113 lines
5.4 KiB
Markdown
113 lines
5.4 KiB
Markdown
---
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title: Sybil Attack
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# katex
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...
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We are able to pretend that bank transactions are instant, because they are
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reversible. And then they get reversed for the wrong reasons, often very bad
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reasons.
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But in general, with irreversible transactions, hard to pretend they go
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through in an instant. Bitcoin times are pretty good, but not good enough to
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pay at the cash register.
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For this reason, all rhocoin transactions are made with an identity. You can
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have as many identities as you please, and these identities are not visible
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to third parties unless one of the parties to the transaction chooses to make
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them visible.
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To support transactions at the cash register, going to need credit rating
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agencies. If you make the payment as an identity that has an adequate credit
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rating, the cash register trusts that the transaction will go through.
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So, how do we stop someone from generating a credit rating by issuing a lot
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of sham transactions from his right hand to his left hand, and then buying
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something and not paying? The credit card problem.
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And how do we stop someone from generating a pile of fake sales, and then
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making a real sale and not delivering? The Ebay problem.
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There is a solution to this problem. Each rating agency will treat payments
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and reviews as a directed graph, and measure graph connectivity. each
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subgraph of fake payments, and fake reviews will have good connectivity to
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itself and poor connectivity to the rest of the graph.
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# SybilGuard
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The [SybilGuard paper] provides graph theory solutions against sybils.
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[SybilGuard paper]:http://www.math.cmu.edu/~adf/research/SybilGuard.pdf
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According to SybilGuard, social networks are empirically determined to be
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“fast mixing”
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“social networks tend to be fast mixing (defined in the next section), which
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necessarily means that subsets of honest nodes have good connectivity to the
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rest of the social network”
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So the legitimate musician with legitimate fans will be connected to the rest
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of the social network – his legitimate fans will have interactions with
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legitimate fans of other legitimate musicians, and will purchase stuff from
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other legitimate musicians, while the fake fans making fake purchases will
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not.
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“in a fast mixing graph, after a small number of hops a random walk is
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equally likely to be traversing any edge in a given hop.”
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The paper argues that Sybil subsets of the network are substantially less
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likely to be entered after a small number of hops.
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“With a length $w$ random walk, clearly the distribution of the ending point
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of the walk depends on the starting point. However, for connected and
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nonbipartite graphs, the ending point distribution becomes independent of the
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starting point when $w → ∞$. This distribution is called the stationary
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distribution of the graph. The mixing time T of a graph quantifies how fast
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the ending point of a random walk approach the stationary distribution. In
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other words, after $Θ(T)$ steps, the node on the random walk becomes roughly
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independent of the starting point. If $T = Θ(log(n))$, the graph is called
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fast mixing.”
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$Θ(f)$ means that the space or running time is always proportional to $f$,
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$O(f)$ means that the common worst case space or running time is proportional
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to $f$. Engineers generally do not care about such differences, and just use
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$O$ everywhere.
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If we mix the graph for a suitable time, legitimate nodes will be well mixed
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with each other, and Sybil nodes will not be well mixed with legitimate nodes.
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Legitimate nodes will have positions that are close to each other in the
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space of distributions of end probabilities, and Sybil nodes will be far from
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legitimate nodes in that space.
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This algorithm is $O(n^2)$ in data size (the distribution for each starting
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point), and $Θ(n^2{log(n)})$ in computational time, which is a lot better
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than non polynomial, though still apt to be inconveniently large. It can
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probably be speeded up considerably by [random dimensional reduction],
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wherein a vector of very large dimension (the probability of reaching each
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end point, the distribution of end points) is randomly mapped to a vector of
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dimension $Θ(log(n))$ log of the original dimension.
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[random dimensional reduction]:recognizing_categories_and_instances_as_members_of_a_category.html#dimensional-reduction
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"recognizing instances as members of a category"
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In this case, the vector space of large dimension is of vectors whose
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elements are the probability of arriving at a given node after a random walk
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of distance $w$, “the distribution of the ending point of the walk”
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Which will, I think, speed it up to $O(n×{{(log (n))}^2})$ which is
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acceptable.
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Mixing for a suitable time, a suitable path distance, divides the graph into
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well mixed regions, one of which is the legitimate nodes. We identify which
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of the regions is non Sybil by checking the region of a small number of known
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legitimate nodes. If they all belong to the same region, our mixing distance
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is sufficient, and chances are the Sybil nodes are not in that region.
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During mixing, the points draw closer to each other in the space of
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distributions. The fan group of a particular musician will coalesce rapidly,
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and then the fan groups of all legitimate coalitions will, more slowly draw
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together, while the fake fan group drifts very slowly indeed. We halt the
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mixing when the known legitimate nodes are all close to each other, because
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by then everyone who is a legitimate node is close to them.
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We then downrank reviews or page ranks that are distant from the coalescence.
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