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

* * * *:{0,1} {0,1} {0,1} {0,1}f 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

* * * *:{0,1} {0,1} {0,1} {0,1}f

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 Gatesa =

a1+a2

+(GF(2))

b = b1+b2

c = c1+c2

a = a1+a2

x(GF(2))

b = b1+b2

c = c1+c2

Addition Gatea =

a1+a2

+(GF(2))

b = b1+b2

c = c1+c2

c1 = a1+b1

c2 = a2+b2

c1+c2 = a1+a2+b1+b2

Multiplication Gate

c1+c2 = (a1+a2)(b1+b2)(c1,c2) should be uniformlychosen amongst all solutionsWe use Oblivious Transfer

a = a1+a2

x(GF(2))

b = b1+b2

c = c1+c2

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:

1 2

1 2 1 pub 1

, , ,

, , , ,RSA-enc ( ), , ,k

i i i k

y y y

e e e e e e

1 2,

2

, ,

Dec( ),Dec( ), ,Dec( )k

k

d d d

y y y

OT Using RSA for semi-honest1 2

1 2 1 pub 1

, , ,

, , , ,RSA-enc ( ), , ,k

i i i k

y y y

e e e e e e

1 2,

2

, ,

Dec( ),Dec( ), ,Dec( )k

k

d d d

y y y

1 2

1 1 2 2

, , ,

hash( ) ,hash( ) ,hash( )k

k k

z z z

d t d t d t

Sender sends Receiver:

Receiver computes:hash( ) hash( ) hash(Dec(RSA-enc( ))i i i i i

i

e z e e t

t

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

1 1 1 1

2 1 1 1

3 1 1 1

4 1 1 1

( 1)

( 1)

( 1)( 1)

t c a b

t c a b

t c a b

t c a b

2 21 2 {1,2,3,4}i a b

Correctness

(a2,b2)(0,0)(0,1)(1,0)(1,1)

i = 1 + 2a2 + b2

1234

Outputc1+a1b

1

c1+a1(b1

+1)c1+b1(a1

+1)c1+

(b1+1)(a1+1)

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

(1 ,1 ) ( , )n n b b

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