lwig                                                           R. Struik
Internet-Draft                               Struik Security Consultancy
Intended status: Informational                         February 22, 2021
Expires: August 26, 2021

            ECDSA Signatures in Verification-Friendly Format


   This document specifies how to represent ECDSA signatures so as to
   facilitate fast verification of single signatures and fast batch
   verification.  We illustrate that this 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.

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

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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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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 . . . . . . . . . . . . . . . . . . . .   5
   6.  Security Considerations . . . . . . . . . . . . . . . . . . .   6
   7.  Privacy Considerations  . . . . . . . . . . . . . . . . . . .   6
   8.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .   6
   9.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .   6
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .   6
     10.1.  Normative References . . . . . . . . . . . . . . . . . .   6
     10.2.  Informative References . . . . . . . . . . . . . . . . .   7
   Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .   7

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.

   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 fill

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   this seeming void 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.

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

   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
   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

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   (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, hen so is (-R,-s), where the latter relies on the
   fact that the conversion function only depends on the x-coordinate of

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 (~30% and up).
   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], 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 generally 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 y-coordinate, change (r,s)
       to (r,-s).

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   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.  In practice, this is the case for all prime-order curves,
   including the NIST prime curves P-256, P-384, P-521, and all
   standardized Brainpool curves.

   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

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 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 if
       possible (and desired);

   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.

   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).

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 specification

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6.  Security Considerations

   The representation conversions described in this document are
   publicly known and, therefore, do not affect security provisions.

7.  Privacy Considerations

   The representation conversions described in this document are
   publicly known and, therefore, do not affect privacy provisions.

8.  IANA Considerations

   With the current draft, no IANA code point assignments are requested.

9.  Acknowledgements

   place holder.

10.  References

10.1.  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-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,

   [RFC7748]  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
              for Security", RFC 7748, DOI 10.17487/RFC7748, January
              2016, <https://www.rfc-editor.org/info/rfc7748>.

   [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,

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   [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, <https://www.rfc-editor.org/info/rfc8174>.

   [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.

10.2.  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,

   [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, 2005.

   [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, 2005.

Author's Address

   Rene Struik
   Struik Security Consultancy

   Email: rstruik.ext@gmail.com

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