# [curves] Is there an established name for the hardness assumption capturing twist security of a curve?

Björn Haase bjoern.m.haase at web.de
Fri Jun 12 07:05:58 PDT 2020

```Hi Rene,

>at negligible incremental cost (a few field multiplies) even if one
implements ECDH using
>Montgomery ladders and is only given the x-coordinate of a point

If you transfer the full point coordinates the cost indeed might be
small. If you spare the communication bandwidth and transfer the
x-coordinate only, you need code and computation for a full field
exponentiation (square root). IMO, this is worth to be spared, at least
on small targets.

Yours,

Björn.

Am 12.06.2020 um 14:55 schrieb Rene Struik:
> Hi Bjorn:
>
> Why not simply check whether the point is on the curve? Within the
> context of DH schemes, this is trivial to do and comes at negligible
> incremental cost (a few field multiplies) even if one implements ECDH
> using Montgomery ladders and is only given the x-coordinate of a point.
>
> Best regards, Rene
>
>
> On 6/12/2020 3:32 AM, Björn Haase wrote:
>> Hi to all,
>>
>> I am currently re-working the security proof for CPace
>> https://datatracker.ietf.org/doc/draft-haase-cpace/ such that tight
>> computational bounds for the adversary could be given.
>>
>> In this context, I am still looking for the name and defininition of the
>> problem that captures the feature of "twist security", i.e. for the
>> tight reduction for the case where an active adversary passes a point on
>> the twist to a honest party.
>>
>> I did not find an established security notion so far that captures this
>> property so that I could re-use it in the re-worked proof.
>> I'd coin it "exponential transfer" and formulate it in the way:
>> Given two groups (modulo negation) J and J' with co-factors c and c' in
>> which the discrete logarithm problem is assumed to be hard in the prime
>> order subgroup and with c' = n * c and d=max(c,c'), the *exponential
>> transfer problem * is defined as:
>> Given two points B,X = B^(d * x) in J: Provide two points B' and X' in
>> J' with X' = B'^(d * x).
>> I'd like to avoid having to newly define it myself. I would very much
>> appreciate if anybody could give me a pointer.
>> Yours,
>> Björn
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>
>
```