[cryptography] [tor-dev] Global semi-passive adversary: suggestion of using expanders

Eugen Leitl eugen at leitl.org
Fri Aug 23 07:49:53 EDT 2013


----- Forwarded message from Paul-Olivier Dehaye <paul-olivier.dehaye at math.uzh.ch> -----

Date: Fri, 23 Aug 2013 03:08:59 +0200
From: Paul-Olivier Dehaye <paul-olivier.dehaye at math.uzh.ch>
To: tor-dev at lists.torproject.org
Subject: [tor-dev] Global semi-passive adversary: suggestion of using expanders
Reply-To: tor-dev at lists.torproject.org

Hello,

Thank you for working on Tor.

I have a suggestion and would appreciate input. Please bear with me as I
have a limited understanding of the design of Tor and all the different
threats that it is meant to mitigate. Below, a (?) indicates a place where
I need some confirmation that my understanding is correct, and N indicates
either the number of Tor nodes, the number of end-users, or the amount of
traffic (I assume these are all linearly related).

As far as I can tell, the main threat by a global passive adversary comes
from traffic analysis (?). This attack should become easier as the number
of Tor nodes increases (?): Tor uses a clique topology, so the number of
edges potentially carrying traffic grows like N^2. A dual way to see this
is that not enough mixing can happen around a node for incoming/outgoing
edge pairs, bar injecting a huge amount of fake traffic.

To compensate, it seems natural to look for a sparse yet highly mixing
network topology. Mathematically, those are called expanders [1]. A typical
example of a family of expanders would be the Erdos-Renyi model [2], and
indeed I have found in the literature suggestions for basing anonymizing
protocols on such a model. The analysis in the presence of an active
adversary becomes very difficult though.

Alternatively, one could use a different method for constructing that
expander topology, working "all at once". This comes from recent
mathematics research (<= 5 years, certainly not my own, see [3]). The graph
is then a Cailey graph [4] in a matrix group (the group is fixed and
determined by an approximation to the number of Tor nodes, such as nearest
third power of a prime number).
In some sense this construction interpolates between mixing chains and Tor,
and can be seen as a lot of mixing chains interwoven.

In the setting of Tor, constructing the Cailey graph would require making
two distributed randomize choices:
- a matching of elements of the group to Tor nodes (possibly 2:1 for some
Tor nodes)
- a small subset of generators for the Cailey graph
>From my understanding of security protocols, it should be easy to do these
two choices safely and fast, as it amounts to choosing a random element in
S_N and filling lots of matrix entries with random elements between 1 and a
prime p, with some rejection.

Once that is done, the network topology is fully determined, and with very
high probability gives an expander. This means that traffic gets mixed up
in very few hops. The number of hops needed grows as log N, with a constant
that can be mitigated by chosing a large generating set above. This is the
only downside I see (apart from difficulty to explain the math behind
this): the latency would increase, from 3 in the current protocol to maybe
10 or so.

I don't know the details of the behaviour of the constants in the last
paragraph, and would appreciate feedback from the list before looking too
much into this.

Paul Dehaye

[1] http://en.wikipedia.org/wiki/Expander_graph
[2] http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model
[3]
http://terrytao.wordpress.com/2011/12/02/245b-notes-1-basic-theory-of-expander-graphs/Exercise
15 and remark below
[4] http://en.wikipedia.org/wiki/Cayley_graph

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