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Date: Tue, 13 Mar 2018 03:25:56 -0500
From: denis bider <denisbider.ietf@...il.com>
To: discussions@...sword-hashing.net
Subject: [PHC] How to: A super-efficient cryptographic accumulator?

Hey everyone!

This problem has been bugging me for a while. It’s relevant to crypto
currency (saving state in a more efficient way than the gargantuan database
produced by Bitcoin) as well as other purposes like checking for weak
passwords, which is relevant to this list. For example, the best measure
for password safety is probably to prevent users from using any of billions
of known revealed passwords, but the compressed database is 5 GB in size:

https://github.com/berzerk0/Probable-Wordlists/blob/master/Downloads.md

Now. I sort of have this intuition that it’s possible to compress this crap
into 20 bytes. That's not a typo – 20 bytes, not megabytes or something.
But I don’t know how.

The basic thinking that supports this intuition is: we're trying to have a
function we can query for any given input, and we want the output to be
either 1 or 0. We want to craft this function so it’s small, and accepts
any given input, and will have zero false negatives, and some small but
acceptable rate of false positives. For cryptocurrency purposes, perhaps
the acceptable false positive rate is 10^(-20). For passwords, perhaps up
to 1/100 is acceptable.

Intuitively, it should be possible to do this because we can do this if we
had an infinitely fast computer. If we have an infinitely fast computer,
then we can simply find X such that:

TruncateTo32Bits ( SHA256 ( X | P1 ) ) == 0x00000000
TruncateTo32Bits ( SHA256 ( X | P2 ) ) == 0x00000000
TruncateTo32Bits ( SHA256 ( X | anything else ) ) == whatever

If we crank the infinitely fast computer long enough, we should be able to
find this X, and it should be small in size. The only reason we should not
be able to find this X is if SHA256 has some structural issue that
prohibits an X like that from existing. In that case we use a better hash
function.

To use this, we calculate TruncateTo32Bits ( SHA256 ( X | Pcandidate) ). If
the result is 0, then we know Pcandidate is “contained” in X with
probability about 2^32 : 1. If the result is any other number, we know
Pcandidate is not “contained” in X.

Now, because we have an infinitely fast computer, we can just do the above,
but for P1 ... P999999999. Ta-daa, we now have a billion passwords stored
in this serendipitous X, which is probably small and on the order of ... 8
bytes? 4 bytes to “store” yes/no information about a billion passwords, and
another 4 for a 2^-32 rate of false positives?

Now, the reason we can’t do that is, the way SHA256 is designed, it
requires us to have an infinitely fast computer.

But what if we design a different function, in some ways similar to SHA256,
but which would allow X to be found more efficiently?

If we can’t design this algorithm for a classical computer – what if we
have a quantum computer?

denis

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