High-entropy High-entropy random selection random selection protocols protocols Michal Michal Koucký Koucký (Institute of Mathematics, Prague) (Institute of Mathematics, Prague) Harry Buhrman, Matthias Harry Buhrman, Matthias Christandl, Zvi Lotker, Boaz Patt- Christandl, Zvi Lotker, Boaz Patt- Shamir, KoliaVereshchagin Shamir, KoliaVereshchagin
High-entropy random selection protocols. Michal Koucký (Institute of Mathematics, Prague) Harry Buhrman, Matthias Christandl, Zvi Lotker, Boaz Patt-Shamir, KoliaVereshchagin. Random string selection: Alice Bob. Goal: Alice and Bob want to agree on a random string r. - PowerPoint PPT Presentation
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High-entropy High-entropy random selection random selection
protocolsprotocolsMichalMichal Koucký Koucký
(Institute of Mathematics, Prague)(Institute of Mathematics, Prague)
Harry Buhrman, Matthias Christandl, Harry Buhrman, Matthias Christandl, Zvi Lotker, Boaz Patt-Shamir, Zvi Lotker, Boaz Patt-Shamir,
KoliaVereshchaginKoliaVereshchagin
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Random string selection:Random string selection:
Alice Alice BobBob
Goal: Goal: Alice and Bob want to agree on a Alice and Bob want to agree on a random random string string rr..
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Goal: Goal: Alice and Bob want to agree on a Alice and Bob want to agree on a random random string string rr..
→→ Measure of randomness:Measure of randomness: Shannon Shannon entropyentropy
Example:Example:random random rr11rr22 … … rrnn/2/2
Alice Alice random random rrnn/2+1/2+1 … … rrnn BobBob
→→ output output rr = = rr11rr22 … … rrnn
H( H( R R ) = ) = nn if Alice and Bob follow the if Alice and Bob follow the protocol.protocol.
H( H( R R ) ) nn/2/2 if one of them if one of them cheatscheats..
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Main results:Main results: Random selection protocol that Random selection protocol that
guaranteesguarantees H( H( RR ) ) nn – O(1)– O(1) even if even if one of the parties cheats. This protocol one of the parties cheats. This protocol runs in log* runs in log* n n rounds and communicates rounds and communicates O( O( n n 2 2 ).).
Three-round protocol that guarantees Three-round protocol that guarantees H( H( R R ) ) ¾ ¾ n n and communicates O( and communicates O( n n ) ) bits.bits.
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Previous work:Previous work: Different variantsDifferent variants
random selection protocol [GGL’95, SV’05, random selection protocol [GGL’95, SV’05, GVZ’06]GVZ’06]
x x ii yy rotation of rotation of x ix i-times.-times. ix ix + + yy xx,, yy FFkk i i FF
F F = GF(2= GF(2log log nn ) ) kk = = nn / / log log nn
ix ix + + yy xx,, y y FF i i H H FFF F = GF(2= GF(2nn ) |) |HH|=|=nn
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Rotations:Rotations:For any For any xx and and yy
( ( x x ii yy ) ) ( ( x x jj y y ) = ) = x x i i x x j j = = x Ax Aijij
where where AAijij has rank has rank n n – 1. – 1.
xx random random n n – 1 – 1 H( H( x Ax Aijij ) ) H( H( x x ii yy , , x x jj y y ))
H( H( R R ) ) nn – log – log nn when Alice cheatswhen Alice cheats H( H( R R ) ) nn /2/2 when Bob cheatswhen Bob cheats
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¾¾n-n-protocol:protocol:
1.1. Pick one half of the string by A-B-A Pick one half of the string by A-B-A “rotating” protocol and the other one “rotating” protocol and the other one by B-A-B “rotating” protocol, i.e., use by B-A-B “rotating” protocol, i.e., use the asymmetry in the cheating powers.the asymmetry in the cheating powers.
2.2. The “line” protocol The “line” protocol ix ix + + yy , where , where xx,, yy [GF(2 [GF(2 nn/4/4 )] )]kk and and kk = 4 = 4
→→ analysis related to the problem of analysis related to the problem of KakeyaKakeya..
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Kakeya Problem:Kakeya Problem:
PP
FFkk
Conj: Conj: PP contains a line in each direction contains a line in each direction ||PP||||FF||k k
( 1( 1– – cc /|/|FF|)|)
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Open problems:Open problems: Better analysis of our candidate Better analysis of our candidate
functions.functions. Other candidate functions?Other candidate functions? Multiple parties.Multiple parties.
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Alice plays honestly, Bob cheats:Alice plays honestly, Bob cheats:
For any For any rr11, , rr2 2 , … , … rrn n , Pr, Prx x [ [ rr11 = = xx11 , … , … rrnn = = xxn n ] = 2 ] = 2 – – nn22
Pr[ Pr[ rr11 = = xx11 y y , … , … rrnn = = xxnn y y ] ] 2 2 n n – – nn22
where where yy is a function of the random is a function of the random xx11, , xx2 2 , … , … xxn n
H( H( xx11 y y , …, , …, xxnn y y ) ) nn 2 2 - - nn