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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

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:

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

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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