what ~1.25 turned out to be or complex poles and dvds
DESCRIPTION
What ~1.25 turned out to be or Complex poles and DVDs. Ilya Mironov Microsoft Research, SVC October 3 rd , 2003. One-to-One Communications. Alice. Bob. One-to-Many Communications. Alice. Bob. Carl. Zing. One-to-Many Communications. Alice. Bob. Carl. Zing. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/1.jpg)
What ~1.25 turned out to What ~1.25 turned out to bebeoror
Complex poles and DVDsComplex poles and DVDsIlya MironovIlya Mironov
Microsoft Research, SVCMicrosoft Research, SVC
October 3October 3rdrd, 2003, 2003
![Page 2: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/2.jpg)
One-to-One One-to-One CommunicationsCommunications
Alice Bob
![Page 3: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/3.jpg)
One-to-Many One-to-Many CommunicationsCommunications
Alice
Bob Carl Zing
![Page 4: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/4.jpg)
One-to-Many One-to-Many CommunicationsCommunications
Alice
Bob Carl Zing
![Page 5: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/5.jpg)
One-to-Many One-to-Many CommunicationsCommunications
Alice
Bob Carl Zing
![Page 6: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/6.jpg)
One-to-Many One-to-Many CommunicationsCommunications
Alice
Bob Carl Zing
![Page 7: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/7.jpg)
BroadcastBroadcast
Alice
Bob Carl Zing
![Page 8: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/8.jpg)
BroadcastBroadcast
Alice
Bob Carl Zing
![Page 9: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/9.jpg)
Real Life Examples of Real Life Examples of BroadcastBroadcast
Pay-per-viewPay-per-view Satellite radio, TV (“dishes”)Satellite radio, TV (“dishes”) DVD playersDVD players
Stateless receivers
![Page 10: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/10.jpg)
Broadcast encryptionBroadcast encryption
source
receivers
k
k k k k k k kk k
k
One rogue user compromises the whole system
Very little overhead
![Page 11: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/11.jpg)
Broadcast encryptionBroadcast encryption
source
receivers
k1, k2, k3, k4, k5,…, kn
k1
k2 k3 k4 k5 k6 k7 kn
…
broadcast E[k1,k], E[k2,k],…, E[kn,k], E[k,M]
![Page 12: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/12.jpg)
Broadcast encryptionBroadcast encryption
source
receivers
k1, k2, k3, k4, k5,…, kn
k1
k2 k3 k4 k5 k6 k7 kn
…
Too many keys Simple user revocation
![Page 13: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/13.jpg)
Botched attemptsBotched attempts
CSS (most famous for the DeCSS CSS (most famous for the DeCSS crack)crack)
CPRM (IBM, Intel, Matsushita, CPRM (IBM, Intel, Matsushita, Toshiba) Can revoke only 10,000 Toshiba) Can revoke only 10,000 devices in 3Mbdevices in 3Mb
![Page 14: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/14.jpg)
Subset-cover frameworkSubset-cover framework (Naor-Naor-Lotspiech’01)(Naor-Naor-Lotspiech’01)
S3
S5
S6
S1
S2S4
S7
S8
![Page 15: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/15.jpg)
Subset-cover frameworkSubset-cover framework (Naor-Naor-Lotspiech’01)(Naor-Naor-Lotspiech’01)
S3
S5
S6
S1
S2S4
S7
S8
k3 k4k5
u
receiver u knows keys:
![Page 16: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/16.jpg)
Key distributionKey distribution
Based on some formal characteristic: Based on some formal characteristic: e.g., DVD player’s serial numbere.g., DVD player’s serial number
Using some real-life descriptors:Using some real-life descriptors:— CMU students/facultyCMU students/faculty— researchersresearchers— Pennsylvania state residentsPennsylvania state residents— college-educatedcollege-educated
![Page 17: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/17.jpg)
Broadcast using subset Broadcast using subset covercover
S3
S5
S6
S1S8
S10
header uses k1, k3, k5, k6, k8, k10
![Page 18: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/18.jpg)
Subtree differenceSubtree differenceAll receivers are associated with theleaves of a full binary tree k0
k00 k01
k0…0 k0…1 k1…1
![Page 19: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/19.jpg)
Subtree differencesSubtree differences
i
j
special set Si,j
![Page 20: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/20.jpg)
Subtree differenceSubtree difference
![Page 21: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/21.jpg)
Subtree differenceSubtree difference
![Page 22: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/22.jpg)
Subtree differenceSubtree difference
![Page 23: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/23.jpg)
Subtree differenceSubtree difference
![Page 24: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/24.jpg)
Subtree differenceSubtree difference
![Page 25: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/25.jpg)
Subtree differenceSubtree difference
![Page 26: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/26.jpg)
Subtree differenceSubtree difference
![Page 27: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/27.jpg)
Subtree differenceSubtree difference
![Page 28: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/28.jpg)
Greedy algorithmGreedy algorithm
Easy greedy algorithm for Easy greedy algorithm for constructing a subtree cover for any constructing a subtree cover for any set of revoked usersset of revoked users
![Page 29: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/29.jpg)
Greedy algorithmGreedy algorithm
Find a node such that both of its Find a node such that both of its children have exactly one revoked children have exactly one revoked descendantdescendant
![Page 30: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/30.jpg)
Greedy algorithmGreedy algorithm
Add (at most) two sets to the coverAdd (at most) two sets to the cover
![Page 31: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/31.jpg)
Greedy algorithmGreedy algorithm
Revoke the entire subtreeRevoke the entire subtree
![Page 32: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/32.jpg)
Greedy algorithmGreedy algorithm
Could be less than two setsCould be less than two sets
![Page 33: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/33.jpg)
Average-case analysisAverage-case analysis
R - number of revoked usersR - number of revoked users
C – number of sets in the coverC – number of sets in the cover
C ≤ 2R-1C ≤ 2R-1 averaged over sets of fixed size averaged over sets of fixed size
[NNL’01][NNL’01]
E[C] ≤ 1.38RE[C] ≤ 1.38R simulation experiments give [NNL’01]simulation experiments give [NNL’01]
E[C] ~ RE[C] ~ R1.251.25
![Page 34: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/34.jpg)
HypothesisHypothesis
1.25… = 5/41.25… = 5/4
![Page 35: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/35.jpg)
Different ModelDifferent Model
Revoke each user independently at Revoke each user independently at random with probability prandom with probability p
![Page 36: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/36.jpg)
Exact formulaExact formula
2 2 2
0
[ ] 1lim 2 2 (1 ) ,
[ ] 1
k kk
nk
E Cq q q
E R q
where 1 1.q p
If a user is revoked with probability If a user is revoked with probability p«1:p«1:
![Page 37: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/37.jpg)
Exact formulaExact formula
2 2 2
0
[ ] 1lim 2 2 (1 ) ,
[ ] 1
k kk
nk
E Cq q q
E R q
where
If a user is revoked with probability If a user is revoked with probability p«1:p«1:
(...)f 1 1.q p
![Page 38: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/38.jpg)
AsymptoticAsymptotic
0
0.25
0.5
0.75
1
1.25
1.5
0 0.25 0.5 0.75 1p
1.245111.24511
E[C]/E[R]
![Page 39: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/39.jpg)
AsymptoticAsymptotic
E[C]/E[R]1.2451134…1.2451134…
1.2451114…1.2451114…
1.24509
1.245095
1.2451
1.245105
1.24511
1.245115
0 0.000005 0.00001
p
![Page 40: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/40.jpg)
Exact formulaExact formula
2 2 2
0
[ ] 1lim 2 2 (1 ) ,
[ ] 1
k kk
nk
E Cq q q
E R q
where 1 .q p
If a user is revoked with probability If a user is revoked with probability p«1:p«1:
(...)f
![Page 41: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/41.jpg)
Singularities of Singularities of ff
Function Function ff cannot be analytically cannot be analytically continued beyond the unit diskcontinued beyond the unit disk
![Page 42: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/42.jpg)
One approachOne approach
5 pages of dense computations – 5 pages of dense computations – series, o, O, lim, etc.series, o, O, lim, etc.
produce only the constant termproduce only the constant term
![Page 43: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/43.jpg)
Mellin transformMellin transform
* 1
0( ) ( ) ( ) sf x f s f x x dx
* * 11( ) ( ) ( )
2
c i s
c if s f s f s x ds
i
![Page 44: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/44.jpg)
ApproximationApproximation
where 1 0x q
For small qFor small q
2 2 2 2 2 2
0 0
( ) 2 (1 ) 2 (1 )k k k kk k x x
k k
f q q q e e
![Page 45: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/45.jpg)
The Mellin TransformThe Mellin Transform
* 1(1 )
1( ) ( )(1 2 3 )
1 2s s
sf s s
Poles at 0, -1, -2, -3, … and 2 / ln 2 1ki
![Page 46: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/46.jpg)
Complex polesComplex poles
0-1-2-3
(1 )
1
1 2 s
1( )(1 2 3 )s ss
2 / ln 2 1ki
…
![Page 47: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/47.jpg)
Mellin transformMellin transform
* 1
0( ) ( ) ( ) sf x f s f x x dx
* * 11( ) ( ) ( )
2
c i s
c if s f s f s x ds
i
![Page 48: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/48.jpg)
ApproximationApproximation
22 log2 2
\{0}
( ) 3log 4 / 3 ( 2 log )
( ),
ki p
k
f q ki p e
o p
where p = 1-qwhere p = 1-q
![Page 49: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/49.jpg)
AsymptoticAsymptotic
E[C]/E[R]1.2451134…1.2451134…
1.2451114…1.2451114…3log2 4/3
1.24509
1.245095
1.2451
1.245105
1.24511
1.245115
0 0.000005 0.00001
p
![Page 50: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/50.jpg)
Average-case analysisAverage-case analysis
R - number of revoked usersR - number of revoked users
C – number of sets in the coverC – number of sets in the cover
If a user is revoked with probability p«1:If a user is revoked with probability p«1:
E[C] ≈ 1.24511 E[R]E[C] ≈ 1.24511 E[R]
![Page 51: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/51.jpg)
Knuth and de BruijnKnuth and de Bruijn
Solution communicated by de Bruijn Solution communicated by de Bruijn to Knuth for analysis of the radix-to Knuth for analysis of the radix-exchange sort algorithm (vol. 3, 1exchange sort algorithm (vol. 3, 1stst ed, p. 131)ed, p. 131)
De Bruijn, Knuth, Rice, “The average De Bruijn, Knuth, Rice, “The average height of planted plane trees,” 1972height of planted plane trees,” 1972
![Page 52: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/52.jpg)
Further readingFurther reading
Flajolet, Gourdon, Dumas, “Mellin Flajolet, Gourdon, Dumas, “Mellin transform and asymptotics: transform and asymptotics: Harmonics sums”, Theor. Comp. Sc., Harmonics sums”, Theor. Comp. Sc., 123(2), 1994123(2), 1994
![Page 53: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/53.jpg)
Back-up slidesBack-up slides
![Page 54: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/54.jpg)
HalevyHalevy--Shamir schemeShamir scheme
Noticed that subtree differences are Noticed that subtree differences are decomposable:decomposable:
![Page 55: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/55.jpg)
HalevyHalevy--Shamir schemeShamir scheme
Fewer special sets reduce memory Fewer special sets reduce memory requirement on receiversrequirement on receivers
![Page 56: What ~1.25 turned out to be or Complex poles and DVDs](https://reader036.vdocuments.us/reader036/viewer/2022062305/56814903550346895db638f1/html5/thumbnails/56.jpg)
ImprovementImprovement
For practical parameters save For practical parameters save additionally 20% compared to the additionally 20% compared to the Halevy-Shamir schemeHalevy-Shamir scheme