Subject: Re: [Bitcoin-development] (space) efficient reusable addr via weil pairing IBE (Re: Bait for reusable addresses)

From: "Jeremy Spilman" <[email-masked]>
Date: Sun, 02 Feb 2014 01:16:20 -0800

>
> Consequently you can now securely and very network/space efficiently
> securely delegate searching a block by computing the private key for the
> IBE pub key that any sender would use for that block, and sending it as
> a query to a random (or node-capture defended random selected node).
> The node can decrypt the encrypted bloom baits with it, but remains
> powerless to correlate with bloom baits to other payments received by
> the same user in bother blocks.
>

I'm not sure I've fully wrapped my head around it.

   d/Q        - Identity key
   E          - Generate an epoch-pubkey: E = Q * H1(epoch)
   r/P        - Ephemeral privkey/pubkey, or discoverable from inputs
   S = r * K  - Shared secret (ECDE)

Payer derives an encryption key H(S), and encrypts M, which is a 1 byte  
bloom bait.

For each epoch, payee generates e = d * H1(epoch) and provides the key to  
a full node for monitoring. So you are providing a per-block or per-epoch  
private key, along with the block ID or epoch ID that the key corresponds  
to.

The full node then uses this privkey to decrypt the same byte in all the  
transactions in that epoch/block which match the expected layout/template,  
e.g. given a certain length OP_RETURN, pull the specific byte and decrypt.

This decrypted byte is then in turn used as bloom bait which may or may  
not cause the transaction to be sent back to the SPV client.

Am I right in saying the full node has no idea if decryption is  
'succeeding' it just feeds the resultant bait into the bloom filter and  
the transaction may match or not? So we do get some level of repudiation  
by the SPV client -- the server doesn't know exactly which transactions  
belong to the SPV client.

The bloom bait specified in the reusable address is still making the  
bandwidth/privacy trade-off, it just doesn't become public information,  
because it's protected by another factor?

What encryption scheme is being used here?

-=-=-=-==

Another approach (inspired by IBE) which narrows the discoverability of  
transactions to the nodes that your SPV client is actually communicating  
with, for the specific blocks/epochs that you specify, would be to use  
PEKS.

PEKS(Q, W) for a public key Q and a keyword W produces a searchable  
encryption of the word W.

The payee holding 'd' (privkey for Q) can create a trapdoor which allows a  
server to search for transactions with W, where the searching party only  
knows if a match is found or not.

Payer:

   d/Q   - Longstanding / identity privkey/pubkey
   r/P   - Ephemeral privkey/pubkey, or discoverable from inputs
   W     - Searchable Keyword
   H1    - Hash function, {0, 1} ∗ → G1
   H2    - Hash function, G2 → {0, 1}p

Secret, as usual per ECDH:

   S = r * Q

For payer to create the searchable encryption of W for Q, called 'Sw':

   Sw = H2(e(H1(W), S))
   OP_RETURN P, Sw

For payee to search for a given 'W', payee calculates a trapdoor 'Tw':

   Tw = d * H1(W)

For a searcher, given a Trapdoor (Tw), check each Transaction (P, Sw):

   H2(e(Tw, P)) ==? Sw
   If the values match, the keyword matches

Without getting into the concepts of e(g,g) and binomial pairing, I think  
of it this way:

   Sw = H2(r * Q * H1(W)), but recall: rQ == dP, so...
      = H2(d * P * H1(W)), which can be written
      = H2(d * H1(W) * P)

Severs finds all transactions with 'P' on relevant parts of the  
blockchain, multiplies by the provided trapdoor 'Tw', applies 'H2', and  
checks for a matching 'Sw' in the transaction;

   Tw = d * H1(W)
   Sw = H2(Tw * P)
        H2(d * H1(W) * P)

PEKS is vulnerable to an offline keyword guessing attack, where you can  
discover the value of the keyword being searched, if the keyword is low  
entropy. The server/attacker can figure out the value of W, but they can't  
generate their own trapdoors to search for other keywords.

But in this case, the 'keyword' can simply be the block ID / epoch ID  
itself, not a secret value at all. In other words, the server can only  
search for your transactions within the blocks/epochs that you specify.

Using blockID/epochID as W, this would allow a server to find all  
transactions belonging to the payer for that blockID / epochID. The SPV  
client would simply provide the trapdoor for each block/epoch to be  
searched.

There are extensions to PEKS which provide for 'fuzzy' matching but they  
are 'fuzzy' within the scope of Q, not across different Q, so that doesn't  
help provide any repudiation. So I see this as only slightly improving  
Peter's original proposal of providing 'Q' to the searcher, but if you  
want repudiation, not as good as Adam's solution.

Perfunctory disclaimer... Hopefully this is close to correct, but please  
don't anyone actually try to implement this!

Thanks,
Jeremy