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Re: [CFRG] Small subgroup question for draft-irtf-cfrg-hash-to-curve

Mike Hamburg <mike@shiftleft.org> Sat, 10 April 2021 20:22 UTC

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From: Mike Hamburg <mike@shiftleft.org>
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Date: Sat, 10 Apr 2021 17:22:48 -0300
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To: "Hao, Feng" <Feng.Hao@warwick.ac.uk>
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Subject: Re: [CFRG] Small subgroup question for draft-irtf-cfrg-hash-to-curve
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> On Apr 10, 2021, at 5:19 PM, Mike Hamburg <mike@shiftleft.org> wrote:
> 
> I was insufficiently precise.  I’d forgotten that PAK refers to several PAKE algorithms; I was thinking of some of the elliptic curve versions, such as PAK-EC from "More efficient password-authenticated key exchange" by Philip MacKenzie, CT-RSA 2001.  I re-proposed a variant of this protocol, not knowing that it had been done before, as "SPAKE2: Elligator Edition”.  This design is slightly slower than SPEKE, but it has a simpler security analysis than SPEKE.  It sends something very roughly of the form aG + hash_to_curve(password), so you want hash_to_curve to be uniform.  Security follows by a very simple argument (compared to most other PAKEs) under ROM + strong DH.

Replying to myself here, because I forgot to mention: as in most other uses of hash/map to curve, PAK-EC is provably secure even for somewhat nonuniform outputs, such as map_to_curve outputs.  It’s just most secure with uniform outputs.

Regards,
— Mike

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