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

Rene Struik <> Tue, 10 March 2020 16:01 UTC

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From: Rene Struik <>
To: Daniel Migault <>,
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Date: Tue, 10 Mar 2020 12:01:16 -0400
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Subject: Re: [Lwip] Iotdir early review of draft-ietf-lwig-curve-representations-08
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Hi Daniel:

I tried to take into account your comments with 

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).


>> 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.


>> 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.
> -- 
>  | Skype: rstruik
> cell: +1 (647) 867-5658 | US: +1 (415) 690-7363

email: | Skype: rstruik
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