Re: [Cfrg] RGLC on draft-irtf-cfrg-pairing-friendly-curves-07

Rene Struik <> Sun, 12 July 2020 23:07 UTC

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To: "Stanislav V. Smyshlyaev" <>, CFRG <>
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From: Rene Struik <>
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Date: Sun, 12 Jul 2020 19:07:47 -0400
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Subject: Re: [Cfrg] RGLC on draft-irtf-cfrg-pairing-friendly-curves-07
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Dear colleagues:

Please find below my review of the pairing draft. If this does not 
display well, please send me an offline note and I can send you the 
original text file (before cut-and-paste into this email).

I hope this helps.

Review of draft-irtf-cfrg-pairing-friendly-curves-07
Reviewer: Rene Struik
Assessment: not ready

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, 
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 
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 
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 of [1] (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 

[1] draft-ietf-lwig-curve-representations-10
[2] R. Lidl, Niederreiter, "Finite Fields", Cambridge University Press, ...

On 2020-06-19 1:50 p.m., Stanislav V. Smyshlyaev wrote:
> Dear CFRG participants,
> This message is starting 2 weeks RGLC on 
> draft-irtf-cfrg-pairing-friendly-curves-07 ("Pairing-Friendly 
> Curves"), that will end on July 4th 2020. If you've read the document 
> and think that it is ready (or not ready) for publication as an RFC, 
> please send a message in reply to this email or directly to CFRG 
> chairs ( <>). If you 
> have detailed comments, these would also be very helpful at this point.
> P.S.: The review on behalf of Crypto Review Panel was done by Chloe 
> Martindale, the comments have been addressed by the authors of the 
> draft, see
> Thank you,
> Stanislav, for CFRG chairs
> _______________________________________________
> Cfrg mailing list

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