Skip List Based Authentication

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Skip List Based Authentication

• ???
• 2008-10-9

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Outline

• Introduction of Skip List
• Comparison with Balanced Tree
• Commutative Hashing
• Skip List Based Authentication

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Introduction of Skip List

• Proposed by William Pugh (University of
  Maryland)
• Commnu. ACM, 33(6)668-676,1990
• A probabilistic alternative to balanced trees
• Simple and efficient
• probabilistic balancing compared to strictly
  enforced balancing
• Balanced trees, such as B tree, is proposed in
  1970s.

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Skip List Example
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Skip List Construction
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Skip List Construction

• Node level of pointer in one node
• Determined at creation time and never change
• According to size rules, there are two kinds of
  skip list
• Random of nodes between two adjacent node with
  equal level is probability based
• Deterministic of nodes between two adjacent
  node with equal level is size rule based, i.e.
  1,3

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Skip List-Query

• Find (k)
• Hop forward
• within the same level from min to max
• until maximal element not greater than k
• Drop down
• One by one
• Example

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Skip List-Insert and Delete

• Two steps query, insert (delete)
• Compared to query, insert has two different
  steps
• Climb up
• Move backward
• A vector update is maintained, so that when the
  search is complete, updatei contains a pointer
  to the rightmost node of level i or one that is
  to the left of the location of the insertion
  (deletion)

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Skip List-Query and Insert
return
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Comparison with Balanced Tree

• Improve query performance at the cost of
  additional pointer (space cost)
• Implementation of skip List is simple, while
  implementation of B-tree is recursive. The
  complexity of implementation is negatively
  related to degree of optimization. For example.
• Skip List can adapt to arbitrary input sequence,
  while B-tree is inefficient when input is ordered.

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Space cost
A fraction p of nodes with level i pointers also
have level i1 pointers
return
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Time comparison

• 216 elements with integer keys
• The value in parenthesis show the results
  relative to the skip list time.
• Time complexity is O(logn). (Constant factor)
• Algorithm B is twice as fast as A, which means
  in the time A takes to process
• a query on a data set of size n, B can process
  a query on a data set of size n2.

return
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Why not Skip List? (My opinion)

• B-tree is at most 3 levels in most cases (a node
  is exactly a page), which makes B-tree I/O
  efficient.
• With the increasing of node levels, Skip list is
  inefficient in I/O cost. So skip list is suitable
  for in-memory data set.

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Commutative Hashing

• Collision resistant hash h(x,y)
• Hard to find (c,d)ltgt(a,b), such that
• h(c,d)h(a,b)
• commutative h(x,y) h(y,x)
• Hard to find (c,d)ltgt(a,b), (d,c)ltgt(a,b), such
  that h(c,d)h(a,b)
• A candidate construction
• h(x,y)f(minx,y, maxx,y)

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Skip List Based Authentication

• Proposed by Michael T. Goodrich (Johns Hopkins
  University), Roberto Tamassia (Brown University)
• In Proc. 2001 DARPA Information Survivability
  Conference and Exposition

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Skip List Based Authentication

• Authentication dictionary includes
• A skip list data structure storing the items of
  S.
• A collection of values f (v) that label each
  node v of the skip list, computed accumulating
  the elements of S with the hash function h, for
  example.
• A statement signed by the source consisting of
  the timestamp of the most recent label f (s) of
  the start node of the skip list.

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Skip List Based Authentication

• Calculation of f (v)
• w right(v), u down(v). If right(v) null,
  then f (v) 0.
• 1. u null, i.e., v is on the base level
• (a) If w is a tower node, then f (v)
  h(elem(v)elem(w)).
• (b) If w is a plateau node, then f (v)
  h(elem(v) f (w)).
• 2. ultgt null, i.e., v is not on the base level
• (a) If w is a tower node, then f (v) f (u).
• (b) If w is a plateau node, then f (v) h( f
  (u) f (w)).

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Skip List Based Authentication
return
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Calculation of Authentication Info

• There are two kinds of authentication
• exists, value, authentication info, for example
• non-exists, boundary value, authentication
  info, for example
• How to get authentication info from query info
• w1right(v1)
• if w1 is a plateau node then x0 f (w1) else
  x0elem(w1)
• X1x, j1
• for i 2 m-1 do
• Wiright(vi)
• if wi is a plateau node then
• jj1
• if wi ltgt vi-1 then xj f (wi)
• else if vi is in the base list S0 then
  Xjelem(vi)
• 
  else uidown(vi), xj f (ui)
• Pj

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Skip List Based Authentication
return
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Skip List Based Authentication
return
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Comparison with MHT

• Update scenario skip list need O(logn), while
  MHT need complete rebuilding.
• Range query skip list cant support, while MHT
• can.
• Authenticate info skip list only need a
  sequence, while MHT need sequence and order info.
  (thats why commutative hashing is needed)

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• Any question?