lamps                                                          R. Struik
Internet-Draft                               Struik Security Consultancy
Intended status: Standards Track                            Oct 25, 2021
Expires: April 28, 2022

            ECDSA Signatures in Verification-Friendly Format


   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 with instantiations with prime-
   order curves.  This facilitates verifying devices to reap the
   significant speed-up potential (ranging from ~1.3x to ~6x) fast
   verification techniques afford.

Requirements Language

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "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.

   Internet-Drafts are working documents of the Internet Engineering
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   This Internet-Draft will expire on April 28, 2022.

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Copyright Notice

   Copyright (c) 2021 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

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   Provisions Relating to IETF Documents
   ( in effect on the date of
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   described in the Simplified BSD License.

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 . . .   7
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .   7
   8.  Privacy Considerations  . . . . . . . . . . . . . . . . . . .   7
   9.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .   8
     9.1.  OIDs for Use with PKIX and CMS  . . . . . . . . . . . . .   8
     9.2.  Algorithm Id for ECDSA* with OpenPGP  . . . . . . . . . .   9
     9.3.  Other Uses  . . . . . . . . . . . . . . . . . . . . . . .   9
   10. Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  10
   11. References  . . . . . . . . . . . . . . . . . . . . . . . . .  10
     11.1.  Normative References . . . . . . . . . . . . . . . . . .  10
     11.2.  Informative References . . . . . . . . . . . . . . . . .  11
   Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  12

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

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   scheme endorsed by regulatory bodies in both the United States and
   the European Union.

   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 instantiated with prime-order curves, 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 (for more details, see, e.g., Appendix Q of
   [I-D.ietf-lwig-curve-representations]).  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

   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

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   (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
   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 ~6x 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).

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   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
   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

   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);

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   b.  Old device: implement flimsy parser that replaces new label by
       old label and proceed as with traditional ECDSA verification.

   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.

   The so-called "Smart Health" framework (FHIR, HL7) uses JSON Web
   Signatures, where signed credentials use ECDSA with NIST curve P-256
   and SHA-256 (ES256).  This framework is rolled out for exchanging
   healthcare-related credentials in the USA and is also considered in
   many contexts where Covid vaccination proof is required for travel
   (e.g., with the IATA Health Pass, NY State Excelsior Pass, QR-code

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   based vaccination records in several Canadian provinces).  The EU
   covid certificate also uses ECDSA-based digital signatures.  All
   these applications could benefit from speedier verification, esp. in
   mass gathering settings, where this draft simply enables this
   functionality, both in single verification and batch verification

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;

   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

   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.

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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};

   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

   As mentioned in Section 4, any ECDSA signature with the old object
   identifier can be changed retroactively to one with the corresponding

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   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 and 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).

9.2.  Algorithm Id for ECDSA* with OpenPGP

   This section registers the ECDSA signature scheme with the modified
   signing procedure of this document as the public-key algorithm ECDSA*
   (with ID=25) in Section 9.1 of [I-D.ietf-openpgp-crypto-refresh] by
   including the following item in Table 15 of that section:

                           |  ID | Algorithm  |
                           |  25 | ECDSA*     |

                  Table 1: Public-Key Algorithm Registry

   As before, the provisions of Section 4 apply.

9.3.  Other Uses

   As suggested in Section 4, any party can retroactively put ECDSA
   signatures into the verification-friendly format, thereby conceivably
   allowing this to be done once and for all for all existing ECDSA
   signatures, no matter the application.  In particular, one could
   apply this to ECDSA-based certificate chains, ECDSA-signed firmware
   updates, COSE, JOSE, etc., etc.  In other words: going forward, never
   use ECDSA signing, always use ECDSA* signing.

   Note: Appendix Q.3.4 of [I-D.ietf-lwig-curve-representations] gives
   an example of how to convert JSON Web Signatures using ECDSA into one
   using ECDSA*, simply by picking the alternative representation where

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   the ephemeral signing key is of the required form (thereby,
   guaranteeing uniqueness in practice).

   Similar techniques can be used to put the German ECGDSA signature
   scheme, the Russian GOST signature scheme, and Chinese SM2 signature
   in a verification-friendly format, although this cannot be done
   retroactively without changing the signature format (it requires one
   extra bit).  Further details are left to a future version of this

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, "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.

              Struik, R., "Alternative Elliptic Curve Representations",
              draft-ietf-lwig-curve-representations-21 (work in
              progress), June 2021.

              Koch, W. and P. Wouters, "OpenPGP Message Format", draft-
              ietf-openpgp-crypto-refresh-04 (work in progress), October

   [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,

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   [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,

   [RFC7515]  Jones, M., Bradley, J., and N. Sakimura, "JSON Web
              Signature (JWS)", RFC 7515, DOI 10.17487/RFC7515, May
              2015, <>.

   [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,

   [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,

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   [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

   Rene Struik
   Struik Security Consultancy


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