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Generating k-independent variables in constant time

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Generating k-independent variables in constant time

Tobias Christiani, Rasmus Pagh

IT University of Copenhagen

Generating k-independent variables in constant time

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Introduction 11010100 01101101 00011011 11111100 10010101 11010001 11100110 Generator Random seed Pseudorandom output sequence

Generating k-independent variables in constant time

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k-independent variables Generator output:

• Sequence of k-independent variables over a finite field F

Generating k-independent variables in constant time

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k-independent variables Generator output:

• Sequence of k-independent variables over a finite field F

Y1 Y2 Y3 Yp · · · Example field:

• The set of integers modulo a prime Fp

Generating k-independent variables in constant time

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k-independent variables Generator output:

• Sequence of k-independent variables over a finite field F

Y1 Y2 Y3 Yp · · · Pr[Yi1 = y1, Yi2 = y2, . . . , Yik = yk] = pk

• k-independence:

– Yi uniformly distributed over Fp – The variables at every set of k distinct positions are independent Example field:

• The set of integers modulo a prime Fp

Generating k-independent variables in constant time

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k-independent variables Generator output:

• Sequence of k-independent variables over a finite field F

Y1 Y2 Y3 Yp · · · Pr[Yi1 = y1, Yi2 = y2, . . . , Yik = yk] = pk

• k-independence:

– Yi uniformly distributed over Fp – The variables at every set of k distinct positions are independent Example field:

• The set of integers modulo a prime Fp
• Many properties of full randomness for k ⌧ p

Generating k-independent variables in constant time

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Model and problem Model:

• Word-RAM model

– Elements of Fp fit in a single word – Constant time addition and multiplication in Fp

Generating k-independent variables in constant time

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Model and problem Problem:

• Construct a data structure that outputs a k-independent sequence
• ver Fp, one variable at the time

– Low time complexity of generating next element – Space usage and initialization time close to k Model:

• Word-RAM model

– Elements of Fp fit in a single word – Constant time addition and multiplication in Fp

Generating k-independent variables in constant time

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Result Explicit data structure for generating the next element of a k-independent sequence in worst case constant time using space and initialization time k poly log k

Main result

Generating k-independent variables in constant time

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Result Explicit data structure for generating the next element of a k-independent sequence in worst case constant time using space and initialization time k poly log k

Main result

• Constant time generation for every choice of k almost as large as p

Generating k-independent variables in constant time

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Result Explicit data structure for generating the next element of a k-independent sequence in worst case constant time using space and initialization time k poly log k

• The generator always outputs a k-independent sequence

Main result

• Constant time generation for every choice of k almost as large as p

Generating k-independent variables in constant time

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Result Explicit data structure for generating the next element of a k-independent sequence in worst case constant time using space and initialization time k poly log k

• The generator always outputs a k-independent sequence

Main result

• Constant time generation for every choice of k almost as large as p
• Space in bits: (k poly log k) · log p

– Logarithmic dependence on length of output sequence

Generating k-independent variables in constant time

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Result Explicit data structure for generating the next element of a k-independent sequence in worst case constant time using space and initialization time k poly log k

• The generator always outputs a k-independent sequence

Main result

• Constant time generation for every choice of k almost as large as p
• Space in bits: (k poly log k) · log p

– Logarithmic dependence on length of output sequence Reference Space Generation time Multipoint evaluation [GG’13] O(k log k) O(log2 k log log k) Expander hashing [Siegel’89] O(pε), ε > 0 O(1) Our generator O(k poly log k) O(1) COMPARISON WITH PREVIOUS RESULTS

Generating k-independent variables in constant time

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k-independent hash functions

• Data structure for random access to a k-independent sequence

Generating k-independent variables in constant time

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k-independent hash functions

• Data structure for random access to a k-independent sequence

Polynomial hash functions [Joffe’74, CW’81]

• Choose k random elements from Fp and use them as coefficients of

a polynomial h(x) = a0 + a1x + a2x2 + · · · + ak1xk1 mod p

Generating k-independent variables in constant time

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k-independent hash functions

• Data structure for random access to a k-independent sequence

h(0) h(1) h(2) h(p 1) · · · Polynomial hash functions [Joffe’74, CW’81]

• Choose k random elements from Fp and use them as coefficients of

a polynomial h(x) = a0 + a1x + a2x2 + · · · + ak1xk1 mod p

• Polynomial hash function as a generator

– Space k to store coefficients – Evaluation time O(k)

• Space usage near-optimal with respect to sequence length

Generating k-independent variables in constant time

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k-independent hashing: lower bound

• What is the best time-space tradeoff we can hope for using hashing?

Generating k-independent variables in constant time

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k-independent hashing: lower bound

• Cell probe lower bound [Siegel’89]

Time t < k ) Space at least p1/t words

k-independent hash function h : Fp ! Fp

• What is the best time-space tradeoff we can hope for using hashing?

Generating k-independent variables in constant time

7

k-independent hashing: lower bound

• Cell probe lower bound [Siegel’89]

Time t < k ) Space at least p1/t words

• Constructions in the word-RAM model still far from lower bound

k-independent hash function h : Fp ! Fp

• What is the best time-space tradeoff we can hope for using hashing?

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound
• Sequence of local hash functions evaluated over small domains

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

? . . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . .

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

? . . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . .

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

? . . .

Y1 Y2 Yp

. . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . . . . .

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

?

Hash function

? . . .

Y1 Y2 Yp

. . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . . . . .

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

?

Hash function

? . . .

Y1 Y2 Yp

. . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . . . . .

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

?

Hash function

?

Hash function

? . . .

Y1 Y2 Yp

. . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . . . . .

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

?

Hash function

?

Hash function

? . . .

Y1 Y2 Yp

. . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . . . . .

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

?

Hash function

?

Hash function

? . . .

Y1 Y2 Yp

. . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . . . . .

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

?

Hash function

?

Hash function

? . . .

Y1 Y2 Yp

. . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . . . . .

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

?

Hash function

? . . .

Y1 Y2 Yp

. . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . . . . . Hash function

? . . .

Generating k-independent variables in constant time

8

k-generators

• Data structure for sequential access to a k-independent sequence
• Circumvent hashing lower bound

Hash function

?

Hash function

? . . .

Y1 Y2 Yp

. . .

Y1 Y2 Yp

• Sequence of local hash functions evaluated over small domains

. . . . . . Hash function

? . . .

Generating k-independent variables in constant time

9

Construction overview Initialization and evaluation of local hash functions:

Generating k-independent variables in constant time

9

Construction overview

. . . . . . Yi+1 Yi+2

. . .

Yi+b

• 1. Multipoint evaluation
• k-independent polynomial hash function h(x)

– Evaluate in k points using time k poly log k – poly log k time per value generated Initialization and evaluation of local hash functions:

Generating k-independent variables in constant time

9

Construction overview

. . . . . . Yi+1 Yi+2

. . .

Yi+b . . .

P

• 1. Multipoint evaluation
• k-independent polynomial hash function h(x)

– Evaluate in k points using time k poly log k – poly log k time per value generated

• 2. Expander hashing
• Unbalanced bipartite expander, degree O(1)

– Generate k poly log k values from k values – Output is Ω(k)-independent Initialization and evaluation of local hash functions:

Generating k-independent variables in constant time

10

Multipoint evaluation

• Assume that Fp contains a primitive kth root of unity ωk

ωk : ωk

k = 1

ωi

k 6= 1 for i = 1, . . . , k 1

Generating k-independent variables in constant time

10

Multipoint evaluation

• Assume that Fp contains a primitive kth root of unity ωk
• We can compute h(1), h(ωk), . . . , h(ωk1

k

) in O(k log k) operations ωk : ωk

k = 1

ωi

k 6= 1 for i = 1, . . . , k 1

h(x) = a0 + a1x + a2x2 + · · · + ak1xk1 mod p

Generating k-independent variables in constant time

10

Multipoint evaluation F⇤

p = {1, ω, . . . , ωk, ωωk, . . . , ωk1 k

, ωωk1

k

, . . . , ωp2}

• Assume that Fp contains a primitive kth root of unity ωk
• We can compute h(1), h(ωk), . . . , h(ωk1

k

) in O(k log k) operations ωk : ωk

k = 1

ωi

k 6= 1 for i = 1, . . . , k 1

h(x) = a0 + a1x + a2x2 + · · · + ak1xk1 mod p

• Let ω be a primitive element that generates F⇤

p = Fp {0}

Generating k-independent variables in constant time

10

Multipoint evaluation F⇤

p = {1, ω, . . . , ωk, ωωk, . . . , ωk1 k

, ωωk1

k

, . . . , ωp2}

• Assume that Fp contains a primitive kth root of unity ωk
• We can compute h(1), h(ωk), . . . , h(ωk1

k

) in O(k log k) operations ωk : ωk

k = 1

ωi

k 6= 1 for i = 1, . . . , k 1

h(x) = a0 + a1x + a2x2 + · · · + ak1xk1 mod p

• Let ω be a primitive element that generates F⇤

p = Fp {0}

h(1) h(ωk) h(ωk1

k

) . . .

Generating k-independent variables in constant time

10

Multipoint evaluation F⇤

p = {1, ω, . . . , ωk, ωωk, . . . , ωk1 k

, ωωk1

k

, . . . , ωp2}

• Assume that Fp contains a primitive kth root of unity ωk
• We can compute h(1), h(ωk), . . . , h(ωk1

k

) in O(k log k) operations ωk : ωk

k = 1

ωi

k 6= 1 for i = 1, . . . , k 1

h(x) = a0 + a1x + a2x2 + · · · + ak1xk1 mod p

• Let ω be a primitive element that generates F⇤

p = Fp {0}

h(ω) h(ωωk) h(ωωk1

k

) . . .

Generating k-independent variables in constant time

10

Multipoint evaluation F⇤

p = {1, ω, . . . , ωk, ωωk, . . . , ωk1 k

, ωωk1

k

, . . . , ωp2}

• Assume that Fp contains a primitive kth root of unity ωk
• We can compute h(1), h(ωk), . . . , h(ωk1

k

) in O(k log k) operations ωk : ωk

k = 1

ωi

k 6= 1 for i = 1, . . . , k 1

h(x) = a0 + a1x + a2x2 + · · · + ak1xk1 mod p

• Let ω be a primitive element that generates F⇤

p = Fp {0}

• Construct h0(x) = Pk1

i=0 aiωixi compute h0(1), h0(ωk), . . . , h0(ωk1 k

) h(ω) h(ωωk) h(ωωk1

k

) . . .

Generating k-independent variables in constant time

10

Multipoint evaluation F⇤

p = {1, ω, . . . , ωk, ωωk, . . . , ωk1 k

, ωωk1

k

, . . . , ωp2}

• Assume that Fp contains a primitive kth root of unity ωk
• We can compute h(1), h(ωk), . . . , h(ωk1

k

) in O(k log k) operations ωk : ωk

k = 1

ωi

k 6= 1 for i = 1, . . . , k 1

h(x) = a0 + a1x + a2x2 + · · · + ak1xk1 mod p

• Let ω be a primitive element that generates F⇤

p = Fp {0}

• General multipoint evaluation: Time O(k log3 k) [GG’13]
• Construct h0(x) = Pk1

i=0 aiωixi compute h0(1), h0(ωk), . . . , h0(ωk1 k

) h(ω) h(ωωk) h(ωωk1

k

) . . .

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ

L R . . . d

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R . . . d

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3
• Expander hashing [Siegel’89]

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Expander hashing [Siegel’89]

– Let f : R ! Fp be dk-independent

• Example for |S| = 3

f(v1) f(v2) f(v3) f(v4) f(v5)

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3

f(v1) f(v2) f(v3) f(v4) f(v5) g(x5)

g(x5) = f(v2) + f(v4) + f(v5) mod p

• Expander hashing [Siegel’89]

– Let f : R ! Fp be dk-independent – g(x) = P

v2Γ({x}) f(v) is

k-independent

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3
• Expander hashing [Siegel’89]

– Let f : R ! Fp be dk-independent – g(x) = P

v2Γ({x}) f(v) is

k-independent

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3
• Expander hashing [Siegel’89]

– Let f : R ! Fp be dk-independent – g(x) = P

v2Γ({x}) f(v) is

k-independent

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3
• Expander hashing [Siegel’89]

– Let f : R ! Fp be dk-independent – g(x) = P

v2Γ({x}) f(v) is

k-independent

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3
• Expander hashing [Siegel’89]

– Let f : R ! Fp be dk-independent – g(x) = P

v2Γ({x}) f(v) is

k-independent

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3
• Expander hashing [Siegel’89]

– Let f : R ! Fp be dk-independent – g(x) = P

v2Γ({x}) f(v) is

k-independent

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3
• Expander hashing [Siegel’89]

– Let f : R ! Fp be dk-independent – g(x) = P

v2Γ({x}) f(v) is

k-independent

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3
• Expander hashing [Siegel’89]

– Let f : R ! Fp be dk-independent – g(x) = P

v2Γ({x}) f(v) is

k-independent

Generating k-independent variables in constant time

11

Expander hashing

• Left d-regular bipartite graph Γ
• Γ is k-unique if every S ✓ L with

1  |S|  k has a unique neighbor L R

• Example for |S| = 3

Claim:

• There exists k-unique Γ with

– |R| = O(k log k) – |L| = (poly log k) · |R| – d = O(1)

• Expander hashing [Siegel’89]

– Let f : R ! Fp be dk-independent – g(x) = P

v2Γ({x}) f(v) is

k-independent

Generating k-independent variables in constant time

12

Example construction Initialization and evaluation of local hash functions:

Generating k-independent variables in constant time

12

Example construction

. . . . . . Yi+1 Yi+2

. . .

Yi+|V |

• 1. Multipoint evaluation
• dk-independent polynomial hash function h(x)

– Generate |R| = Ω(k log k) values using time O(k log4 k) Initialization and evaluation of local hash functions:

Generating k-independent variables in constant time

12

Example construction

. . . . . .

• 1. Multipoint evaluation
• dk-independent polynomial hash function h(x)

– Generate |R| = Ω(k log k) values using time O(k log4 k) Initialization and evaluation of local hash functions: R

. . .

Generating k-independent variables in constant time

12

Example construction

. . . . . . . . .

• 1. Multipoint evaluation
• dk-independent polynomial hash function h(x)

– Generate |R| = Ω(k log k) values using time O(k log4 k)

• 2. Expander hashing
• Construct random Γ with:

– Constant degree d = 8 – |L| = (log4 k)|R| Initialization and evaluation of local hash functions: R L P

. . .

Generating k-independent variables in constant time

12

Example construction

. . . . . . . . .

• 1. Multipoint evaluation
• dk-independent polynomial hash function h(x)

– Generate |R| = Ω(k log k) values using time O(k log4 k)

• 2. Expander hashing
• Construct random Γ with:

– Constant degree d = 8 – |L| = (log4 k)|R| Initialization and evaluation of local hash functions: R L

• Generation procedure

P

. . .

Generating k-independent variables in constant time

12

Example construction

. . . . . . . . .

• 1. Multipoint evaluation
• dk-independent polynomial hash function h(x)

– Generate |R| = Ω(k log k) values using time O(k log4 k)

• 2. Expander hashing
• Construct random Γ with:

– Constant degree d = 8 – |L| = (log4 k)|R| Initialization and evaluation of local hash functions: R L

• Generation procedure

P

. . .

Generating k-independent variables in constant time

12

Example construction

. . . . . . . . .

• 1. Multipoint evaluation
• dk-independent polynomial hash function h(x)

– Generate |R| = Ω(k log k) values using time O(k log4 k)

• 2. Expander hashing
• Construct random Γ with:

– Constant degree d = 8 – |L| = (log4 k)|R| Initialization and evaluation of local hash functions: R L

• Generation procedure

P

. . .

Generating k-independent variables in constant time

12

Example construction

. . . . . . . . .

• 1. Multipoint evaluation
• dk-independent polynomial hash function h(x)

– Generate |R| = Ω(k log k) values using time O(k log4 k)

• 2. Expander hashing
• Construct random Γ with:

– Constant degree d = 8 – |L| = (log4 k)|R| Initialization and evaluation of local hash functions: R L

• Generation procedure

P

. . .

Generating k-independent variables in constant time

12

Example construction

. . . . . . . . .

• 1. Multipoint evaluation
• dk-independent polynomial hash function h(x)

– Generate |R| = Ω(k log k) values using time O(k log4 k)

• 2. Expander hashing
• Construct random Γ with:

– Constant degree d = 8 – |L| = (log4 k)|R| Initialization and evaluation of local hash functions: R L

• Generation procedure

P

. . . . . .

Generating k-independent variables in constant time

13

Explicit k-generator

. . . . . . Yi+1 Yi+2

. . .

Yi+b . . . . . .

• No known explicit k-unique Γ

with constant degree and poly log k imbalance . . . P

Generating k-independent variables in constant time

13

Explicit k-generator

. . . . . . Yi+1 Yi+2

. . .

Yi+b . . . . . .

• No known explicit k-unique Γ

with constant degree and poly log k imbalance

• Explicit k-unique Γ [CRVW’02]

– |L| = c|R| for constant c > 1 – |R| = O(k) – d = O(1) . . . P

Generating k-independent variables in constant time

13

Explicit k-generator

. . . . . . Yi+1 Yi+2

. . .

Yi+b . . . . . .

• No known explicit k-unique Γ

with constant degree and poly log k imbalance

• Explicit k-unique Γ [CRVW’02]

– |L| = c|R| for constant c > 1 – |R| = O(k) – d = O(1)

• Cascading composition of

O(log log k) expanders suffices for constant time generation . . . . . . P P . . . . . . P

Generating k-independent variables in constant time

13

Explicit k-generator

• No known explicit k-unique Γ

with constant degree and poly log k imbalance

• Explicit k-unique Γ [CRVW’02]

– |L| = c|R| for constant c > 1 – |R| = O(k) – d = O(1)

• Cascading composition of

O(log log k) expanders suffices for constant time generation . . . . . . P

• Generation procedure

k poly log k-ind. . . . . . . P . . .

Generating k-independent variables in constant time

13

Explicit k-generator

• No known explicit k-unique Γ

with constant degree and poly log k imbalance

• Explicit k-unique Γ [CRVW’02]

– |L| = c|R| for constant c > 1 – |R| = O(k) – d = O(1)

• Cascading composition of

O(log log k) expanders suffices for constant time generation . . . . . . P

• Generation procedure

k poly log k-ind. . . . . . . P . . .

Generating k-independent variables in constant time

13

Explicit k-generator

• No known explicit k-unique Γ

with constant degree and poly log k imbalance

• Explicit k-unique Γ [CRVW’02]

– |L| = c|R| for constant c > 1 – |R| = O(k) – d = O(1)

• Cascading composition of

O(log log k) expanders suffices for constant time generation . . . . . . P

• Generation procedure

k poly log k-ind. . . . . . . dk-ind. P . . .

Generating k-independent variables in constant time

13

Explicit k-generator

• No known explicit k-unique Γ

with constant degree and poly log k imbalance

• Explicit k-unique Γ [CRVW’02]

– |L| = c|R| for constant c > 1 – |R| = O(k) – d = O(1)

• Cascading composition of

O(log log k) expanders suffices for constant time generation . . . . . . P

• Generation procedure

k poly log k-ind. . . . . . . k-ind. dk-ind. P . . .

Generating k-independent variables in constant time

14

Experiments k Polynomial hash fct. Multipoint [GM’10] Our generator 25 5.650 4.115 66.667 210 0.169 2.227 40.000 215 0.005 1.567 14.493 220 0.000 1.157 5.714 MILLIONS OF 64-BIT VALUES GENERATED PER SECOND,

RANDOMIZED CONSTRUCTION

Generating k-independent variables in constant time

14

Experiments k Polynomial hash fct. Multipoint [GM’10] Our generator 25 5.650 4.115 66.667 210 0.169 2.227 40.000 215 0.005 1.567 14.493 220 0.000 1.157 5.714 MILLIONS OF 64-BIT VALUES GENERATED PER SECOND,

RANDOMIZED CONSTRUCTION

• Millions of highly independent words per second

– Potential application to Monte Carlo experiments

Generating k-independent variables in constant time

14

Experiments k Polynomial hash fct. Multipoint [GM’10] Our generator 25 5.650 4.115 66.667 210 0.169 2.227 40.000 215 0.005 1.567 14.493 220 0.000 1.157 5.714 MILLIONS OF 64-BIT VALUES GENERATED PER SECOND,

RANDOMIZED CONSTRUCTION

• Gao-Mateer’s additive FFT + Haswell support for F264

– Evaluate a degree 220 polynomial in 220 points in < 1 second

• Millions of highly independent words per second

– Potential application to Monte Carlo experiments

Generating k-independent variables in constant time

14

Experiments k Polynomial hash fct. Multipoint [GM’10] Our generator 25 5.650 4.115 66.667 210 0.169 2.227 40.000 215 0.005 1.567 14.493 220 0.000 1.157 5.714 MILLIONS OF 64-BIT VALUES GENERATED PER SECOND,

RANDOMIZED CONSTRUCTION

• Gao-Mateer’s additive FFT + Haswell support for F264

– Evaluate a degree 220 polynomial in 220 points in < 1 second

• Millions of highly independent words per second

– Potential application to Monte Carlo experiments

• Slowdown of our generator as k increases is due to cache effects

Generating k-independent variables in constant time

15

Summary Result:

• Explicit constant time generation using space k poly log k

Generating k-independent variables in constant time

15

Summary Result:

• Explicit constant time generation using space k poly log k

Technique:

• Combination of multipoint evaluation and a cascading composition
• f explicit expanders

Generating k-independent variables in constant time

15

Summary Result:

• Explicit constant time generation using space k poly log k

Technique:

• Combination of multipoint evaluation and a cascading composition
• f explicit expanders

Open questions:

• Explicit expanders with

– |R| = O(k poly log k) – |L| (poly log k)|R| – d = O(1)

Generating k-independent variables in constant time

15

Summary Result:

• Explicit constant time generation using space k poly log k

Technique:

• Combination of multipoint evaluation and a cascading composition
• f explicit expanders

Open questions:

• Explicit expanders with

– |R| = O(k poly log k) – |L| (poly log k)|R| – d = O(1)

• Constant time generation with O(k) space

Generating k-independent variables in constant time

16

Multipoint evaluation

• A k-independent polynomial hash function h(x)
• Faster algorithms exist if:

– x1, x2, . . . , xk lie in arithmetic/geometric progression – F contains a highly composite kth root of unity

• We can evaluate h(x) in k arbitrary points x1, x2, . . . , xk using

O(k log3 k) operations p1(x) = Qk/2

i=1(x xi)

p2(x) = Qk

i=(k/2)+1(x xi)

r1(x) = h(x) mod p1(x) r2(x) = h(x) mod p2(x) h(x) = q(x)p(x) + r(x) r1(xi) = h(xi) for 1  i  k/2

Generating k-independent variables in constant time

17

New results

• We only require sequential evaluation of Γ
• Constant time generator for k-unique

Γ : L ! Rd using space O(k logε k)

. . . . . . Yi+1 Yi+2

. . .

Yi+b . . .

• Tabulating Γ requires at least

|R| = Ω(k log2+ε k) words

• New recursive constructions of k-independent

hash functions that come close to Siegel’s lower bound