use the old label during ECDSA signing and only change this to
the corresponding new label afterwards, whereby verifying devices
always replace the new label (since simply a pseudonym) by the
corresponding old label before processing the ECDSA signature. This
ensures that the signature semantics are not impacted and that old
devices' ECDSA verification implementations (after reinstating old
labels) work as is, while still being able to flag verification-
friendly ECDSA signature formatting.
5. Implementation Status
[Note to the RFC Editor] Please remove this entire section before
publication, as well as the reference to [RFC7942].
The ECDSA* signature scheme has been implemented in V2V
specifications [P1609.2], where ECDSA is used with the NIST curves
P-224 and P-256.
6. Informal Comparison with Speed-ups for EdDSA Signatures
The main message of this draft is as follows (no crypto required,
except believing that the third step below works):
a. EdDSA [RFC8032] does allow speedy signature verification and
batch verification, since the signature is (R,s), i.e., it
represents the ephemeral signing key R as part of the signature;
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b. With ECDSA, the signature is (r,s), where r is derived from the
signing key R (essentially, r is the x-coordinate of R if the
curve has co-factor h=1). However, generally, one cannot go back
and get (r,s) --> (R,s), at least not efficiently;
c. If one uses the modified ECDSA signing procedure of Section 3,
one can, though, thereby allowing similar accelerations (30% and
up) for signature verification as EdDSA does. This can be viewed
as "point compression" (since it determines which of R and -R
apply);
d. The rest is detail, where the ideas underlying the speed-ups
informally described in Section 3 are described in detail in the
papers [SAC2005] and [SAC2010].
7. Security Considerations
The signature representation change described in this document is
publicly known and, therefore, does not affect security provisions.
Obviously, any adversary could change the signature value in a
malicious way, so as to make signature verification fail. This does,
however, not extend capabilities the adversary already had.
8. Privacy Considerations
The signature representation change described in this document is
publicly known and, therefore, does not affect privacy provisions.
9. IANA Considerations
This section requests the following IANA code point assignments.
Editorial Note: the approach below is simply one way of realizing
ECDSA* functionality. Other options to consider include, e.g.,
introducing a non-critical extension as label, where old devices can
simple ignore this. This will be elaborated upon further in next
versions of this draft, after feedback.
9.1. OIDs for Use with PKIX and CMS
This section registers the following object identifiers for the
verification-friendly version of ECDSA introduced in this document:
a. id-ecdsa-star-with-sha256 ::= {iso(1) identified-organization(3)
thawte (101) (100) 81};
b. id-ecdsa-star-with-sha384 ::= {iso(1) identified-organization(3)
thawte (101) (100) 82};
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c. id-ecdsa-star-with-sha512 ::= {iso(1) identified-organization(3)
thawte (101) (100) 83};
d. id-ecdsa-star-with-shake128 ::= {iso(1) identified-
organization(3) thawte (101) (100) 84};
e. id-ecdsa-star-with-shake256 ::= {iso(1) identified-
organization(3) thawte (101) (100) 85}.
Each of these object identifiers indicates the use of ECDSA with the
indicated hash function, as the corresponding object identifiers
without the "-star-" substring specified in [RFC5480] (for ECDSA with
SHA2-hash family members) and in [RFC8692] (for ECDSA with SHAKE
family members) do, where the "-star-" substring simply indicates
that the modified signing procedure specified in Section 3 of this
document was indeed used.
These new object identifiers are used with PKIX certificates and CMS
in the same way as the corresponding object identifiers without the
"-star-" substring, except that verifying devices now have the option
to implement ECDSA signature verification as if ECDSA* signatures had
been used, since the new object identifiers indicate the modified
signing operation was followed, as illustrated in Section 3 of this
document.
As mentioned in Section 4, any ECDSA signature with the old object
identifier can be changed retroactively to one with the corresponding
new object identifier, provided one has assurance that the modified
ECDSA signing procedure was indeed followed and, conversely, any
ECDSA signature with the new object identifier can be changed to one
with the corresponding old object identifier, without change in
semantics (assuming these object identifiers are not part of the
message that is signed).
With [RFC5280], the signature algorithm is indicated twice: once as
signatureAlgorithm field of the Certificate and once as the Signature
field of the sequence tbsCertificate, where the former is not part of
the message to be signed, whereas the latter is. Moreover, these two
fields are stipulated to be the same (see Sections 4.1.1.2 and
4.1.2.3 of [RFC5280]). In this case, old and new labels MUST be used
as indicated in the NOTE of Section 3, where the two fields
indicating the signature algorithm are always both changed at the
same time (thereby, strictly complying with MUST behavior of PKIX
that these two fields should be the same).
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10. Acknowledgements
Thanks to Rich Salz for suggesting to informally compare speed-ups
with ECDSA* with those of EdDSA (now in Section 6).
11. References
11.1. Normative References
[FIPS-186-4]
FIPS 186-4, "Digital Signature Standard (DSS), Federal
Information Processing Standards Publication 186-4", US
Department of Commerce/National Institute of Standards and
Technology, Gaithersburg, MD, July 2013.
[I-D.ietf-lwig-curve-representations]
Struik, R., "Alternative Elliptic Curve Representations",
draft-ietf-lwig-curve-representations-19 (work in
progress), December 2020.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[RFC5280] Cooper, D., Santesson, S., Farrell, S., Boeyen, S.,
Housley, R., and W. Polk, "Internet X.509 Public Key
Infrastructure Certificate and Certificate Revocation List
(CRL) Profile", RFC 5280, DOI 10.17487/RFC5280, May 2008,
.
[RFC5480] Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk,
"Elliptic Curve Cryptography Subject Public Key
Information", RFC 5480, DOI 10.17487/RFC5480, March 2009,
.
[RFC7942] Sheffer, Y. and A. Farrel, "Improving Awareness of Running
Code: The Implementation Status Section", BCP 205,
RFC 7942, DOI 10.17487/RFC7942, July 2016,
.
[RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017,
.
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[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, .
[RFC8692] Kampanakis, P. and Q. Dang, "Internet X.509 Public Key
Infrastructure: Additional Algorithm Identifiers for
RSASSA-PSS and ECDSA Using SHAKEs", RFC 8692,
DOI 10.17487/RFC8692, December 2019,
.
[SEC1] SEC1, "SEC 1: Elliptic Curve Cryptography, Version 2.0",
Standards for Efficient Cryptography, , June 2009.
[SEC2] SEC2, "SEC 2: Elliptic Curve Cryptography, Version 2.0",
Standards for Efficient Cryptography, , January 2010.
11.2. Informative References
[ECC] I.F. Blake, G. Seroussi, N.P. Smart, "Elliptic Curves in
Cryptography", Cambridge University Press, Lecture Notes
Series 265, July 1999.
[GECC] D. Hankerson, A.J. Menezes, S.A. Vanstone, "Guide to
Elliptic Curve Cryptography", New York: Springer-Verlag,
2004.
[P1609.2] IEEE 1609.2-2013, "IEEE Standard for Wireless Access in
Vehicular Environments-Security Services for Applications
and Management Messages", IEEE Vehicular Technology
Society, New York: IEEE, 2013.
[SAC2005] A. Antipa, D.R. Brown, R. Gallant, R. Lambert, R. Struik,
S.A. Vanstone, "Accelerated Verification of ECDSA
Signatures", SAC 2005, B. Preneel, S. Tavares, Eds.,
Lecture Notes in Computer Science, Vol. 3897, pp. 307-318,
Berlin: Springer, 2006.
[SAC2010] R. Struik, "Batch Computations Revisited: Combining Key
Computations and Batch Verifications", SAC 2010, A.
Biryukov, G. Gong, D.R. Stinson, Eds., Lecture Notes in
Computer Science, Vol. 6544, pp. 130-142, Berlin-
Heidelberg: Springer, 2011.
Author's Address
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Rene Struik
Struik Security Consultancy
Email: rstruik.ext@gmail.com
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