Re: [CFRG] [Cfrg] I-D Action: draft-irtf-cfrg-pairing-friendly-curves-08.txt

Yumi Sakemi <yumi.sakemi@lepidum.co.jp> Thu, 25 February 2021 11:38 UTC

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From: Yumi Sakemi <yumi.sakemi@lepidum.co.jp>
Date: Thu, 25 Feb 2021 20:38:16 +0900
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To: "cfrg@irtf.org" <cfrg@irtf.org>
Cc: Tetsutaro Kobayashi <tetsutaro.kobayashi.dr@hco.ntt.co.jp>, SAITO Tsunekazu <tsunekazu.saito.hg@hco.ntt.co.jp>, "Riad S. Wahby" <rsw@cs.stanford.edu>
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Subject: Re: [CFRG] [Cfrg] I-D Action: draft-irtf-cfrg-pairing-friendly-curves-08.txt
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Dear Rene

I'm so sorry for the late reply.
We have not been able to reply to the comments on the Normative
References received from you.
The following is our understanding as authors.

The current I-D is according to the understanding that Normative
References are described in RFC3967 as follows.
(We have also checked RFC 4897 and RFC 8067 which are updated versions
of RFC 3967.)

Section 1 in RFC 3967
---
 Broadly speaking, a normative reference specifies a document that
must be read to fully understand or implement the subject matter in
the new RFC, or whose contents are effectively part of the new RFC, as
its omission would leave the new RFC incompletely specified.
---

Because of the above, the current Normative References are carefully
selected to include ONLY the important documents (such as standards
and conference papers) that we need to understand the subject of the
draft or read to implement it.
As a result, our draft is in line with the idea expressed in RFC 3967,
and we strongly believe that there are no problems.

Also, RFC 2119 and RFC 8174 which are mentioned in the comments are
included in the Normative References in the current I-D, so we
consider that necessary normative references that you pointed out are
already included in the current version.

We appreciate your comments.

Best regards,
Yumi

2020年11月12日(木) 10:13 Rene Struik <rstruik.ext@gmail.com>om>:
>
> One more note: except for RFC 2119 and RFC 8174, none of the normative
> references are, in fact, normative (since, e.g., referring to a
> conference paper).  Whether all 7 pages of informative references are
> informative is in the eye of the beholder (it does hike up quotation
> numbers, though).
>
>
> On 2020-11-11 5:10 p.m., Rene Struik wrote:
> > Dear Yumi et al:
> >
> > Please find below my review of
> > draft-irtf-cfrg-pairing-friendly-curves-08. My apologies for the
> > getting back to you this late.
> >
> > Best regards, Rene
> >
> > ==
> >
> > Review of draft-irtf-cfrg-pairing-friendly-curves-08
> > Reviewer: Rene Struik
> > Assessment: again, not ready (perhaps, ditch?)
> > Note: page numbers in this review refer to rev-08
> >
> > General comments:
> > 1) It seems that at least half of my main technical comments (apart
> > from editorials) with my review of the rev-07 draft have not been
> > addressed. Moreover, it is somewhat hard to get back to the authors on
> > the mailing list, since they buried their comment resolutions in a
> > somewhat nebulous github structure. This makes it a little bit hard to
> > get enthusiastic about doing yet another review.
> > 2) It seems that the authors have not done any effort to disentangle
> > technical specification, availability of libraries, and actual
> > deployment (see my comment #01 of my rev07 review), where there still
> > is a somewhat misleading tilt due to equating all these disparate
> > things and where there is still an unusual prominence of company
> > names, etc. Perhaps, some of this can be taken care of by including a
> > section with "implementation status" (which disappears once this
> > becomes an RFC), as other IETF drafts have done. Nevertheless, this
> > would fly in the face of selection policies for pairing-based curves
> > which are based on conflated notions. This begs the question what the
> > purpose of this draft is.
> > 3) The draft, as it stands, still contains lots of material that is in
> > a state with much to be desired for. As an example, Appendix A
> > contains quite a few errors (Ate pairing) and lacks sufficient
> > explanatory context and Appendix C is such an adhoc hack that (I feel)
> > this should never be endorsed by CFRG. For details, see some of my
> > comments on rev08 (where I did focus on the appendices).
> > 4) It may be best if CFRG ditches this document, since it still seems
> > far off from being a useful document, written clearly so as to educate
> > the CFRG and the wider IETF community. (Given the lack of feedback on
> > this document over time, perhaps nobody would care.). Should CFRG
> > still wish to give this draft a chance to become a CFRG-issued RFC, I
> > believe it needs a major rewrite to get rid of nontechnical spin
> > material. I also think it would be beneficial not to have yet more
> > specifications come up with their own representations and procedures
> > where there is no real technical justification for this. Should the
> > authors wish to push this "as is", they could always pursue the
> > "independent stream" for this (where one could argue that nothing can
> > be changed). If this should be pushed furthr, I can imagine a slim
> > draft, without umpteen tables and zillions of references, where one
> > simply provides domain parameters and explains how mappings work
> > without all the ads for libraries and start-up companies with
> > non-empty github pages.
> >
> > For some detailed technical and editorial comments, see below. (I have
> > more comments, but I think the below suffices to show this is far from
> > ready.)
> >
> > I know this is not what authors want to hear and does not make me
> > popular, but so be it.
> >
> > Detailed comments:
> > 1) Appendix A: the description of the so-called line function is very
> > unclear for non-pairing people (which I presume is most of the CFRG),
> > where expressing this in a style resembling elliptic curve notation
> > more would certainly help. Some suggestions: express the line function
> > as Line_function(P1,P2,Q), where P1:=(x1,y1) and P2:=(x2,y2) are
> > affine points of G2 (curve defined over GF(p^2)) and where Q:=(X,Y) is
> > an affine point of G1 (curve defined over GF(p)), mapping to subgroup
> > G_T of GF(p^k)^*. This allows expressing the number l if P1=P2 as
> > l:=lambda:=(3*x1^2+a)/(2*y1), which is familiar in point doubling
> > formulas for short-Weierstrass curves (where, here, domain parm a is
> > zero) and, similarly, if P1<> (+/-) P2, as l:=lambda:=(y2-y1)/(x2-x1)
> > (see, e.g., [3]). Here, it would also help explaining what happens in
> > case of non-affine inputs (point-at-infinity) and why the output would
> > be an element of the prime-order multiplicative subgroup G_T of
> > GF(p^k)^*.
> > 2) Appendix A, last para before A.1: one should mentions the image of
> > the point-at-infinity under this Frobenius map.
> > 3) Appendix A.1: this section seems to be full of ambiguities and
> > errors: (a) should one assume the parameter t is the same as the one
> > definining the prime p, etc.? (if so, please say so); (b) write
> > c:=c0+c1*2+ ... + c_{L}*2^{L} and explain that c_L is nonzero and that
> > the Ate pairing procedure works no matter the choice of the
> > coefficients in the range {-1,0,1} (if not, much explanation is
> > needed, since, e.g., c=0+1=2-1 may have different results; (c) fix the
> > loop to "for i:=L-1 downto 0"; (d) replace the silly "|" language in
> > the if condition "c_i=1 | c_i=-1" by "c_i nonzero" (you are writing
> > this for humans, not for parsers); (d) the Line_function has three
> > arguments, but in the call Line_function(T, ci*Q) it only has two:
> > should this include P as third argument?; (e) how does one know the
> > line function never receives non-affine points as input?; (f) the line
> > with frobenius maps seems to compute Q1:=frob(Q); Q2:=frob(Q1) and, if
> > so, doesn't one get Q2=frob(frob(Q))=Q since Q is a point of G2 and,
> > thereby a point of a curve defined over GF(p^2)?
> > 4) Appendix A.1: how does one know that f is never the zero element 0
> > (if so, it is not in GF(p^k)*)?
> > 5) Appendix A.1: the Ate pairing seems to use integer c written in
> > NAF-notation. If so, shouldn't one at least say something about side
> > channels, in case the inputs or outputs of the Ate pairing are
> > non-public parameters? What about the huge final exponentiation in
> > GF(p^k)^*, where the exponent h:=(p^k-1)/r has an enormous size (~
> > p^{k-1})? (Note RS: I did ask about side-channels in my review of the
> > rev07-draft [rev07-comment #10], but that comment was never addressed.)
> > 6) Appendix A.2: the beginning of that section is virtually the same
> > as Appendix A.1's, but misses some language (e.g., on "sum of
> > (i=0,1,...,L)" --> equals c, order r --> of G1, etc. Remarks on some
> > of the steps of the listed procedure similar to those with Appendix
> > A.1 apply.
> > 7) Appendix A would benefit from a much better explanation of the role
> > of the Ate pairing in pairing-based crypto. Otherwise, this seems just
> > loose sand, without context. (If the goal is to educate the CFRG
> > audience about coolness of pairing-based crypto, why not provide
> > sufficient detail to raise interest?; otherwise, why pursuing this
> > draft?)
> > 8) Appendix B: the relationship between elements of GF(p^k) and the
> > output sequences (e_0,...,e_{11}), resp. (e_0,...,e_{47}) is not
> > described in sufficient detail.
> > 9) Appendix C: this appendix is so poorly written and the construction
> > so adhoc that this should be abandoned right away (see also my
> > previous comment in my rev-07 draft review (rev07-comments#21/#22).
> > Here, simply observe that this point encoding only works for BLS12-381
> > and not for, e.g., BN462, purely because of poor design choices made.
> > Please also see Note 2 of Appendix I.8 of
> > draft-ietf-lwig-curve-representations-13 (which describes SEC1
> > representations, but now for any field of odd characteristic). Simply
> > using the SEC1-encoding, where one squeezes-in the 1-octet identifier
> > for affine points (0x04), point-at-infinity (0x00), and compressed
> > y-coordinate (0x02 or 0x03) in the leftmost 2 bits of the tight
> > MSB/msb-order representation of elements of GF(p), where the
> > bit-length of p has two fixed bit positions for BLS12-381 and BN462
> > case does the job.
> > 10) Appendix C: while I think this appendix should be ditched right
> > away and replaced by a more systematic approach that, in principle,
> > applies to all pairing-based curve domain parameters (see prevous
> > comment), there are also some problems with the text itself, including
> > (a) the high-level remarks are confusing and do not seem to contain
> > any useful information, so might easily be removed; (b) The I2OSP and
> > OS2IP functions in RFC 8017 do describe integer-to-octet-string and
> > octet-string-to-integer conversions, but do not specify the
> > bit-ordering (which is needed to describe the encoding of integers as
> > bit-strings and vice-versa);  (c) Step 2 (in Appendix C.1) seems to
> > confuse specification and implementation (since seemingly aimed at
> > machines rather than humans); (d) Step 3 (in Appendix C.1) seems to
> > suggest that an element x=x0 + u*x1 of GF(p^2) is represented in the
> > order "highest coefficient first", whereas in Section 2.5 this seems
> > to be in the order "lowest coefficient first" (although this is hard
> > to figure out due to the recursive definition in terms of towers of
> > fields) - it is hard to figure out what "see discussion of vector
> > representation above" refers to, since there does not seem to have
> > been one; (e) Step 8 (in Appendix C.2) may result in output P:=(x,-y),
> > where the second coordinate is a negative integer (and, thereby, not
> > in the interval [0,p)).
> >
> >
> > Editorial comments:
> > 1) Appendix A, last para before A.1: this para refers to
> > Barreto-Naehrig curves and should be moved to the beginning of
> > Appendix A.1.
> > 2) Appendix A, last para before A.1: why not expressing the Frobenius
> > map in a more human-friendly way, e.g., frob(x,y):=(x^p,y^p), where
> > one leaves the parameter p implicit?
> > 3) Appendix B, l. 8: in "where u is a indeterminate", change "a" to "an".
> > 4) Appendix B, l. 8: check for consistent naming of BLS48-581, etc.
> > (underscore vs. dash language). Also elsewhere. BTW - it would help to
> > add the curve name to section headings to make these easier to find,
> > e.g., adding BLS12_381 to the title of Section 4.2.1, etc.
> > 5) Appendix C, p. 49: replace "sign_F_p" by "sign_GF(p)", etc.
> > 6) Section 4.2.1, last para: I am not sure what adding the new (in
> > rev08) advertisement for another cfrg draft (hash-to-curve) adds to
> > this document, since it seems to be just PR, without any context.
> > Shouldn't one shy away from cross-marketing?
> >
> > Ref:
> > [3] D. Hankerson, A. Menezes, S. Vanstone, Guide to ECC, Springer, 2003.
> >
> >
> > [review posted to CFRG mailing list: Sun July 12, 2020, 7.07pm EDT]
> > Review of draft-irtf-cfrg-pairing-friendly-curves-07
> > Reviewer: Rene Struik
> > Assessment: not ready
> >
> > Summary:
> > The draft provides domain parameters for a few curves used with
> > pairing-based cryptography and provides some test vectors.
> >
> > General remarks:
> > a) While the draft suggests multiple times that the exTNFS attack
> > (2016) negatively impacted the bit-security level of various proposed
> > domain parameter sets for pairing-based curves, it does not provide
> > any detail on this attack itself, nor any reassurances that potential
> > extensions of these (only 4-yr old!) attacks on the general discrete
> > logarithm problem of composite degree extension fields GF(p^t) for
> > small composite t>1 would not be in the cards. This diminishes trust
> > in the claimed security of these curves and their "fitness for use" in
> > practice. the only use of these attacks is simply providing lengthy
> > tables with claimed revised security levels. Why should one have
> > confidence in this, do advances in solving the DLP in small degree
> > extension fields provide the kiss of death for pairings (or if not:
> > why not?), etc?
> > b) The draft has 6 1/2 pages of references and 3 1/2 pages of tables
> > of "adopted parameters" on a total of 27 pages of main body text. It
> > seems one should be able to considerably prune both tables and
> > references (which now come across as unwieldy). A good starting point
> > may be to consider that availability of a library or standardization
> > does not necessarily imply that schemes are actually deployed.
> > c) It is unclear what motivated change in co-authorship of this draft,
> > e.g., when considering changes between rev04 and rev07 of this draft.
> > It seems more customary to credit minor contributions in the
> > acknowledgement section than by change of authorship.
> > d) Lots of specification details seems adhoc and even motivated by a
> > particular company's conventions. For CFRG, it may be more appopriate
> > to specify curves, finite fields, and objects that live herein, in a
> > more systematic way, rather than reinventing the wheel, thereby
> > fostering reuse and maintainability.
> > e) I am curious why this draft's intended status is "experimental"
> > (vs. "informational", as is far more common for IRTF documents).
> >
> > Detailed comments:
> > 1) Section 1.2 (and also elsewhere) seems to conflate standardization
> > of parameters, availability of libraries, and actual deployment, where
> > there is an unusual prominence (for an IRTF document) of company
> > names. Wouldn't it make more sense to describe potential applications
> > of pairing based cryptography in more technical terms, e.g., in terms
> > of facilitating aggregate signatures, remote attestation, etc., and
> > provide a brief description (and a technical reference)?
> > 2) Section 1.3: the main point of the extTNFS attack is that it tries
> > and solve the DLP in low-degree extension field GF(p^t) for small
> > composite t>1 faster. To bring this point accross, I suggest changing
> > reference to "by the attack" (l. 5 of Section 1.3) by something more
> > closely reflecting this.
> > 3) Section 2.3, 4th para (top of p. 8): elaborate on the "BN curves
> > always have order 6 twists" remark (this seemed to have been copied ad
> > verbatim from the [BD18] paper).
> > 4) Section 2.4, 4th para: it would be good to mention that parameter t
> > must be 1 (mod 3), since otherwise p is not an integer.
> > 5) Section 2.5: the representation conventions are highly confusing,
> > esp. for extension fields. Why not define everything in terms of a
> > prime field GF(p) and extension field GF(p^t), with fixed irreducible
> > polynomial f(z) of degree t over GF(p)? This has been successfully
> > used with elliptic curve specifications (NIST, ISO, ANSI, BSI) not
> > tailored to pairing based crypto. This would also avoids questions
> > that now immediately come up (such as whether defining GF(p^{d'}) in
> > terms of GF(p^{d}) and "inductively applying the above convention"
> > does yield an unambigous definition. Since all finite fields of fixed
> > size are isomorphic, it would be much easier to stick to the standard
> > way of doing this. This would also avoid messy tower of extension
> > field arguments and messy representations of elements hereof (e.g., in
> > Section 4.4). See also Appendix J on data conversions of the lwig
> > draft referenced in this draft. As final  note, the data conversions
> > in that lwig draft require specifying bit/octet encodings (which, in
> > the pairing case, seem to be most-significant-byte-first (MSB) and
> > most-significant-bit-first (msb).
> > 6) Section 3.1, 4th para: it is unclear what the meaning of "best
> > known" is: is this "best-known" (i.e., most well-known) or "simply the
> > best"? Given the description, the first meaning should be the correct
> > one...
> > 7) Section 3.2: what is missing is a section that actually describes
> > the attack, rather than simply plugging in some implied numbers based
> > on a paper (presumably [BD18]). Why not add some verbiage that
> > explains this in simple but roughly accurate terms ("The exTNFS
> > Attack", or, better, "Attacks on DLP over Low-Degree Extension
> > Fields", or even better, "Solving the DLP in Finite Fields" (there has
> > been lots of progress there for mid-size base fields too)). This is a
> > CFRG document, so one would expect something that provides insight,
> > rather than simply a bombardment of tables, some selection criteria,
> > and a filtered list. Wouldn't the objective of this whole effort be to
> > educate the CFRG audience on pairing-based crypto, rather than (say,
> > obtaining an RFC number from an SDO and marketing this as an implied
> > "authoritative" approval stamp)?
> > 8) Section 3.2: I remember that Francisco Rodriguez-Henriquez (fix
> > name in references) presented attacks at the CHES 2013 rump sessions,
> > where question was whether this spelled trouble for pairing-based
> > crypto. While I do not have his presentation then on file, it may be
> > good to dig this up or ask him, and contemplate on wider implications
> > of DLP progress in general (see also first general review remark).
> > 9) Section 3.2, 2nd para: A natural question is what one could say
> > about DLP complexity for GF(p^k), in terms of dependency on p and k.
> > Unfortunately, this section does not provide any insight on this (it
> > only provides a single numerical value for BLS48 curves, without any
> > context). I would suspect the reader audience would appreciate such
> > insight, without need to wed through a whole forest (6 1/2 pages: far
> > too much reference stuffing!) of references by himself without any
> > guidance as to whether this would be time well-spent or lost.
> > 10) Section 4: as stated before, the selection criteria seem somewhat
> > arbitrary, since conflate specification text, libraries, with actual
> > deployment. Moreover, the most important criteria should probably be
> > security and speed given particular security strength, and potential
> > support for finite field arithmetic on platforms (speaking of which:
> > why not devote a section on whether one can actually implement GF(p^k)
> > securely using finite field libraries, including modular reductions,
> > side channels, etc.?). The term "adoption status" seems, in any event,
> > misleading.
> > 11) Section 4.1, Table on pp. 12-15: if one strikes out domain parms
> > rendered immediately suspect by the exTNFS paper on DLP, this kills
> > off 8 out of 10 entries on p. 12 (and far more if one stikes out
> > suspect values accross the entire table). This makes me wonder what
> > the technical reason is for including this entire table. To the casual
> > reader it now suggests that there are huge numbers of implementations
> > out there, whereas - perhaps - most of those should be switched off
> > immediately...
> > 12) Section 4.1.2: once again, I am wondering why there is so much
> > emphasis on libraries here. Isn't this an IRTF/CFRG document?
> > 13) Section 4.2.1: The extension field GF(p^{12}) can also be
> > described via GF(p) and degree-12 polynomial f(w):=(w^6-1)^2+1. This
> > would allow using simple conventions used in Lidl et al's finite field
> > book [2]. It also allows description of an element x of GF(p^{12}) as
> > vector (x_{11}, ..., x_1, x_0) of coefficients of GF(p) with this
> > irreducible polynomial. Note here that u+1:=w^6 and v:=w^2, so one can
> > easily internally use the more complex tower field stuff in the draft,
> > while sticking to simple and easy to maintain standard constructions
> > (known for 2 centuries) for specifications (so, nobody looses out if
> > one has a slim interface that does this conversion back and forth, if
> > necessary).
> > 14) Section 4.2.1, bottom of p. 18: shouldn't the cofactor h be such
> > that h*r= # E'(GF(p^2))? {please also fix Et and use E' notation as
> > elsewhere in the draft}
> > 15) Section 4.2.1, parameter b' (bottom of p. 19): if one uses the
> > complicated tower construction, why then not also mention the value of
> > u in the enumeration? Is one actually sure this value is uniquely
> > defined (e.g., I did not check but wondered what would happen if one
> > replaces u by -u in the tower construction)? Same with end of Section
> > 4.2.2...
> > 16) Section 4.2.2, 5th para: the statement "CP8_544 is more efficient"
> > is hard to interpret without context (e.g., half the cost, cost-1,
> > cost - o(log log log n)).
> > 17) Section 4.2.2, bottom of p. 20: with parms BN462, why not simply
> > introduce base point G, parameter n, h, etc., once and for all in
> > Section 2.1, without trying to repeat their definition in Section
> > 4.2.2 (and later sections)?
> > 18) Section 4.2.2, top of p. 21: the formula for h seems incorrect (G2
> > is defined over GF(p^2), whereas the formula refers to a curve defined
> > over GF(p^8).
> > 19) The security consideration section (Section 5) is rather slim:
> > (speculation on my side) is the reason to label this draft as
> > "experimental" that design strength vs. actual strength is somewhat in
> > limbo due to progress on DLP problem? Why suddenly squeeze in a point
> > validation routine if no context is given at all of where and how
> > pairing based crypto is used? Wit point validation, what if an octet
> > representation is outside GF(p) boundary? (while the forelast para
> > seems to imply an attack one is completely left in the dark what is at
> > stake here). Not sure whether it is the role of CFRG documents to
> > legitimize "BN254 use ... to keep interoperability". The end of the
> > first para ("as of 2020, as far as we know, there are no fatal attacks
> > that significantly reduce the security of pairing-friendly curves
> > after exTNFS") is entirely non-reassuring to me: it is only four years
> > after the DLP attack that necessitated to strike out half of the
> > entries in the table earlier on in the paper, not that many people
> > work on pairing-based algorithmic cryptanalysis in the first place,
> > etc., etc. Where does this confidence come from (shouldn't one be more
> > modest here, technically speaking???). The Cheon attack is not
> > explained and cannot be evaluated at all, since no context on elliptic
> > arithmetic is given at all in the draft.
> > 20) Secion 5, 3rd-last para: Why would the Montgomery ladder suddenly
> > come to the rescue to salvage side channel resistance? Why refer to
> > RFC 7748: whereas pairing-friendly curves are all Weierstrass curves,
> > the curves in RFC 7748 are all Montgomery curves with completely
> > different underlying detail on differential-addition chains). It seems
> > that an entire section should be devoted to how implementations coul
> > avoid SCA attacks, esp. since some operations take place in huge
> > extension fields GF(p^t)...
> > 21) Appendix C seems to convey a particular encoding used by ZCash. I
> > don't think it is the role of CFRG to make those the standard way of
> > doing things. This being said, technically that representation is a
> > tiny tweak of what the SECG1 specification already stipulated in 2001
> > (with SEC1, one can extract the affine/compressed encoding of an
> > affine point and whether this relates to the point at infinity from
> > the leftmost octet (0x04, 0x02 or 0x03, vs. 0x00), which more or less
> > 1-1 corresponds to the (C_bit, I_bit, S_bit) combinations. If so,
> > reinventing yet another representation is hard to defend.
> > 22) The parity bit notation for finite fields is highly non-standard,
> > compared to, e.g., what has been standard usage with compressed points
> > for curves over prime fields or binary extension fields. Even if this
> > would have some uses, why not defining things once and for all for all
> > extension fields of an odd prime field, so that this is a simple
> > extension. See also Appendix H (parity function for any field GF(p^k),
> > where p odd). This should also help in limiting side channel leakage
> > from the first-half vs. last-half of [0,p-1] test.
> >
> > Editorial comments:
> >
> > 1) Section 2.1, first para: replace "F_q" by GF(q), stipulate that
> > extension degree n>0.
> > 2) Section 2.1, first para: with defining equation, use more common
> > domain parameters a and b (i.e., lowercase instead of upper case).
> > Elsewhere, do use common nomenclature used with NIST, ANSI, SECG, ISO
> > for 2 decades, including n for prime-order subgroup, h for co-factor,
> > mention irreducible polynomial f(z) with extension field, denote fixed
> > base point by P (instead of BP), etc. If one wishes, refer to Appendix
> > B.1 of [1].
> > 3) Section 2.1, 2nd para: "point on" should be "point of".
> > 4) Section 2.1, first/3rd para: isn't it simpler to define curve over
> > GF(q) and then introduce curve with same domain parms, but then
> > defined over extension field GF(q^k)?
> > 5) Section 2.1, 3rd para: "group law which" should be "group law
> > that", "reflection about x-axis" should be "reflection around x-axis",
> > with "unique third point of intersection [R]" (i.e., give this a name,
> > here R), with [a]P, stipulate that [0]P is the identity element and
> > that [-a]P=-([a]P), etc.
> > 6) Section 2.1, terminology: fix E(F_{q^k}) (i.e., add paranthesis),
> > fix that this refers to GF(q^k)-rational points (rather than
> > GF(q)-rational points, same with cardinalities.
> > 7) Section 2.1, terminology: with co-factor h, doesn't one need
> > gcd(h,n)=1 (so, as to ensure unique order-n subgroup)?
> > 8) Sectioon 2.2, 2nd para "is called embedding degree of E over GF(q)"
> > (i.e., add curve and field over which this is defined)
> > 9) Section 2.2, 2nd para: the term "twist" is not defined (but often
> > used elsewhere in the draft), neither is the term GF(p^k)* (nonzero
> > elements of GF(p^k).
> > 10) Section 2.3, 2nd para: replace "prime p" by "prime number p (where
> > p at least five)".
> > 11) Section 2.3, 3rd para (top of p. 8): write "the multiplicative
> > group..." or, better still, simply state that b is a primitive element
> > of GF(p) (and add this to terminology).
> > 12) Section 2.4, 4th para: "parameterized" should read "parameters".
> > 13) Section 3.1, 3rd para: "paiting-based" should be "pairing-based"
> > (i.e., fix the typo "t" --> "r").
> > 14) Section 3.1, 4th para: "to solve" should read "for solving".
> > 15) Section 3.2, 1st para: "the security level(s)" (i.e., make
> > plural), "... correspond" (i.e., use corresponding verb conjugation).
> > 16) Section 4.2, 2nd para: reword "more prudent option" as "more
> > conservative option".
> > 17) Section 4.2.1, 4th para: "categorized as M-type" ("as" instead of
> > "into").
> >
> >
> >
> > Ref:
> > [1] draft-ietf-lwig-curve-representations-10
> > [2] R. Lidl, Niederreiter, "Finite Fields", Cambridge University
> > Press, ...
> >
> >
>
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>
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-- 
Yumi Sakemi, Ph. D.
Lepidum Co. Ltd.

E-Mail: yumi.sakemi@lepidum.co.jp