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

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

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:
OT Using RSA for semi-honest

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