Re: [CFRG] Questions regarding draft-irtf-cfrg-hash-to-curve-10

[Mike Hamburg <[e-mail]>]

Hello Björn,

At some point (2013?) I wrote up an article on uniform hashing specifically for Elligator, but it got rejected for being too similar to previous work, and I never bothered to post it on ePrint.  I’ll do that now.  Please find attached an analysis of two constructions, one of which is hash-twice-and-add.



Regards,
— Mike

> On Dec 2, 2020, at 11:58 AM, Björn Haase <[e-mail]> wrote:
> 
> Hello Riad, Hello Christopher,
>  
> As requested, I have filed an issue in the GIT draft regarding an update of our CPace security analysis.
>  
> I am currently reviewing one other aspect regarding the hash_to_curve construction, where the result of two mappings is added.
>  
> hash_to_curve(msg)
> Input: msg, an arbitrary-length byte string.
> Output: P, a point in G.
>  
> Steps: 
> 1. u = hash_to_field(msg, 2)
> 2. Q0 = map_to_curve(u[0])
> 3. Q1 = map_to_curve(u[1])
> 4. R = Q0 + Q1 # Point addition
> 5. P = clear_cofactor(R)
> 6. return P
>  
> The important aspect would be, that the result P comes from a uniform distribution.
>  
> I am aware of a result from Coron, Icart, brier and Madore “Efficient Indifferentiable Hashing into Ordinary Elliptic Curves.” where they saw the need for using rather something of the type of 
> P = Q0 + x * Q1.
>  
> In “Indifferentiable Deterministic Hashing to Elliptic and Hyperelliptic Curves” there is a discussion that, even if uniformity could not be guaranteed, at least some weaker property of “well-distributed encodings” holds, which they show for SWU.
>  
> https://eprint.iacr.org/2010/539.pdf <https://eprint.iacr.org/2010/539.pdf>
>  
> I am having now the questions: Are you aware of a result that extends this to Elligator2? If I understood the paper correctly, the case of SSWU should be covered as part of the general properties of SWU. The guarantees seem to be linked to the property of the map that it is “well distributed” (where I did not yet understand the full implication of the character sums definition …).
>  
> Yours,
>  
> Björn.
> Mit freundlichen Grüßen I Best Regards 
> 
> Dr. Björn Haase 
> 
> Senior Expert Electronics | TGREH Electronics Hardware
> 
> Endress+Hauser Liquid Analysis
> 
> Endress+Hauser Conducta GmbH+Co.KG | Dieselstrasse 24 | 70839 Gerlingen | Germany
> Phone: [phone] | Fax: [phone]
> [e-mail] <mailto:[e-mail]> |  www.ehla.endress.com <http://www.ehla.endress.com/> 
> 
> Endress+Hauser Conducta GmbH+Co.KG
> Amtsgericht Stuttgart HRA 201908
> Sitz der Gesellschaft: Gerlingen
> Persönlich haftende Gesellschafterin:
> Endress+Hauser Conducta
> Verwaltungsgesellschaft mbH
> Sitz der Gesellschaft: Gerlingen
> Amtsgericht Stuttgart HRA 201929
> Geschäftsführer: Dr. Manfred Jagiella
> 
> Gemäss Datenschutzgrundverordnung sind wir verpflichtet, Sie zu informieren, wenn wir personenbezogene Daten von Ihnen erheben.
> 
> Dieser Informationspflicht kommen wir mit folgendem Datenschutzhinweis <https://www.de.endress.com/de/cookies-endress+hauser-website> nach.
> 
>  
> Disclaimer: 
> 
> The information transmitted is intended only for the person or entity to which it is addressed and may contain confidential, proprietary, and/or privileged
> material. Any review, retransmission, dissemination or other use of, or taking of any action in reliance upon, this information by persons or entities
> other than the intended recipient is prohibited. If you receive this in error, please contact the sender and delete the material from any computer.
> This e-mail does not constitute a contract offer, a contract amendment, or an acceptance of a contract offer unless explicitly and conspicuously designated or stated as such.
> 
>  
> _______________________________________________
> CFRG mailing list
> [e-mail] <mailto:[e-mail]>
> https://www.irtf.org/mailman/listinfo/cfrg <https://www.irtf.org/mailman/listinfo/cfrg>