Secure Computation and theCombinatorics of Hidden Diversity

Juan Garay (AT&T Research)David Johnson (AT&T Research) Aggelos Kiayias (U. Athens) Moti Yung (Google)

Hidden Diversity and Secure Multiparty Computation

Resource-based Corruptions Adversaries corrupt parties...

…for FREE!

Prover Verifier

Hidden Diversity and Secure Multiparty Computation

Resource-based Corruptions (cont’d)

How much does corruption cost?• Different parties may require different “resources” to get

corrupted

Can “anonymity” be used to raise those costs?

Our new questions:

A focal point : Corruption diversity

Given that corruptions happen in different ways and based on different parameters, they can require a different amount of resources

How to model corruption diversity?

Resource-based corruptions

s1s2

s3

s4

s5

Budget b(with “tokens”)

Adversary’s

Goal :

Hidden Diversity and IndistinguishabilitySuppose different parties require different resources for

corruption but externally appear the same

s2

s3s4

s5

s1

?Adversary will need to waste more resources for subverting the system!

A Combinatorial Game

GIVEN: Set B1, B2, …, Bn of buckets, with bucket Bi having non-negative integer size si, and a target fraction α, 0 < α < 1.

GOAL: Fill αn of the buckets using as few balls as possible, where a bucket of size si is filled if it receives si balls.

Hidden Diversity and Secure Multiparty Computation

n = 5, α = ½, αn = 3

Balls and Buckets (cont’d)

Hidden Diversity and Secure Multiparty Computation

Only Feedback from Placing a Ball: “Bucket Now Full” or

“Bucket Not Yet Full”

Balls and Buckets (cont’d)

Hidden Diversity and Secure Multiparty Computation

How many balls?

Hidden Diversity and Secure Multiparty Computation

In this work Framework for realization of above abstraction

• Computational corruptions Sufficient conditions for abstraction

• Information-Effort-Preserving (IEP) functions

• Hardness Indistinguishability

• Exact Hardness

Hidden Diversity and Secure Multiparty Computation

Candidate Functions Random oracle Exponentiation

• f : Zq → S; q: λ-bit prime number; S: (generic) multiplicative group

Multiplication

• fmult : Pλ x Pλ → N

Hidden Diversity and Secure Multiparty Computation

In this work Framework for realization of above abstraction

• Computational corruptions Sufficient conditions for abstraction

• Information-Effort-Preserving (IEP) functions

• Hardness Indistinguishability

• Exact Hardness

Much is to be gained : MPC

• Security: unbounded additional adversarial effort

• Efficiency: force corruption threshold to drop from 1/2 to 1/3, and run information-theoretic MPC protocol

Thanks!