# Privacy-Preserving Computation ——secure Multi-Party Computation (II)

Before technique details, one should define the security goal, which is called the security model in cryptography. The semi-honest model and malicious model are widely considered to capture the capabilities of adversaries.

In the semi-honest setting, the players strictly follow the instructions of the protocol, and are still curious to learn information of other inputs from the transactions. This is meaningful in some enterprise-to-enterprise business scenarios, where the enterprises must strictly follow the procedure and the codes are run in some safe place that stuffs can not touch. The malicious setting is more realistic, which means the players could do what they can to get private information of other inputs. A secure MPC protocol means that in both setting, all the players can not learn any additional information of the outputs. Note that “additional information” here does not mean “any information” of the input, anything could directly derived from the output result is allowed to be learned by the players.

Many specific techniques are proposed to construct semi-honest or malicious MPC protocols, such as garbled circuit, secret sharing, oblivious transfer, homomorphic encryption and their combinations. In this article, we introduce the classic Yao’s garbled circuit in the two-party setting, it is also one of the most efficient constructions by now.

Alice and Bob who have private input x and y respectively, want to jointly compute a function f and both obtain the result z=f(x,y) without revealing anything else. One should first use a circuit compiler to convert the function f into boolean circuit that consists of AND and XOR gate (as shown in Figure 1). Since most real-world programs written in advanced language contain complex data structures, this is a highly non-trivial task.

For the easy understanding of the description, we assume that x,y are both 2-bit strings, and x feeds into the first two wire in the left, and y feeds into the last two wires in the figure above.

Alice takes as input the circuit, and writes down all the truth table of each gate. Then it chooses two uniformly random string, which is called labels, for each gate to represent 0 or 1 as in Figure 2. Note that, although Alice does not know the input of C,D wires, she still chooses labels to represent 0 and 1 in each wire.

After choosing the labels, Alice replaces the truth table with labels according to the topology of the circuit. Take the left-up gate as an example. It has A and B wires as inputs and E wire as output. Then Alice replaces the first truth table as:

After replacing all the truth tables, Alice gets a “label table” as in figure 3.

For each gate Alice garbles the gates using the labels. I.e., for each row of the label table, Alice encrypts the output label with the input label as keys to a double encryption. More specifically, for the first row in the first label table, Alice uses A_0 and B_0 as keys to AES and encrypts E_0 as AES_A0(AES_B0(E_0)), all the other rows and gates are doing in the similar way. Then Alice gets a garbled circuit as in Figure 4.

Suppose Alice’s input x = 10, Bob’s input y=01. Alice then sends the garbled circuit (all the garbled gate) to Bob, together with A_1 and B_0, according to her input bits, and the meaning of output labels (i.e. the relations I_0 →0; I_1 →1). Note that Bob can not get any input information of Alice from A_1 and B_0, since they are just random strings, and the relation of output labels will not leak any additional information of the output value.

Before Bob evaluates the garbled circuit, he has to obtain the input labels according to his input information. Since the labels are sampled by Alice, he has to run a “transfer protocol” with Alice to get the labels without telling her the inputs in plain. This “transfer protocol” is called oblivious transfer in cryptography, since it is a very fundamental cryptographic primitive, we will not get into the details, but list the properties below.

1. Alice takes labels X_0, X_1 as input, and Bob takes bit b=0/1 as input. At the end of the protocol, Bob will get X_b.

2. Alice does not know which label is chosen by Bob.

3. Bob does not know the other label X_1-b.

After running the oblivious transfer protocol with Alice, Bob gets the labels C_0, D_1 according to his input, and then he could evaluate the garbled circuit. For each garbled gate, Bob decrypts four ciphertexts with the input labels as AES keys, note that he could only get one meaningful output label, and then iteratively decrypts the garbled gate to get the output label I_0 . Since Bob knows the meaning of I_0, he will get the output 0, and send it to Alice.The whole evaluation process is described as in Figure 5.