One-way protocols and combinatorial designs

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One-way protocols and combinatorial designs

• Mike Atkinson
• Joint work with
• Michael Albert, Hans van Ditmarsch, Robert
  Aldred, Chris Handley

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The plan

• Description of problem
• Modelling the problem
• Solutions

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The 2000 Moscow Mathematical Olympiad

• Players Alice, Bob, Crow draw cards from a 7 card
  deck. A receives 3 cards, B receives 3 cards, C
  receives 1 card
• How can A, in a single public announcement, tell
  B what her cards are without C learning a single
  card of A or Bs holding?

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First thoughts

• A could make some very complex announcement (I
  hold card 2 or card 4 if I hold card 3 I dont
  hold card 5 if I hold any consecutive numbered
  cards then one is prime,.)
• B, knowing his own cards, finds As announcement
  useful
• C, knowing only his card, cant use it

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Pitfalls

• Suppose A held 0,1,2 she could say I hold 0,1,2
  or 3,4,5
• B would successfully learn As hand because only
  one of those possibilities can be consistent with
  his own hand
• But, for all A knows, C might hold 3 and then C
  could infer As holding (note A would be safe if
  C held 6)

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Second thoughts

• No matter how complex is As announcement it is
  tantamount to saying My holding is one of the
  following
• As announcement must be effective for B and
  ineffective for C no matter what B and C hold

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First solution

• A says Modulo 7 my total is x.
• The 35 possible holdings for A come in 7 groups
  of 5 corresponding to their sum mod 7
• Modulo 7 my total is 3 is tantamount to saying
  I hold 012, 136, 145, 235, or 046
• B can now work out Cs card and therefore work
  out As holding
• C can only work out As sum modulo 7 and Bs sum
  modulo 7 he cant work out any one card of A or
  B.

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Second solution

• A could announce (supposing that she holds 0,1,2)
  I hold one of 012,056,034,145,136,235,246
• Exhaustive check. E.g. suppose B held 345 then he
  could deduce A holds 012 since all other
  possibilities intersect his own holding. But C
  (holding 6) can deduce only that As holding is
  one of 012,034,145,235 and no card of A is
  revealed.

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Other solutions

• All solutions involve an announcement of 5 or 6
  or 7 possible holdings
• More than 7 makes it too hard for B
• Less than 5 makes it too easy for C

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Reveal as little as possible

• If A wishes to reveal as little as possible she
  should choose to present 7 possible holdings
  rather than 5
• How are the optimal solutions found?

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Structure of the solution 012,056,034,145,136,235
,246

• The 7 triples are the lines of the 7 point
  projective plane

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The general problem

• A holds a cards, B b cards, C c cards from a deck
  of vabc cards
• A must make one public announcement from which B
  can infer As holding but C cannot infer any card
  of either A or B
• For which a, b, c is this possible?
• If it is possible, what are the most and least
  informative announcements?
• Find a suitable announcement!

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Communication protocols

• A protocol is a series of messages by various
  parties to communicate information E.g. A might
  send a message to B, B might answer with another
  message, A might send yet another message,.
  Eventually the required information is
  communicated.
• We are studying one-way protocols

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The one-way restriction

• Suppose a2, b4, c1 (and v7)
• No one-way protocol is possible
• There is a 2 message protocol
• B first announces a number of possible holdings
  for himself that allows A to deduce Bs holding
  whereas C learns no card of either A or B
• A now knows Cs card and announces it this tells
  C nothing further but allows B to infer As
  holding

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The one-way restriction

• Suppose a2, b4, c1 (and v7)
• No one-way protocol is possible
• There is a 2 message protocol
• B (holding, say, 1236) could announce he holds
  one of 3456, 0156, 1245, 1236, 0134, 0235, 0246.
  A (holding, say, 05) could then infer Bs holding
• A now knows Cs card is 4 and announces it B can
  now deduce that A holds 05

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Combinatorial conditions

• A collection L of a-subsets of 0,1,..,v-1 is a
  one-way protocol if and only if
• For all L1,L2 in L , L1 ? L2 a-c-1
• For all c-sets X the set of members of L disjoint
  from X have empty intersection and their union
  contains every point not in X

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Combinatorial problems

• For given a,b,c find a suitable collection L of
  a-subsets of 0,1,,v-1.
• Find upper and lower bounds on the size of L.
• Find general constructions valid for a range of
  (a,b,c) values.

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Bounds on L

• L
• L v(c1)/a
• Some other bounds also known
• Sometimes the bounds prove that no one-way
  protocol exists
• Occasionally, they pin down L uniquely
• e.g. if b2, c1 then L (a2)(a3)/6

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General construction

• Let D be a set of a integers such that among the
  (non-zero) differences d1-d2 no value occurs more
  than e times.
• Let L be the set i D i 0 v-1 (arithmetic
  mod v)
• L realises the parameter set
• a,v-2ae1,a-e-1

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Examples

• Many one-way protocols seem to have no further
  combinatorial interest
• Those for which L is maximal are often more
  interesting
• v 13 (all the spades), a 4, b 7, c 2, L
  is the set of 13 lines of the 13 point projective
  plane
• v 11, a 5, b 5, c 1, L is the set of 66
  blocks of the Steiner system 4-(5,11,1) whose
  automorphism group is M11

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Examples cont.

• a4, b3, c1. Code the 8 cards as vectors in Z2
  ? Z2 ? Z2. Let L be the 7 subgroups of order 4
  and their complements