Re: [MLS] Re-randomized TreeKEM

[Joel Alwen <[e-mail]>]

I think the challenge with the hash-forward approach is how to do that
homomorphically. I.e. what we need are two algorithms; one to refresh
the PK without knowing the SK (but possibly knowing a secret
rerandomizer delta if needed) and one to update SK (again possibly using
delta). So to use a hash-forward approach their must be:

 1) a way to evolve PK forward to PK' and
 2) a *one-way* method to evolve SK forward to SK' compatible PK'.

One-wayness is what gives us Forward Secrecy and "compatibility" between
the two key evolution methods is what allows for asynchronous (i.e. 1
packet) updates.

Currently we use a secret re-randomizer delta to ensure the SK update
method is one-way. That is, without the delta you cant "undo" the
update. But that would break if we (at least naively) used some public
delta, say hash(ciphertext). So I think this is the challenge that we'd
have to overcome. Basically, make sure we SK evolution is one-way but
also compatible the public evolution of PK.



Now one way sweet way to get all this (and more) would be to use a HIBE.

Initial, PK for a ratchet tree node (i.e. its "identity" since this is a
HIBE now) is simply the empty vector PK := () while the secret key is
the master public key for a fresh HIBE instance SK := MSK. We also
include, as a second component of the nodes PK, the master public key
PK_0 = MPK. To "hash forward" / "re-randomize" when sending a ciphertext
C to that node we can do:

 PK' := (PK, hash(C)).
 SK' := DeriveHIBEKey(PK, SK).

So simply append hash(C) to the identity for that node and derive the
corresponding HIBE key.

Ignoring the problems with using HIBE for a second, this is a very cool
solution. We don't need to send out the updated PK since everyone in the
group (and even the adversary) can compute it for themselves. We also
dont need a re-randomize delta as part of the plaintext because we're
using delta := hash(ciphertext) so the plaintext is shorter again.
Moreover, HIBE security means that learning SK' doesn't tell you
anything interesting about SK. In particular, we have forward security.
(In fact, FS will hold even if hash(C) were chosen *completely*
adversarially, say, as part of a malicious update in an insider attack!)

Of course, the problem with this solution is that we're using HIBE.
Worse, with unbounded depth because each new ciphertext sent to a node
results in going one depth further into the hierarchy. AFAIK all HIBE
constructions have pretty horrible (read exponential) efficiency as a
function of their depth. (And I won't mention the state of
standardization and open implementations for HIBE.)

Now there could be a totally different approach that entirly avoids
HIBE. But even with this approach there's at least some glimer of hope
to improve on it because, if we don't wory about insider attacks we can
assume C is honestly generated which means hash(C) really has a ton of
entropy. So we dont seem to need the full expresivity HIBE identities
allow us. Rather we only need HIBE for "random" identities. Still, that
seems like a pretty slim hope for major efficiency improvement. It also
doesn't do anything to address the lack of implementations and standards.

- Joël

On 17/10/2019 16:37, Karthik Bhargavan wrote:
> Thanks Yevgeniy,
> 
> This helps a lot.
> 
> To further my understanding, another question:
> 
>>  Intuitively, the sender will not only encrypt the message, but also a
>> random Delta value. It will change public key using homomorthism by
>> multiplying with g^Delta (in specific DH based scheme), while the
>> recipient will decrypt Delta (using old secret key), and add it to the
>> old secret key to get there new one. So now corrupting (old sk plus
>> Delta) will not help decrypting the ciphertext just decepted, emailing
>> forward secrecy. 
> 
> I see that in the DH-based scheme, this Delta needs to be private,
> otherwise the adversary can compute sk once it knows sk+Delta.
> But, in general, is it possible to conceive of a UPKE scheme where the
> recipient effectively “hashes forward” its symmetric key,
> where this one way hash-forward function does not have to rely on an
> externally chosen secret value?
> 
> Best,
> Karthik
> 
>> This is the high level, hope it makes sense.
>> Thanks for your question,
>> Yevgeniy
>>
>> On Thu, Oct 17, 2019, 1:43 AM Karthik Bhargavan
>> <[e-mail]
>> <mailto:[e-mail]>> wrote:
>>
>>     Hi Joel,
>>
>>     This looks very interesting. It is new to me since I was not at
>>     the interim.
>>     After reading the paper and the slides, I am still a bit fuzzy
>>     about what the recipient of an update needs to do.
>>
>>     For example, for the running example in your slide deck, it would
>>     help if I could see:
>>     - what secret keys does each leaf need to keep
>>     - how do these secrets change when an update from some other node
>>     is received.
>>     Just working this out for one update is enough.
>>
>>     I know that this is made precise in the eprint, but it would be
>>     faster if you could help us understand it :)
>>
>>     Best,
>>     Karthik
>>
>>     > On 16 Oct 2019, at 23:51, Joel Alwen <[e-mail]
>>     <mailto:[e-mail]>> wrote:
>>     >
>>     > <FS-TreeKEM.pdf>
>>
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