Re: [Lwip] Iotdir early review of draft-ietf-lwig-curve-representations-08

Rene Struik <> Mon, 10 August 2020 19:13 UTC

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To: Daniel Migault <>, Daniel Migault <>
Cc: "" <>, "" <>, "" <>
References: <> <> <> <> <> <>
From: Rene Struik <>
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Subject: Re: [Lwip] Iotdir early review of draft-ietf-lwig-curve-representations-08
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Hi Daniel:

Thanks for your additional feedback.

To address your nits (except the last one), I slightly reworded the
first section as below. I suggest I incorporate this in an official new
upload once this document moves along further (just to avoid a
proliferation of tiny deltas). {I will then also try and rethink your
last nit.} I hope this works.

BTW - all switches between curve models work equally well for any curve
in any of the models (as long as the mathematical object does exist). In
particular, it works equally well for Curve25519 and Curve448 flavors
and with protocols that use these as building blocks. {It also works for
pairing-based crypto and, e.g., supersingular curves used with
isogeny-based post-quantum schemes. (No need to worry about the latter:
I simply wanted to express that the exposition in the draft is general
enough to cover all of those, even if one only cares about traditional
elliptic curves.)}

Slightly reworded first section:

1.  Fostering Code Reuse with New Elliptic Curves

   Elliptic curves can be represented using different curve models.
   Recently, IETF standardized elliptic curves that are claimed to have
   better performance and improved robustness against "real world"
   attacks than curves represented in the traditional "short"
   Weierstrass curve model.  These so-called CFRG curves [RFC7748] use
   the Montgomery curve model and the model of twisted Edwards curves.

   In this document, we specify these curves using the traditional
   "short" Weierstrass model and also define how to efficiently switch
   between representations in these different curve models.  In
   particular, we specify Wei25519, which allows an alternative
   representation of points of Curve25519 (a Montgomery curve) and of
   points of Edwards25519 (a twisted Edwards curve), as points of a
   corresponding "short" Weierstrass curve.  Similarly, we specify
   Wei448, which allows an alternative representation of points of
   Curve448 (a Montgomery curve) and of points of Ed448 (an Edwards
   curve), as points of a corresponding "short" Weierstrass curve.

   Use of Wei25519 and Wei448 allows easy definition of new
   instantiations of signature schemes and key agreement schemes already
   specified for traditional NIST prime curves, thereby allowing easy
   integration with existing specifications, such as NIST SP 800-56a
   [SP-800-56a], FIPS Pub 186-4 [FIPS-186-4], and ANSI X9.62-2005
   [ANSI-X9.62], and fostering code reuse on platforms that already
   implement some of these schemes using elliptic curve arithmetic for
   curves in "short" Weierstrass form (see Appendix C.1).  To illustrate
   this, we specify how to use Wei25519 and Wei448 with co-factor ECDH

   and with ECDSA, thereby giving rise to the key agreement schemes
   ECDH25519 and ECDH448 and the signature schemes ECDSA25519 and
   ECDSA448.  In all these cases, implementors may use the curve
   arithmetic for the curve model of their choosing (where they can
   efficiently switch between representations in different curve models,
   if required).

   For ease of exposition, we consider Wei25519 first and introduce
   Wei448 simply as an illustration of how to create other "offspring"
   objects and protocols (see Section 4.4).  We also provide extensive
   background material that we hope may be useful for implementors of
   elliptic curve cryptography or for cross-referencing with future
   specification work.

On 2020-08-10 10:35 a.m., Daniel Migault wrote:
> Hi, 
> I expected to provide additional comments after the IETF but I
> happened to be sick last week which explains the slow response. I
> apology for it.
> The document seems to me very clear and nicely written.  This version
> addresses my earlier comments. 
> Some nits:
> In the Introduction
> I think it would be nice to specify that the document provides
> guideline to do similar work for 448 curves or if it also specifies
> Wei448.
> From the current text it seems not.
> A nit the sentence "This document ... " is pretty long and it maybe
> would be clearer to say "This document specifies Wei25519, an
> alternative ...." but that is very personal ;-)
> ECDSA25519 is also not mentioned at all in the introduction. It seems
> to me important to position it briefly. More specifically, - at least
> this is my understanding of it -  Wei25519 was motivated to enable the
> implementation of Montgomery curve Curve25519 operations, re-using
> implementations for the NIST curves while ECDSA25519 enables the
> implementation of ECDSA with Montgomery curve Curve25519. This seems
> to me slightly different. 
> just before section 10.2.1:  s/scheme scheme/scheme
> I have the impression section 4.1 - 4.4 are more than Examples ;-)
> Yours, 
> Daniel
> On Mon, Jul 20, 2020 at 9:54 AM Daniel Migault
> <
> <>> wrote:
>     Hi Rene, 
>     Thank you for the feed back - I am just back from vacation. As far
>     as I remember, my comments mostly concerned some clarifications
>     and should not be seen as preventing the document to move forward
>     - especially for an early review. I did appreciated the way you
>     handled the comments and believe your are the best person to  know
>     and decide how to handle these comments. 
>     I have not reviewed the latest version, but I can commit to review
>     the draft in August. I hope that does not prevent the document to
>     be moved forward. 
>     Yours, 
>     Daniel 
>     ------------------------------------------------------------------------
>     *From:* Rene Struik <
>     <>>
>     *Sent:* Tuesday, July 14, 2020 1:43 PM
>     *To:* Daniel Migault <
>     <>>;
>     < <>>
>     *Cc:* <> <
>     <>>;
>     <>
>     <
>     <>>
>     *Subject:* Re: Iotdir early review of
>     draft-ietf-lwig-curve-representations-08
>     Hi Daniel:
>     It has been a while that you provided your early IoTDir review
>     comments. I do not know whether such early reviews are a gate
>     keeper for sealing off the lwig curve draft. Nevertheless, it may
>     be good to know if you are reasonably happy for this to move forward.
>     For some details on how I tried to take your comments into
>     account, please see my March 10th email below, where the mailing
>     list link is
>     For the current draft, see
>     I am looking forward hearing back from you..
>     Best regards, Rene
>     On 2020-03-10 12:01 p.m., Rene Struik wrote:
>>     Hi Daniel:
>>     I tried to take into account your comments with
>>     draft-ietf-lwig-curve-representations-09.
>>     I will send out a separate email to the list highlighting main
>>     changes.
>>     Below, I will explain how I tried and accommodate your previous
>>     comments on the 08-draft, now that the new draft is out.
>>     Important note:
>>     Based on offline communications with SecDir people, I did include
>>     Curve448-related material with the 09 version. I tried to do this
>>     in a way that would create the least editorial upheaval and would
>>     not take away from the main objectives of the document: to this
>>     end, I only introduced Curve448-related material in Section 4.4
>>     [thereby avoiding having to sprinkle in curve448-related language
>>     in each section and blurring the message of the document]. The
>>     only other sections where Curve448 plays a role is IANA Section
>>     10 and the appendices that give the parameters and test vectors
>>     for the Curve448 family members. {Note: I probably should have
>>     added a note on ECDSA448 in the security consideration section,
>>     though (I made a note on this and will fix).}
>>     On 10/25/2019 4:26 PM, Rene Struik wrote:
>>>     Hi Daniel:
>>>     Thanks very much for your constructive review.
>>>     I provided brief feedback on some of your comments - see below
>>>     (RS>> <<RS). Where I did not provide a comment yet, I will
>>>     reflect on these later (and this may benefit from looking at the
>>>     results from the pending SecDir review, once ready, at the same
>>>     time).
>>>     Best regards, Rene
>>>     On 10/25/2019 2:38 PM, Daniel Migault via Datatracker wrote:
>>>>     Reviewer: Daniel Migault
>>>>     Review result: Almost Ready
>>>>     Hi,
>>>>     I have reviewed this document as part of the IoT directorate's
>>>>     ongoing effort to review all IETF documents being processed by the
>>>>     IESG.  These comments were written primarily for the benefit of the
>>>>     internet area directors.  Document editors and WG chairs should treat
>>>>     these comments just like any other last call comments.
>>>>     The summary of the review is Almost Ready
>>>>     Overall I believe the document is well written with a lot of useful
>>>>     information. Though I have not carefully reviewed the annexes, I believe the
>>>>     document could be re-organized to  provide more implementation details of
>>>>     Wei25519 and its potentially the implementation of the isogeny between
>>>>     Wei25519.-3 and Wei25519. Note that the latest point may depend on the
>>>>     implementation status of NIST curves.
>>>>     Please find my review, which I hope will help to move the document to be moved
>>>>     forward.
>>>>     Yours,
>>>>     Daniel
>>>>     Abstract
>>>>        This document specifies how to represent Montgomery curves and
>>>>        (twisted) Edwards curves as curves in short-Weierstrass form and
>>>>        illustrates how this can be used to carry out elliptic curve
>>>>        computations using existing implementations of, e.g., ECDSA and ECDH
>>>>        using NIST prime curves.
>>>>     <mglt>
>>>>     Overall I have the impression the document's intent is to define Wei25519 for
>>>>     COSE. The current document is very well documented and nicely written on the
>>>>     generic methodology used, that is using the Weierstrass models of curves using
>>>>     other models that is in our case twisted-Edward/Montgomery curves. My reading
>>>>     of the document is that it might benefit by being re-focused on Wei25519 and
>>>>     provide the necessary details in the main body - that is not the annex to
>>>>     implement: * Wei25519 - the domains parameters * the isogeny between
>>>>     Wei25519.-3 and Wei25519 - as this is necessary for reusing NIST curve
>>>>     implementation when a hard coded. That said, this depends on the implementation
>>>>     status of NIST curves and may be left in the annex.
>>>>     Specification is sufficient to assign code points which is sufficient with an
>>>>     Informational status.
>>>>     I would even have Wei25519 in the title. In my opinion the title is too generic
>>>>     and the scope should be narrow down a bit. </mglt>
>>>>     <mglt>
>>>>     The document introduces some notations, I believe these notations should be
>>>>     specified, and maybe use the same notation as RFC7748, if possible to avoid
>>>>     that we have one notation per document. </mglt>
>>>     RS>> Elliptic-curve and Finite Field-related notation is
>>>     introduced in Annex B,
>>>     see
>>>     This notation is well-aligned with conventions in ANSI, NIST,
>>>     SECG, and BSI specifications. It is also in line with RFC 6090.
>>>     <<RS
>>     RS1>> I did revisit notational convention, also based on
>>     reviewing the draft NIST SP 800-186 and draft FIPS Pub 186-5
>>     documents (out for public review Oct 31, 2019 till Jan 29, 2020;
>>     closed now). Based on this, I slightly edited Appendix B
>>     (backgrounder on curve terminology and finite fields). I did
>>     decide not to align with some recent cfrg notation, since -
>>     frankly - I believe it is a mess and the only outlier w.r.t.
>>     ANSI, NIST, SECG,and BSI specs. <<RS1
>>>>     1.  Fostering Code Reuse with New Elliptic Curves
>>>>        It is well-known that elliptic curves can be represented using
>>>>        different curve models.
>>>>     <mglt>
>>>>     nits: It might be preferred to remain factual and simply say "Elliptic curves
>>>>     can be represented using different models." </mglt>
>>>>        Recently, IETF standardized elliptic curves
>>>>        that are claimed to have better performance and improved robustness
>>>>        against "real world" attacks than curves represented in the
>>>>        traditional "short" Weierstrass model.
>>>>     <mglt>
>>>>     Not being a native english speaker, the sentence above may lead the reader to
>>>>     think that Weierstrass representation is one reason for less robustness and
>>>>     performance. If that were the case, the reader may wonder why do we want to
>>>>     move to that representation.
>>>>     I believe that before Montgomery or Edwards curves have been introduced curves
>>>>     with Weierstrass model were widely deployed and implemented. </mglt>
>>>     RS>>  The reason to use the word "claimed" is that, from my
>>>     perspective, it is unclear that the claim actually holds. {My
>>>     opinion on the matter is irrelevant for this document, though.}
>>>     As to performance, this depends on implementation details, see,
>>>     e.g., remarks on existing hardware implementations (see the
>>>     forelast para of 
>>>     <>).
>>>     As to robustness, see the second para of
>>>     <>).
>>>     As to different perspectives on performance vs. robustness, see
>>>     also Item 1.3b of p. 77 of
>>>     <>
>>>     To my knowledge, Weierstrass curves, such as the NIST prime
>>>     curves are still very much prevalent to-date (so "were widely
>>>     deployed" is probably something that actually still is).
>>>     <<RS
>>     RS1>> I kept the language as is, based on assessment above. <<RS1
>>>>        This document specifies an
>>>>        alternative representation of points of Curve25519, a so-called
>>>>        Montgomery curve, and of points of Edwards25519, a so-called twisted
>>>>        Edwards curve, which are both specified in [RFC7748], as points of a
>>>>        specific so-called "short" Weierstrass curve, called Wei25519.  We
>>>>        also define how to efficiently switch between these different
>>>>        representations.
>>>>     <mglt>
>>>>     I understand the switch can be performed in two ways, it might help the reader
>>>>     to distinguish two scenario one with a being a parameter and one with a being
>>>>     hard coded.
>>>>     We may also briefly expose why Curve25519 cannot be "proxied" by Ed-to-Wei
>>>>     followed by Wei25519. In which case Wei25519 would be an implementation detail.
>>>>     This would justify the need to add code points for Wei25519. </mglt>
>>>>        Use of Wei25519 allows easy definition of new signature schemes and
>>>>        key agreement schemes already specified for traditional NIST prime
>>>>        curves, thereby allowing easy integration with existing
>>>>        specifications, such as NIST SP 800-56a [SP-800-56a], FIPS Pub 186-4
>>>>        [FIPS-186-4], and ANSI X9.62-2005 [ANSI-X9.62], and fostering code
>>>>        reuse on platforms that already implement some of these schemes using
>>>>        elliptic curve arithmetic for curves in "short" Weierstrass form (see
>>>>        Appendix C.1).
>>>>     <mglt>
>>>>     Not being a cryptographer at all. I am wondering if the definition of "new"
>>>>     signature scheme is correct. At least it sounds a dangerous slope. I think the
>>>>     text wants to say it allows to re-use signature scheme defines for the NIST
>>>>     curves using another curves Wei25519, namely ECDSA in our case. </mglt>
>>>     RS>> New refers to a new instantiation, e.g., ECDSA with the
>>>     Wei25519 curve simply defines a new curve to be used with an
>>>     existing generically specified ECDSA specification. <<RS 
>>     RS1>> I tried and accommodate this with the following text in
>>     Section 4.3 (with [...] deleted from this email):
>>     FIPS Pub 186-4 [FIPS-186-4] specifies the signature scheme ECDSA
>>     and can be instantiated not just with the NIST prime curves, but
>>     also with other Weierstrass curves (that satisfy additional
>>     cryptographic criteria). In particular, one can instantiate this
>>     scheme with the Weierstrass curve Wei25519 and the hash function
>>     SHA-256 [FIPS-180-4], [...]
>>     We denote by ECDSA25519 the instantiation of ECDSA with SHA-256
>>     and with curve Wei25519, where the signature (r,s) is represented
>>     as the right-concatenation of the integers r and s, each
>>     represented as fixed-size strings with tight MSB/msb ordering
>>     (see Appendix I).
>>     <<RS1
>>>>     2.  Specification of Wei25519
>>>>        For the specification of Wei25519 and its relationship to Curve25519
>>>>        and Edwards25519, see Appendix E.
>>>>     <mglt>
>>>>     If we are defining Wei25519, it seems that at least the definition of Wei25519
>>>>     needs to be part of the main draft. A deep explanation as well as other
>>>>     information may be left in the annexe. I would thus probably recommend to add
>>>>     the necessary domain parameters that are needed to instantiate Wei25519. </mglt>
>>>>        For further details and background
>>>>        information on elliptic curves, we refer to the other appendices.
>>>>        The use of Wei25519 allows reuse of existing generic code that
>>>>        implements short-Weierstrass curves, such as the NIST curve P-256, to
>>>>        also implement the CFRG curves Curve25519 and Edwards25519.  We also
>>>>        cater to reusing of existing code where some domain parameters may
>>>>        have been hardcoded, thereby widening the scope of applicability.  To
>>>>        this end, we specify the short-Weierstrass curves Wei25519.2 and
>>>>        Wei25519.-3, with hardcoded domain parameter a=2 and a=-3 (mod p),
>>>>        respectively; see Appendix G.  (Here, p is the characteristic of the
>>>>        field over which these curves are defined.)
>>>>     <mglt>
>>>>     The text needs here a bit more more explanations. As I understood, we are not
>>>>     simply changing the representation but also considering the constraint of the
>>>>     NIST curves implementations. My understanding is that NIST curves have hard
>>>>     coded a=-3, so we need to work around this with the isogeny. The construction
>>>>     is, in my opinion, a bit different from what simple isomorphism. It might also
>>>>     be good to have that overview stated in the introduction section.
>>>>     Annexes are good, but the necessary information for an implementation may be
>>>>     moved up to the main body.
>>>>     </mglt>
>>     RS1>>  I reflected on this and investigated the impact on the
>>     integrity and editorial cohesion of the document. Main problem
>>     with moving the actual domain parameters of Wei25519 to the main
>>     body is that one still would have to understand where these come
>>     from, thereby requiring a look at the mappings from Montgomery to
>>     Weierstrass curves and vice-versa (Annex D), their instantiation
>>     in this particular case (Annex E), where for isogenies one would
>>     have to delve into Annex F (general informative text) and Annex G
>>     (instantiation in this particular case). With the added
>>     Curve448-related material, this would become even messier, esp.
>>     since - in Annex M.3 - I had to add a note that some parameters
>>     in RFC7748 and some mappings in there are actually incorrect.
>>     Since the main body focuses on benefit of switching between
>>     different curve models, even if one does not care about a
>>     particular curve in question, I thought it best to stick to
>>     emphasizing that point (see Section 3 - Use of Representation
>>     Switches) and not drive potential readers away with long
>>     parameter lists and minutiae in the main body. If there is one
>>     thing I would like to stick with readers of the draft, it is how
>>     representation switches work in general and how these could help
>>     in code reuse of specification reuse.
>>     <<RS1
>>>>     3.  Use of Representation Switches
>>>>        The curve Wei25519.-3 (which has hardcoded domain parameter a=-3 (mod
>>>>        p)) is not isomorphic to the curve Wei25519, but is related in a
>>>>        slightly weaker sense: the curve Wei25519 is isogenous to the curve
>>>>        Wei25519.-3, where the mapping of Appendix G.2 is an isogeny of
>>>>        degree l=47 that maps the specified base point G of Wei25519 to the
>>>>        specified base point G' of Wei25519.-3 and where the so-called dual
>>>>        isogeny (which maps Wei25519.-3 to Wei25519) has the same degree
>>>>        l=47, but does not map G' to G, but to a fixed multiple hereof, where
>>>>        this multiple is l=47.  Consequently, a public-private key pair
>>>>        (k,R:=k*G) for Wei25519 corresponds to the public-private key pair
>>>>        (k, R':= k*G') for Wei25519.-3 (via the l-isogeny), whereas the
>>>>        public-private key pair (k, R':=k*G') corresponds to the public-
>>>>        private key pair (l*k, l*R=l*k*G) of Wei25519 (via the dual isogeny).
>>>>        (Note the extra scalar l=47 here.)
>>>>     <mglt>
>>>>     This is just a comment that ascii does not help reading 1=47... but there is no
>>>>     much we can do. </mglt>
>>     RS1>> Agreed. However, lots of variables already have a
>>     connotation, so sticking to l (length or degree) seemed best. <<RS1
>>>>        Alternative curve representations can, therefore, be used in any
>>>>        cryptographic scheme that involves computations on public-private key
>>>>        pairs, where implementations may carry out computations on the
>>>>        corresponding object for the isomorphic or isogenous curve and
>>>>        convert the results back to the original curve (where, in case this
>>>>        involves an l-isogeny, one has to take into account the factor l).
>>>>        This includes use with elliptic-curve based signature schemes and key
>>>>        agreement and key transport schemes.
>>>>     <mglt>I believe we should specify somewhere that changing the representation
>>>>     does not ease the ECDLP for a given prime/curve</mglt>
>>>     RS>> This is a good observation. I captured this generally in
>>>     the Security Considerations section (see first para of
>>>     <>)
>>>     and also reminded the reader of this when introducing each
>>>     isomorphism discussed in the document. See
>>>     <>
>>>     where I added this remark for the isomorphic mapping from
>>>     twisted Edwards to Montgomey (Annex D.1, end of first para),
>>>     Montgomery to Weierstrass (Annex D.2, end of first para). I did
>>>     not repeat this for the twisted Edwards to Weierstrass mapping
>>>     (Annex D.3) or with the low-degree isogenies (Annex F), but
>>>     could certainly sprinkle this in there as well, if so desired.
>>>     <<RS
>>     RS1>> If you feel I should add more than I did with rev-09
>>     already, please let me know and I can sprinkle this in. <<RS1
>>>>     4.  Examples
>>>>     <mglt>
>>>>     I do not believe these are example regarding the focus of the document. Of
>>>>     course, the document could be more generic, but here we focus on 25519, so I
>>>>     would rename Example as Implementation. Or move subsection to one level, that
>>>>     is 4.1 -> 4. 4.2->5... </mglt>
>>     RS1>> I kept the header "examples" in (sorry), since these are
>>     examples of how representation switches can be used to define
>>     what in Section 4.4 I call "offspring protocols". That is only a
>>     section heading, though. <<RS1
>>>>     8.  Security Considerations
>>>>        Elliptic curves are generally used as objects in a broader
>>>>        cryptographic scheme that may include processing steps that depend on
>>>>        the representation conventions used (such as with, e.g., key
>>>>        derivation following key establishment).  These schemes should
>>>>        (obviously) unambiguously specify fixed representations of each input
>>>>        and output (e.g., representing each elliptic curve point always in
>>>>        short-Weierstrass form and in uncompressed tight MSB/msb format).
>>>>     <mglt>
>>>>     I think it might be worth mentioning here that while the scheme used in this
>>>>     document may be applied for other curves that 25519, not all modules are
>>>>     isomorphic to Weierstrass. In addition not all Weierstrass representation can
>>>>     be switched to a Montgomery representation. </mglt>
>>>     RS>> As you suggested above, not all curves can be expressed in Montgomery or twisted Edwards format (see also the 6th para of Appendix B.1: This being said, all [non-binary] curves *are* isomorphic to one in short-Weierstrass format, so that the short-Weierstrass format can be thought of as a "catch all" representation format. 
>>>     <<RS
>>     RS1>> As explained above. <<RS1
>>>>        To prevent cross-protocol attacks, private keys SHOULD only be used
>>>>        with one cryptographic scheme.  Private keys MUST NOT be reused
>>>>        between Ed25519 (as specified in [RFC8032]) and ECDSA25519 (as
>>>>        specified in Section 4.3).
>>>>        To prevent intra-protocol cross-instantiation attacks, ephemeral
>>>>        private keys MUST NOT be reused between instantiations of ECDSA25519.
>>>     -- 
>>>     email: <> | Skype: rstruik
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>>     -- 
>>     email: <> | Skype: rstruik
>>     cell: +1 (647) 867-5658 | US: +1 (415) 287-3867
>     -- 
>     email: <> | Skype: rstruik
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> Daniel Migault
> Ericsson

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