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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

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

• 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