[PPT] - Analysis of the Parallel Distinguished Point Tradeoff Jin Hong, *Ga PowerPoint Presentation

SLIDE 1

Analysis of the Parallel Distinguished Point Tradeoff

Jin Hong, *Ga Won Lee, Daegun Ma

Seoul National University

13/12/2011

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 1 / 24

SLIDE 2

The Inversion Problem

N : key space with size N. F : N → N : one-way function The inversion problem For a given inversion target F(x) = y, Find x.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 2 / 24

SLIDE 3

The Inversion Problem

N : key space with size N. F : N → N : one-way function The inversion problem For a given inversion target F(x) = y, Find x.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 2 / 24

SLIDE 4

The Inversion Problem

N : key space with size N. F : N → N : one-way function The inversion problem For a given inversion target F(x) = y, Find x. Two extreme methods Exhaustive search : T=N, M=1, Dictionary attack : T=1, M=N, where T is total online time, M is storage size.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 2 / 24

SLIDE 5

The Inversion Problem Time Memory Tradeoff(Hellman)

Pre-computation phase :

pre-compute sufficiently many (a, F(a)) pairs, and store a digest of the computation in a table smaller than the complete dictionary.

Online phase :

given an inversion target, using the table, find the answer in time shorter than required by exhaustive search.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 3 / 24

SLIDE 6

The DP Tradeoff : pre-computation phase

R. Rivest

Choose parameters m, t satisfying mt2 = N.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 4 / 24

SLIDE 7

The DP Tradeoff : pre-computation phase

R. Rivest

Choose parameters m, t satisfying mt2 = N. DP(distinguished point) is an element satisfying a certain pre-set

property. Here, the prob. of DP occurrence is set to 1

t .

(ex. X ≡ 0 mod t)

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 4 / 24

SLIDE 8

The DP Tradeoff : pre-computation phase

R. Rivest

Choose parameters m, t satisfying mt2 = N. DP(distinguished point) is an element satisfying a certain pre-set

property. Here, the prob. of DP occurrence is set to 1

t .

(ex. X ≡ 0 mod t)

1. Construct t many DP matrices using F.
each chain is set to end on a DP.

m              SP1 = ◦ F → ◦ F → · · · · · ·

F

→ ◦ = EP1 : DP SP2 = ◦ F → ◦ F → · · · · · ·

F

→ · · · · · · F → ◦ = EP2 : DP . . . SPm = ◦ F → · · · · · · F → ◦ = EPm : DP : A single DP matrix

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 4 / 24

SLIDE 9

The DP Tradeoff : pre-computation phase

R. Rivest

Choose parameters m, t satisfying mt2 = N. DP(distinguished point) is an element satisfying a certain pre-set

property. Here, the prob. of DP occurrence is set to 1

t .

(ex. X ≡ 0 mod t)

1. Construct t many DP matrices using F.
each chain is set to end on a DP.

m              SP1 = ◦ F → ◦ F → · · · · · ·

F

→ ◦ = EP1 : DP SP2 = ◦ F → ◦ F → · · · · · ·

F

→ · · · · · · F → ◦ = EP2 : DP . . . SPm = ◦ F → · · · · · · F → ◦ = EPm : DP : A single DP matrix

2. Store {(SPj, EPj)}m

j=1 only, throwing the rest out.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 4 / 24

SLIDE 10

The DP Tradeoff : online phase

Given an inversion target y = F(x)

1. Online chian creation

Create online chain from y. y F → ◦ F → ◦ F → · · · · · · F → • : DP

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 5 / 24

SLIDE 11

The DP Tradeoff : online phase

Given an inversion target y = F(x)

1. Online chian creation

Create online chain from y. y F → ◦ F → ◦ F → · · · · · · F → • : DP Check if it matches an ending point in {EPj}.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 5 / 24

SLIDE 12

The DP Tradeoff : online phase

Given an inversion target y = F(x)

1. Online chian creation

Create online chain from y. y F → ◦ F → ◦ F → · · · · · · F → • : DP Check if it matches an ending point in {EPj}. m              SP1 = ◦ F → ◦ F → · · · · · ·

F

→ ◦ = EP1 : DP SP2 = ◦ F → ◦ F → · · · · · ·

F

→ · · · · · · F →

alarm!!

= EP2 : DP

. . . SPm = ◦ F → · · · · · · F → ◦ = EPm : DP

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 5 / 24

SLIDE 13

The DP Tradeoff : online phase

2. pre-computed chain regeneration

Expectation : SP2 = ◦ F → ◦ F → · · · · · · x F → y F → ◦ · · · · · · F →

alarm!!

= EP2

x F → y F → ◦ · · · · · · F →

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 6 / 24

SLIDE 14

The DP Tradeoff : online phase

2. pre-computed chain regeneration

Expectation : SP2 = ◦ F → ◦ F → · · · · · · x F → y F → ◦ · · · · · · F →

alarm!!

= EP2

x F → y F → ◦ · · · · · · F →

pre-computed chain regeneration :

SP2 = ◦

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 6 / 24

SLIDE 15

The DP Tradeoff : online phase

2. pre-computed chain regeneration

Expectation : SP2 = ◦ F → ◦ F → · · · · · · x F → y F → ◦ · · · · · · F →

alarm!!

= EP2

x F → y F → ◦ · · · · · · F →

pre-computed chain regeneration :

SP2 = ◦ F → ◦

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 6 / 24

SLIDE 16

The DP Tradeoff : online phase

2. pre-computed chain regeneration

Expectation : SP2 = ◦ F → ◦ F → · · · · · · x F → y F → ◦ · · · · · · F →

alarm!!

= EP2

x F → y F → ◦ · · · · · · F →

pre-computed chain regeneration :

SP2 = ◦ F → ◦ F →

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 6 / 24

SLIDE 17

The DP Tradeoff : online phase

2. pre-computed chain regeneration

Expectation : SP2 = ◦ F → ◦ F → · · · · · · x F → y F → ◦ · · · · · · F →

alarm!!

= EP2

x F → y F → ◦ · · · · · · F →

pre-computed chain regeneration :

SP2 = ◦ F → ◦ F → · · · · · · F →

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 6 / 24

SLIDE 18

The DP Tradeoff : online phase

2. pre-computed chain regeneration

Expectation : SP2 = ◦ F → ◦ F → · · · · · · x F → y F → ◦ · · · · · · F →

alarm!!

= EP2

x F → y F → ◦ · · · · · · F →

pre-computed chain regeneration :

SP2 = ◦ F → ◦ F → · · · · · · F → x

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 6 / 24

SLIDE 19

The DP Tradeoff : online phase

2. pre-computed chain regeneration

Expectation : SP2 = ◦ F → ◦ F → · · · · · · x F → y F → ◦ · · · · · · F →

alarm!!

= EP2

x F → y F → ◦ · · · · · · F →

pre-computed chain regeneration :

SP2 = ◦ F → ◦ F → · · · · · · F → x F → y

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 6 / 24

SLIDE 20

The DP Tradeoff : online phase

2. pre-computed chain regeneration

Expectation : SP2 = ◦ F → ◦ F → · · · · · · x F → y F → ◦ · · · · · · F →

alarm!!

= EP2

x F → y F → ◦ · · · · · · F →

pre-computed chain regeneration :

SP2 = ◦ F → ◦ F → · · · · · · F → x F → y

’x’ is just found!!!

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 6 / 24

SLIDE 21

The DP Tradeoff : online phase However,

Most case : Since F is not injective, ´ x = x SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր x F → y F → ◦ · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 7 / 24

SLIDE 22

The DP Tradeoff : online phase However,

Most case : Since F is not injective, ´ x = x SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր x F → y F → ◦ · · ·

It is called a false alarm.
J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 7 / 24

SLIDE 23

The DP Tradeoff : online phase However,

Most case : Since F is not injective, ´ x = x SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր x F → y F → ◦ · · ·

It is called a false alarm.

pre-computed chain regeneration : SP2 = ◦ F → · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 7 / 24

SLIDE 24

The DP Tradeoff : online phase However,

Most case : Since F is not injective, ´ x = x SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր x F → y F → ◦ · · ·

It is called a false alarm.

pre-computed chain regeneration : SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 7 / 24

SLIDE 25

The DP Tradeoff : online phase However,

Most case : Since F is not injective, ´ x = x SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր x F → y F → ◦ · · ·

It is called a false alarm.

pre-computed chain regeneration : SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 7 / 24

SLIDE 26

The DP Tradeoff : online phase However,

Most case : Since F is not injective, ´ x = x SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր x F → y F → ◦ · · ·

It is called a false alarm.

pre-computed chain regeneration : SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · ·

F

→

= EP2
J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 7 / 24

SLIDE 27

The DP Tradeoff : online phase However,

Most case : Since F is not injective, ´ x = x SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր x F → y F → ◦ · · ·

It is called a false alarm.

pre-computed chain regeneration : SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · ·

F

→

= EP2
Whole pre-computed chain is re-generated, but ’x’

cannot be found.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 7 / 24

SLIDE 28

The Rainbow Method : the rainbow matrix

Oechslin Choose parameters m, t satisfying mt = N.(Recall. In DP, mDt2

D = N)

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 8 / 24

SLIDE 29

The Rainbow Method : the rainbow matrix

Oechslin Choose parameters m, t satisfying mt = N.(Recall. In DP, mDt2

D = N)

1. Create one big m × t matrix using F and reduction functions ri.

SP1 = ◦ F1 → ◦ F2 → · · · · · · Ft → ◦ = EP1 SP2 = ◦ F1 → ◦ F2 → · · · · · · Ft → ◦ = EP2 . . . . . . . . . SPm = ◦ F1 → ◦ F2 → · · · · · · Ft → ◦ = EPm : A single rainbow matrix ,where Fi = ri ◦ F.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 8 / 24

SLIDE 30

The Rainbow Method : the rainbow matrix

Oechslin Choose parameters m, t satisfying mt = N.(Recall. In DP, mDt2

D = N)

1. Create one big m × t matrix using F and reduction functions ri.

SP1 = ◦ F1 → ◦ F2 → · · · · · · Ft → ◦ = EP1 SP2 = ◦ F1 → ◦ F2 → · · · · · · Ft → ◦ = EP2 . . . . . . . . . SPm = ◦ F1 → ◦ F2 → · · · · · · Ft → ◦ = EPm : A single rainbow matrix ,where Fi = ri ◦ F.

2. Store {(SPj, EPj)}m

j=1 only, throwing the rest out.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 8 / 24

SLIDE 31

Previous Results : online time complexity [HM10]

The DP tradeoff, mt2 = DmscN

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 9 / 24

SLIDE 32

Previous Results : online time complexity [HM10]

The DP tradeoff, mt2 = DmscN The expected number of distinct points in a single DP matrix is Dcrmt, where Dcr = 2 √1 + 2Dmsc + 1.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 9 / 24

SLIDE 33

Previous Results : online time complexity [HM10]

The DP tradeoff, mt2 = DmscN The expected number of distinct points in a single DP matrix is Dcrmt, where Dcr = 2 √1 + 2Dmsc + 1. The success probability of the DP tradeoff is Dps = 1 − e−DcrDpc, with pre-computation cost DpcN.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 9 / 24

SLIDE 34

Previous Results : online time complexity [HM10]

The DP tradeoff, mt2 = DmscN The expected number of distinct points in a single DP matrix is Dcrmt, where Dcr = 2 √1 + 2Dmsc + 1. The success probability of the DP tradeoff is Dps = 1 − e−DcrDpc, with pre-computation cost DpcN. The time memory tradeoff curve for the DP tradeoff is TM2 = DtcN2, where Dtc = (2 + 1 Dmsc ) 1 D3

cr

Dps{ln(1 − Dps)}2.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 9 / 24

SLIDE 35

Previous Results : online time complexity [HM10]

The rainbow method, mt = RmscN, l tables

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 10 / 24

SLIDE 36

Previous Results : online time complexity [HM10]

The rainbow method, mt = RmscN, l tables The success probability of the rainbow method is Rps = 1 −

2

2 + Rmsc 2l .

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 10 / 24

SLIDE 37

Previous Results : online time complexity [HM10]

The rainbow method, mt = RmscN, l tables The success probability of the rainbow method is Rps = 1 −

2

2 + Rmsc 2l . The time memory tradeoff curve for the rainbow method is TM2 = RtcN2, where Rtc =

l3 (2l+1)(2l+2)(2l+3)

{(2l − 1) + (2l + 1)Rmsc}(2 + Rmsc)2

−4{(2l − 1) + l(2l + 3)Rmsc}

2

2+Rmsc

2l .

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 10 / 24

SLIDE 38

Previous Results : efficient use of storage [HM10]

mDtD2 = N = mRtR

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 11 / 24

SLIDE 39

Previous Results : efficient use of storage [HM10]

mDtD2 = N = mRtR For each entry (SPi, EPi) in the DP tradeoff and the rainbow method, logm bits for the starting point, very small bits for the ending point. are required.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 11 / 24

SLIDE 40

Previous Results : efficient use of storage [HM10]

mDtD2 = N = mRtR For each entry (SPi, EPi) in the DP tradeoff and the rainbow method, logm bits for the starting point, very small bits for the ending point. are required. Typically, logmR ≈ logmD + logtD and logtR ≈ logtD So, logmR logmD ≈ logmD + logtD logmD ≈ 2

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 11 / 24

SLIDE 41

Previous Results : efficient use of storage [HM10]

mDtD2 = N = mRtR For each entry (SPi, EPi) in the DP tradeoff and the rainbow method, logm bits for the starting point, very small bits for the ending point. are required. Typically, logmR ≈ logmD + logtD and logtR ≈ logtD So, logmR logmD ≈ logmD + logtD logmD ≈ 2 and MR = 2 MD

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 11 / 24

SLIDE 42

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 12 / 24

SLIDE 43

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. Most case : false alarm SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր

x F

→ y F → ◦ · · ·

recorded
J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 12 / 24

SLIDE 44

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. Most case : false alarm SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր

x F

→ y F → ◦ · · ·

recorded

pre-computed chain regeneration : SP2 = ◦

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 12 / 24

SLIDE 45

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. Most case : false alarm SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր

x F

→ y F → ◦ · · ·

recorded

pre-computed chain regeneration : SP2 = ◦ F → ◦

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 12 / 24

SLIDE 46

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. Most case : false alarm SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր

x F

→ y F → ◦ · · ·

recorded

pre-computed chain regeneration : SP2 = ◦ F → ◦ · · · ´ x

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 12 / 24

SLIDE 47

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. Most case : false alarm SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր

x F

→ y F → ◦ · · ·

recorded

pre-computed chain regeneration : SP2 = ◦ F → ◦ · · · ´ x F → ◦

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 12 / 24

SLIDE 48

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. Most case : false alarm SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր

x F

→ y F → ◦ · · ·

recorded

pre-computed chain regeneration : SP2 = ◦ F → ◦ · · · ´ x F → ◦ F → ◦

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 12 / 24

SLIDE 49

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. Most case : false alarm SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր

x F

→ y F → ◦ · · ·

recorded

pre-computed chain regeneration : SP2 = ◦ F → ◦ · · · ´ x F → ◦ F → ◦ · · · F →

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 12 / 24

SLIDE 50

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. Most case : false alarm SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր

x F

→ y F → ◦ · · ·

recorded

pre-computed chain regeneration : SP2 = ◦ F → ◦ · · · ´ x F → ◦ F → ◦ · · · F → •

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 12 / 24

SLIDE 51

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. Most case : false alarm SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · ·

F

→ • · · · F →

alarm!!

= EP2

ր

x F

→ y F → ◦ · · ·

recorded

pre-computed chain regeneration : SP2 = ◦ F → ◦ · · · ´ x F → ◦ F → ◦ · · · F → • (Recall : In the original DP tradeoff, SP2 = ◦ F → · · · ´ x F → ◦ F → ◦ · · · F → • · · · F →

= EP2 )
J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 12 / 24

SLIDE 52

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the original DP tradeoff] In the online phase,

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 13 / 24

SLIDE 53

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the original DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ 2 → ◦ 3 → ◦ 4 → · · ·

s

→ DP 2nd DP table y s+1 → ◦ s+2 → · · · 3rd DP table . . . t-th DP table

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 13 / 24

SLIDE 54

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the original DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ 2 → ◦ 3 → ◦ 4 → · · ·

s

→ DP 2nd DP table y s+1 → ◦ s+2 → · · · 3rd DP table . . . t-th DP table

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 13 / 24

SLIDE 55

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the original DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ 2 → ◦ 3 → ◦ 4 → · · ·

s

→ DP 2nd DP table y s+1 → ◦ s+2 → · · · 3rd DP table . . . t-th DP table

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 13 / 24

SLIDE 56

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the original DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ 2 → ◦ 3 → ◦ 4 → · · ·

s

→ DP 2nd DP table y s+1 → ◦ s+2 → · · · 3rd DP table . . . t-th DP table

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 13 / 24

SLIDE 57

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the parallel DP tradeoff] In the online phase,

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 14 / 24

SLIDE 58

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the parallel DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ t+1 → ◦ · · · 2nd DP table y

2

→ ◦ t+2 → ◦ · · · 3rd DP table y

3

→ ◦ t+3 → ◦ · · · . . . t-th DP table y

t

→ ◦ t+t → ◦ · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 14 / 24

SLIDE 59

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the parallel DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ t+1 → ◦ · · · 2nd DP table y

2

→ ◦ t+2 → ◦ · · · 3rd DP table y

3

→ ◦ t+3 → ◦ · · · . . . t-th DP table y

t

→ ◦ t+t → ◦ · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 14 / 24

SLIDE 60

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the parallel DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ t+1 → ◦ · · · 2nd DP table y

2

→ ◦ t+2 → ◦ · · · 3rd DP table y

3

→ ◦ t+3 → ◦ · · · . . . t-th DP table y

t

→ ◦ t+t → ◦ · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 14 / 24

SLIDE 61

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the parallel DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ t+1 → ◦ · · · 2nd DP table y

2

→ ◦ t+2 → ◦ · · · 3rd DP table y

3

→ ◦ t+3 → ◦ · · · . . . t-th DP table y

t

→ ◦ t+t → ◦ · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 14 / 24

SLIDE 62

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the parallel DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ t+1 → ◦ · · · 2nd DP table y

2

→ ◦ t+2 → ◦ · · · 3rd DP table y

3

→ ◦ t+3 → ◦ · · · . . . t-th DP table y

t

→ ◦ t+t → ◦ · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 14 / 24

SLIDE 63

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the parallel DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ t+1 → ◦ · · · 2nd DP table y

2

→ ◦ t+2 → ◦ · · · 3rd DP table y

3

→ ◦ t+3 → ◦ · · · . . . t-th DP table y

t

→ ◦ t+t → ◦ · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 14 / 24

SLIDE 64

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the parallel DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ t+1 → ◦ · · · 2nd DP table y

2

→ ◦ t+2 → ◦ · · · 3rd DP table y

3

→ ◦ t+3 → ◦ · · · . . . t-th DP table y

t

→ ◦ t+t → ◦ · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 14 / 24

SLIDE 65

The Parallel DP Tradeoff(The pD Tradeoff)

Variant of the DP tradeoff (Hoch, Shamir 09), A full record of the online chain is maintained during the online phase, The DP tables processed in parallel, rather than serially. [the parallel DP tradeoff] In the online phase, 1st DP table y

1

→ ◦ t+1 → ◦ · · · 2nd DP table y

2

→ ◦ t+2 → ◦ · · · 3rd DP table y

3

→ ◦ t+3 → ◦ · · · . . . t-th DP table y

t

→ ◦ t+t → ◦ · · ·

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 14 / 24

SLIDE 66

The pD Tradeoff : the online time complexity

mt2 = DmscN

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 15 / 24

SLIDE 67

The pD Tradeoff : the online time complexity

mt2 = DmscN The online chain creation of the pD Tradeoff require t2 Dps DmscDcr invocations of F.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 15 / 24

SLIDE 68

The pD Tradeoff : the online time complexity

mt2 = DmscN The online chain creation of the pD Tradeoff require t2 Dps DmscDcr invocations of F. The number of iterations required by the pD tradeoff in dealing with alarms is t2 ln(1 − Dps) Dcr 1 (1 − Dps)1−u ln u du.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 15 / 24

SLIDE 69

The pD Tradeoff : the tradeoff curve

mt2 = DmscN T=the total online time complexity M= storage size

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 16 / 24

SLIDE 70

The pD Tradeoff : the tradeoff curve

mt2 = DmscN T=the total online time complexity M= storage size The time memory tradeoff curve for the pD tradeoff is TM2 = pDtcN2, where pDtc = ln(1 − Dps) Dps 1 (1 − Dps)1−u ln u du+ 1 Dmsc 1 D3

cr

Dps{ln(1−Dps)}2.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 16 / 24

SLIDE 71

The pD Tradeoff : the tradeoff curve

mt2 = DmscN T=the total online time complexity M= storage size The time memory tradeoff curve for the pD tradeoff is TM2 = pDtcN2, where pDtc = ln(1 − Dps) Dps 1 (1 − Dps)1−u ln u du+ 1 Dmsc 1 D3

cr

Dps{ln(1−Dps)}2. Recall : In the original DP tradeoff, Dtc = ( 2 + 1 Dmsc ) 1 D3

cr

Dps{ln(1 − Dps)}2.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 16 / 24

SLIDE 72

pD versus DP

Since ln(1 − Dps) Dps 1 (1 − Dps)1−u ln u du < 1 < 2, DP < pD the pD tradeoff will outperform the original DP tradeoff.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 17 / 24

SLIDE 73

pD versus Rainbow

Xtc = TM2

N2

is a measure of how efficiently the algorithm balances

nline time against storage requirements.

◮ A smaller Xtc implies a more efficient tradeoff.

However, a better tradeoff efficiency usually requires a higher pre-computation cost and is not always desirable in practice. ⇒ We have to consider both Xtc and Xpc for comparison.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 18 / 24

SLIDE 74

pD versus Rainbow

Xtc = TM2

N2

is a measure of how efficiently the algorithm balances

nline time against storage requirements.

◮ A smaller Xtc implies a more efficient tradeoff.

However, a better tradeoff efficiency usually requires a higher pre-computation cost and is not always desirable in practice. ⇒ We have to consider both Xtc and Xpc for comparison. In a fair manner, compare Dtc with 4Rtc, since MR = 2MD.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 18 / 24

SLIDE 75

pD versus Rainbow

The pD tradeoff pDtc = ln(1 − Dps) Dps 1 (1 − Dps)1−u ln u du + 1 Dmsc 1 D3

cr

Dps{ln(1 − Dps)}2 The rainbow method[HM10] Rtc =

l3 (2l+1)(2l+2)(2l+3)

{(2l − 1) + (2l + 1)Rmsc}(2 + Rmsc)2

−4{(2l − 1) + l(2l + 3)Rmsc}

2

2+Rmsc

2l ,where Rps = 1 −

2

2 + Rmsc 2l , Dps = 1 − e−DcrDpc.

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 19 / 24

SLIDE 76

pD versus Rainbow

Dpc : pDtc , Rpc : 4Rtc

Figure: the pD(line) and the rainbow(bullet)

25

0.30 0.32 0.34 0.36 0.38 0.0 0.1 0.2 0.3 0.4 0.5 0.6

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 20 / 24

SLIDE 77

pD versus Rainbow

Dpc : pDtc , Rpc : 4Rtc

Figure: the pD(line) and the rainbow(bullet)

50

0.70 0.75 0.80 0.85 0.90 1 2 3 4 5 6

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 21 / 24

SLIDE 78

pD versus Rainbow

Dpc : pDtc , Rpc : 4Rtc

Figure: the pD(line) and the rainbow(bullet)

90

2.4 2.6 2.8 3.0 3.2 20 40 60 80 100 120

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 22 / 24

SLIDE 79

pD versus Rainbow

Dpc : pDtc , Rpc : 4Rtc

Figure: the pD(line) and the rainbow(bullet)

99

4.5 5.0 5.5 6.0 6.5 100 200 300 400 500

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 23 / 24

SLIDE 80

Conclusion

There are two added conditions in the pD in comparison with the DP.

◮ online chain record ◮ parallel processing

⇒ In the online phase, cost for resolving alarms is reduced more than half. The pD tradeoff is not likely to be preferable over the rainbow method under most situations. The only exception is when the success rate requirement is very low.

◮ example. multi-target time memory tradeoff

J. Hong, G.W. Lee, D. Ma (SNU)

Analysis of the pD tradeoff 13/12/2011 24 / 24