Introduction to modern cryptography lecture 12
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Introduction to Modern Cryptography, Lecture 12. Secure Multi-Party Computation. We want to emulate a trusted party. Imagine that the parties send their inputs to a trusted party (no eavesdroping)

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Introduction to Modern Cryptography, Lecture 12

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Introduction to Modern Cryptography, Lecture 12

Secure Multi-Party Computation


We want to emulate a trusted party

  • Imagine that the parties send their inputs to a trusted party (no eavesdroping)

  • The trusted party computes the “functional” (not a function): a random process that maps m inputs to m outputs

  • The trusted party gives every party its output (again no eavesdroping)

  • We want to do without a trusted party


General Two-Party Computation

  • A 2 party protocol problem is a random process that maps pairs of inputs (one per party) to pairs of outputs

  • Special cases of interest:

  • f(x,y) = (g(x,y),g(x,y))

  • f(x,y) = uniformly distributed over ((0,0),(1,1))


Conventions

  • The protocol problem has to be solved only for inputs of the same length

  • The functionality is computable in time polynomial in the length of the input

  • Security is measured in terms of the length of the input (use inputs 1n)


The semi-honest model

  • A semi-honest party is one who follows the protocol with the exception that it keeps all its intermediate computations

    • In particular, when the protocol calls for tossing a fair coin, the semi-honest party will indeed toss a fair coin

    • Also, the semi-honest party will send all messages as instructed by the protocol

  • Actually, it suffices to keep the internal coin tosses and all messages received


Privacy in the semi-honest model

  • A protocol privately computes

    if whatever a semi-honest party can obtain after participating in the protocol, it could obtain from its input and output


Security in the semi-honest model

  • The “ideal” execution makes use of a trusted third party

  • A semi-honest protocol is secure if the results of the protocol can be simulated in the ideal model

  • In the semi-honest model, security = privacy


The Malicious Model

  • There are three things we cannot hope to avoid:

    • Parties refusing to participate

    • Parties substituting their local input

    • Parties aborting the protocol prematurely

  • Security in the malicious model: the protocol emulates the ideal model (with a trusted third party)


Secure Protocols for the Semi-Honest model

  • Produce a Boolean circuit representing the functionality

  • Use a “circuit evaluation protocol” which scans the circuit from the inputs wires to the output wires

  • When entering a basic step, the parties hold shares of the values of the input wires, and when exiting a basic step, the parties hold shares of the output wires

NOTE: ONLY DETERMINISTIC SO FAR


What gates?

  • It suffices to consider AND and XOR gates of fan-in 2

  • Use arithmetic over GF(2) where multiplication = AND and addition = XOR

    • 1*1=1, 1*0=0, 0*0=0, 0*1=0

    • 1+1=0, 1+0=1, 0+1=1, 0+0=0


Required Gates


Addition Gate

c1 = a1+b1

c2 = a2+b2

c1+c2 = a1+a2+b1+b2


Multiplication Gate

c1+c2 = (a1+a2)(b1+b2)

(c1,c2) should be uniformly

chosen amongst all solutions

We use Oblivious Transfer


Oblivious transfer in the case of semi-honest parties

  • Sender has t1, t2, …, tk (bits)

  • Receiver chooses some 1 ≤ i ≤ k

  • Goal: Receiver gets ti, Sender does not know i


OT Using RSA for semi-honest

  • Sender chooses RSA keys, sends public key to Receiver

  • Receiver chooses random e1, e2, …, ek

  • Receiver computes RSApub(ei)

  • Receiver sends Sender:

  • Sender computes:


OT Using RSA for semi-honest

Sender sends Receiver:

Receiver computes:


Privately computing c1+c2=(a1+a2)(b1+b2)

  • We use Oblivious transfer with four shares

  • Party 1 chooses a random c1 in 0,1

  • Party 1 has a1, b1, and plays the OT sender with

  • Party 2 has a2, b2, and plays the OT receiver with


Correctness


The circuit evaluation protocol

  • Do a topological sort of all wires in the circuit

  • Input wires: every player “shares” the value of her input wire with the other player

  • Once the shares of the circuit output wires are computed, every party sends its share of wires for the other party


How to force semi-honest behavior

  • Theorem: suppose that trapdoor permutations exist (e.g., RSA), then any two party functionality can be securely computable in the MALICIOUS MODEL.


Problems with Malicious parties

  • Different input (nothing to do)

  • Does not use truly random bits (I happen to have chosen at random the ace) – use coin tossing in a well

  • Send messages other than the messages it should send via the protocol – use zero knowledge proofs


Coin tossing in a well

  • A coin tossing in a well protocol is a two party protocol for securely computing (in the malicious model) the randomized functionality

Where b is uniformly distributed on 0,1


Simple solution

  • Use an encoding of 0’s and 1’s

  • Alice chooses a random encoding of a random bit b and sends Bob the one-way function (or more exactly bit commitment) of the bit

  • Bob sends a random bit c to Alice

  • Alice reveals the commitment to b

  • The common random bit is b+c


Alice does not want Bob to know her coin tosses, only to prove that they are honest:

  • Alice chooses many random bits b1, b2, …and sends Bob the bit commitments

  • Bob sends Alice random bits c1, c2, …

  • Alice uses the bits bi + ci in her computation

  • Alice gives Bob a zero knowledge proof that the computation uses these bits, based upon the commitments to the bits that Bob already has


Alice’s other inputs

  • Alice needs to be consistent in her inputs, we cannot force Alice not to lie about her input, but at least we can force her to be consistent


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