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Revisiting Square Root ORAM Efficient Random Access in Multi-Party Computation Samee Zahur Jack Doerner David Evans Xiao Wang Jonathan Katz Mariana Raykova Adrià Gascón oblivc.org/sqoram

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Page 1: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

RevisitingSquareRootORAMEfficientRandomAccessinMulti-PartyComputation

SameeZahurJackDoernerDavidEvans

XiaoWangJonathanKatz

MarianaRaykova Adrià Gascón

oblivc.org/sqoram

Page 2: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Securemulti-partycomputationapplications

Setintersection[FNP04]

Linearridge-regression[NWIJBT13]

Mediancomputation[AMP04]

Iriscodematching[LCPLB12]

Matrixfactorizationforrecommendations

[NIWJTB13]

Page 3: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

RandomAccess

Page 4: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Hidingaccesspattern

Linearscan

AccesseveryelementPer-accesscost:Θ 𝑛

ObliviousRAM

ContinuallyshuffleelementsaroundPer-accesscost:Θ(log'𝑛)

Page 5: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Figurefrom:Wang,Chan,Shi.CircuitOram.CCS’15

Linearscan

Page 6: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

6

(our work)

Page 7: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Approach:revisitoldschemes

Classic“squareroot”schemebyGoldreich andOstrovsky (1996).

ConsideredslowforMPCbecauseofper-accesshashevaluation.

Per-accessamortizedcost:Θ 𝑛� log𝑛

Page 8: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Four-elementORAM LargerSizes

Page 9: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

4-BlockORAM

Cost:5𝐵 +𝐵+2𝐵+3𝐵 +…=11𝐵 every3accesses

Page 10: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Comparison

Linearscan

Cost:4𝐵 =12𝐵/3

Ourscheme

Cost:11𝐵/3

Page 11: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Four-elementORAM LargerSizes

Page 12: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Positionmap

3 0 2 1

0 1 2 3

1 3 0 2

0 1 2 3

Page 13: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Creatingpositionmap

Page 14: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Creatingpositionmap

Page 15: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Inversepermutation

𝜋C

𝑝

𝜋C ⋅ 𝑝𝜋F = 𝜋C ⋅ 𝑝

Page 16: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Inversepermutation

𝜋CBobcomputes𝜋FLM = 𝑝LM ⋅ 𝜋CLM

𝜋C

𝜋FLM ⋅ 𝜋C= 𝑝LM ⋅ 𝜋CLM ⋅ 𝜋C= 𝑝LM

𝜋F = 𝜋C ⋅ 𝑝

𝜋F

Page 17: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Rinseandrepeat

1. Shuffleelements2. Recreatepositionmap3. Service𝑇 = 𝑛 log𝑛� accesses

Page 18: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Accesstime

Page 19: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Initializationcost

Page 20: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Benchmarks

Task Parameters Linearscan CircuitORAM

Square-rootORAM

Binarysearch 210 searches215 elements 1020s 5041s 825s

Breadth-firstsearch

210 vertices213 edges 4570s 3750s 680s

Stablematching 29 pairs - 189000s 119000sscrypt hashing N =214 ≈7days 2850s 1920s

Page 21: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Conclusion

Werevisitedawell-knownschemeanduseditto• Lowerinitializationcost• Improvebreakevenpoint

Showsthatasymptoticcostsarenotthefinalword,concretecostsrequiremoreconsideration.

Page 22: Revisiting Square Root ORAM - ieee-security.org · Benchmarks Task Parameters Linear scan Circuit ORAM Square-root ORAM Binary search 210 searches 215 elements 1020 s 5041 s 825 s

Download

oblivc.org/sqoram

Contactforhelp:SameeZahur <[email protected]>


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