computationally data-independent memory hard functions · 2020. 7. 9. · 1 4 5,7 6 8 10 memory...
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Computationally Data-Independent Memory Hard
Functions
MOHAMMAD HASSAN AMERI
JEREMIAH BLOCKI
SAMSON ZHOU
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MHF is a function that has high cumulative memory complexity (𝑐𝑐) when computed by any (parallel) algorithm
Memory Hard Functions [ABMW05, Percival09]
Time
Memory in Use
𝑐𝑐
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MHF is a function that has high cumulative memory complexity (𝑐𝑐) when computed by any (parallel) algorithm
Amortization: Metric that scales to the memory cost of computing function on 𝑚 inputs [AS15]
Application: Password hashing (want to ensure cost of checking millions/billions of password guesses is prohibitively high for attacker)
Memory Hard Functions [ABMW05, Percival09]
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iMHFs
In a data-independent memory hard functions (iMHFs) 𝑓, the memory access pattern for the evaluation algorithm of the MHF is static (information theoretically independent of the input)
3
2
9
1
4 5,7
6
8
10
memory access pattern
for 𝑓(𝑥)
RAM
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iMHFs
In a data-independent memory hard functions (iMHFs) 𝑓, the memory access pattern for the evaluation algorithm of the MHF is static (information theoretically independent of the input)
3, 3
2, 2
9, 9
1, 1
4, 4 5, 5, 7, 7
6, 6
8, 8
10, 10
same memory access pattern
for 𝑓(𝑦)
RAM
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dMHFs
In a data-dependent memory hard functions (dMHFs) 𝑓, the memory access pattern is dynamic and depends on the input
3
2
9
1
4 5,7
6
8
10
memory access pattern
for 𝑓(𝑥)
RAM
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dMHFs
In a data-dependent memory hard functions (dMHFs) 𝑓, the memory access pattern is dynamic and depends on the input
differentmemory
access pattern for 𝑓(𝑦)
RAM
3 4
2 2
9
9 1
4, 8 5,7
6 3
6
1
8, 5
7
10, 10
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iMHFs vs. dMHFs
iMHFs: Resists side channel attacks, but 𝑐𝑐(𝑓) =
𝑂𝑁2 log log 𝑁
log 𝑁[AB16] (𝑁 is the sequential evaluation
algorithm runtime)
dMHFs : Exist constructions with 𝑐𝑐(𝑓) = Ω 𝑁2
[Percival09, ACP+17], but vulnerable to side channel attacks [Bernstein05]
optimal
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iMHFs vs. dMHFs
iMHFs: Resists side channel attacks, but 𝑐𝑐(𝑓) =
𝑂𝑁2 log log 𝑁
log 𝑁[AB16] (𝑁 is the sequential evaluation
algorithm runtime)
dMHFs : Exist constructions with 𝑐𝑐(𝑓) = Ω 𝑁2
[Percival09, ACP+17], but vulnerable to side channel attacks [Bernstein05]
optimal
Can we design functions 𝑓 with 𝑐𝑐(𝑓) = Ω 𝑁2 AND resist side channel
attacks?
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ciMHFs
Intuition: Attacker cannot distinguish between memory access patterns for inputs 𝑥 and 𝑦
Resists side channel attacksCan we get maximally hard ciMHF constructions?
Naïve approach: ORAM hides memory access patterns but incurs Ω(log𝑁) overhead time, so the new runtime is Ω(𝑁 log𝑁) and
does not achieve effective 𝑐𝑐(𝑓) = Ω 𝑁2
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Our Contributions (I)
We construct a family of “𝑘-restricted dynamic graphs” where 𝑘 =𝑜 𝑁𝜖 for any constant 0 < 𝜖 < 1 with 𝑐𝑐(𝑓𝐺,𝐻) = Ω 𝑁2
For each 𝐺 that is “amenable to shuffling”, there exists a computationally data-independent sequential evaluation algorithm computing an MHF based on the graph 𝐺 that runs in time 𝑂 𝑁
There exists a family of ciMHFs 𝐺 with 𝑐𝑐(𝑓𝐺,𝐻) = Ω 𝑁2
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Our Contributions (II)
Let 𝐺 be any family of 𝑘-restricted dynamic graphs with constant indegree. Then
𝑐𝑐(𝑓𝐺,𝐻) = 𝑂𝑁2
log log 𝑁+ 𝑁
2−1
2 log log 𝑁 𝑘1−
1
log log 𝑁
Thus for 𝑘 = 𝑜 𝑁1/ log log 𝑁 , we have 𝑐𝑐(𝑓𝐺,𝐻) = 𝑜 𝑁2
Results essentially characterize the spectrum of 𝑘-restricted
dynamic graphs, i.e., 𝑘 = 𝑜 𝑁𝜖 and 𝑘 = 𝑜 𝑁1/ log log 𝑁
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Assumption
Evaluation algorithm has (small) data structure that attacker cannot see
E.g., tiered memory architecture, where attacker can see access patterns to RAM but not cache
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Memory Hard Functions [ABMW05, Percival09]
ℓ1 ℓ2 ℓ3 ℓ5
ℓ4
pwd
𝑓𝐺,𝐻
hash
ℓ1 = 𝐻(𝑝𝑤𝑑), ℓ2 = 𝐻(ℓ1), ℓ3 = 𝐻 ℓ1, ℓ2 , ℓ4 = 𝐻(ℓ2), ℓ5 = 𝐻 ℓ3, ℓ4
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Dynamic and Static Graphs
ℓ1 ℓ2 ℓ3 ℓ4 ℓ5dMHF
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Dynamic and Static Graphs
ℓ1 ℓ2 ℓ3 ℓ4 ℓ5dMHF
ℓ6
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Dynamic and Static Graphs
ℓ1 ℓ2 ℓ3 ℓ4 ℓ5dMHF
ℓ6 ℓ7
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Dynamic and Static Graphs
ℓ1 ℓ2 ℓ3 ℓ4 ℓ5dMHF
ℓ6 ℓ7 ℓ8
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Dynamic and Static Graphs
ℓ1 ℓ2 ℓ3 ℓ4 ℓ5dMHF
ℓ6 ℓ7 ℓ8 ℓ9
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Dynamic and Static Graphs
ℓ1 ℓ2 ℓ3 ℓ4 ℓ5dMHF
ℓ6 ℓ7 ℓ8 ℓ9 ℓ10
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Dynamic and Static Graphs
ℓ1 ℓ2 ℓ3 ℓ4 ℓ5iMHF
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𝑘-Restricted Dynamic Graphs
ℓ1 ℓ2 ℓ3 ℓ4 ℓ5
ℓ1 ℓ2 ℓ3 ℓ4 ℓ5
dMHF
iMHF
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Maximally Hard 𝑘-Restricted Dynamic Graphs
Family of 𝑘-restricted dynamic graphs where 𝑘 = 𝑜 𝑁𝜖 for anyconstant 0 < 𝜖 < 1 with 𝑐𝑐(𝑓𝐺,𝐻) = Ω 𝑁2
… 𝑁
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Maximally Hard 𝑘-Restricted Dynamic Graphs
Family of 𝑘-restricted dynamic graphs where 𝑘 = 𝑜 𝑁𝜖 for anyconstant 0 < 𝜖 < 1 with 𝑐𝑐(𝑓𝐺,𝐻) = Ω 𝑁2
… 2𝑁
… 𝑁
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Maximally Hard 𝑘-Restricted Dynamic Graphs
Family of 𝑘-restricted dynamic graphs where 𝑘 = 𝑜 𝑁𝜖 for anyconstant 0 < 𝜖 < 1 with 𝑐𝑐(𝑓𝐺,𝐻) = Ω 𝑁2
… 2𝑁
…1 2 … 𝑘
Block 1
…
Block 2
…
Block 𝑁/𝑘
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Maximally Hard 𝑘-Restricted Dynamic Graphs
Family of 𝑘-restricted dynamic graphs where 𝑘 = 𝑜 𝑁𝜖 for anyconstant 0 < 𝜖 < 1 with 𝑐𝑐(𝑓𝐺,𝐻) = Ω 𝑁2
… 2𝑁
…1 2 … 𝑘
Block 1
…
Block 2
…
Block 𝑁/𝑘
![Page 27: Computationally Data-Independent Memory Hard Functions · 2020. 7. 9. · 1 4 5,7 6 8 10 memory access pattern for 𝑓( ) RAM. iMHFs In a data-independent memory hard functions (iMHFs)](https://reader035.vdocuments.us/reader035/viewer/2022071107/5fe1c6567d38506586271034/html5/thumbnails/27.jpg)
Maximally Hard 𝑘-Restricted Dynamic Graphs
To compute labels 𝑁 + 1 to 2𝑁, attacker should either (1) keep labels on nodes 1 to 𝑁 throughout or (2) recompute labels of nodes 1 to 𝑁when necessary
… 2𝑁
…1 2 … 𝑘
Block 1
…
Block 2
…
Block 𝑁/𝑘
![Page 28: Computationally Data-Independent Memory Hard Functions · 2020. 7. 9. · 1 4 5,7 6 8 10 memory access pattern for 𝑓( ) RAM. iMHFs In a data-independent memory hard functions (iMHFs)](https://reader035.vdocuments.us/reader035/viewer/2022071107/5fe1c6567d38506586271034/html5/thumbnails/28.jpg)
Maximally Hard 𝑘-Restricted Dynamic Graphs
To compute labels 𝑁 + 1 to 2𝑁, attacker should either (1) keep labels on nodes 1 to 𝑁 throughout or (2) recompute labels of nodes 1 to 𝑁when necessary
Cost is 𝑁 × 𝑁 = Ω(𝑁2)
… 2𝑁
…1 2 … 𝑘
Block 1
…
Block 2
…
Block 𝑁/𝑘
![Page 29: Computationally Data-Independent Memory Hard Functions · 2020. 7. 9. · 1 4 5,7 6 8 10 memory access pattern for 𝑓( ) RAM. iMHFs In a data-independent memory hard functions (iMHFs)](https://reader035.vdocuments.us/reader035/viewer/2022071107/5fe1c6567d38506586271034/html5/thumbnails/29.jpg)
Maximally Hard 𝑘-Restricted Dynamic Graphs
To compute labels 𝑁 + 1 to 2𝑁, attacker should either (1) keep labels on nodes 1 to 𝑁 throughout or (2) recompute labels of nodes 1 to 𝑁when necessary
Design the graph on nodes 1 to 𝑁 to be very expensive to recompute!
… 2𝑁
…1 2 … 𝑘
Block 1
…
Block 2
…
Block 𝑁/𝑘
![Page 30: Computationally Data-Independent Memory Hard Functions · 2020. 7. 9. · 1 4 5,7 6 8 10 memory access pattern for 𝑓( ) RAM. iMHFs In a data-independent memory hard functions (iMHFs)](https://reader035.vdocuments.us/reader035/viewer/2022071107/5fe1c6567d38506586271034/html5/thumbnails/30.jpg)
Maximally Hard 𝑘-Restricted Dynamic Graphs
… 2𝑁
…1 2 … 𝑘
Block 1
…
Block 2
…
Block 𝑁/𝑘Grates [Sch83]
Superconcentrators[Pip77,LT82]
To compute labels 𝑁 + 1 to 2𝑁, attacker should either (1) keep labels on nodes 1 to 𝑁 throughout or (2) recompute labels of nodes 1 to 𝑁when necessary
Design the graph on nodes 1 to 𝑁 to be very expensive to recompute!
Amenable to shuffling
![Page 31: Computationally Data-Independent Memory Hard Functions · 2020. 7. 9. · 1 4 5,7 6 8 10 memory access pattern for 𝑓( ) RAM. iMHFs In a data-independent memory hard functions (iMHFs)](https://reader035.vdocuments.us/reader035/viewer/2022071107/5fe1c6567d38506586271034/html5/thumbnails/31.jpg)
Maximally Hard 𝑘-Restricted Dynamic Graphs
… 2𝑁
…1 2 … 𝑘
Block 1
…
Block 2
…
Block 𝑁/𝑘Grates [Sch83]
Superconcentrators[Pip77,LT82]
To compute labels 𝑁 + 1 to 2𝑁, attacker should either (1) keep labels on nodes 1 to 𝑁 throughout or (2) recompute labels of nodes 1 to 𝑁when necessary
Design the graph on nodes 1 to 𝑁 to be very expensive to recompute!
Amenable to shuffling
ciMHF with
optimal 𝑐𝑐
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Attack on 𝑘-Restricted Dynamic Graphs
For 𝑘 = 𝑜 𝑁1/ log log 𝑁 , we have 𝑐𝑐(𝑓𝐺,𝐻) = 𝑜 𝑁2
To compute labels 𝑁 + 1 to 2𝑁, attacker should either (1) keep labels on nodes 1 to 𝑁 throughout or (2) recompute labels of nodes 1 to 𝑁when necessary
Previously: Design the graph on nodes 1 to 𝑁 to be very expensive to recompute labels??
Attack: always recompute labels because no longer possible to be very expensive for small 𝑘, similar strategy to [AB16]
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For 𝑘 = 𝑜 𝑁1/ log log 𝑁 , we have 𝑐𝑐(𝑓𝐺,𝐻) = 𝑜 𝑁2
To compute labels 𝑁 + 1 to 2𝑁, attacker should either (1) keep labels on nodes 1 to 𝑁 throughout or (2) recompute labels of nodes 1 to 𝑁when necessary
Previously: Design the graph on nodes 1 to 𝑁 to be very expensive to recompute labels??
Attack: always recompute labels because no longer possible to be very expensive for small 𝑘, similar strategy to [AB16]
Attack on 𝑘-Restricted Dynamic Graphs
Valiant’s Lemma [Val77]: Keep labels of a small
number of key locations
![Page 34: Computationally Data-Independent Memory Hard Functions · 2020. 7. 9. · 1 4 5,7 6 8 10 memory access pattern for 𝑓( ) RAM. iMHFs In a data-independent memory hard functions (iMHFs)](https://reader035.vdocuments.us/reader035/viewer/2022071107/5fe1c6567d38506586271034/html5/thumbnails/34.jpg)
Summary
We construct a family of 𝑘-restricted dynamic graphs where 𝑘 = 𝑜 𝑁𝜖 for any constant 0 <𝜖 < 1 with 𝑐𝑐 𝑓𝐺,𝐻 = Ω 𝑁2
and give a ciMHF implementation of 𝑓𝐺,𝐻
We show that 𝑐𝑐(𝑓𝐺,𝐻) =𝑜 𝑁2 for 𝑘 = 𝑜 𝑁1/ log log 𝑁
![Page 35: Computationally Data-Independent Memory Hard Functions · 2020. 7. 9. · 1 4 5,7 6 8 10 memory access pattern for 𝑓( ) RAM. iMHFs In a data-independent memory hard functions (iMHFs)](https://reader035.vdocuments.us/reader035/viewer/2022071107/5fe1c6567d38506586271034/html5/thumbnails/35.jpg)
Future Directions
Fully characterize and tighten bounds for the spectrum of 𝑘-restricted dynamic graphs
Optimal ciMHFs without cache hierarchy assumptions
Show pebbling reduction for dMHFs