Suppose there are $u$ users and each user $i$ possesses $x_i\in\{0,1\}^n$ and a function $F_i:\{0,1\}^{nu} \rightarrow \{0,1\}^m$. Then we wish to construct a protocol such that at its completion, each user $i$ knows $F_i(x_1,...,x_u)$ but knows nothing more about $x_j$ for $j \ne i$.

Clearly this could be done with a trusted third party, but we want to do it without one.

Security models:

• Honest-but-curious: all $u$ parties follow the protocol honestly, and a protocol is $t$-private if any $t$ parties who collude at the end of the protocol learn anything beyond their own outputs from their transcripts.

To prove a protocol is $t$-private, we build a simulator that, when given inputs and outputs of $t$ colluding parties, generates $t$ transcripts from the same distribution as the actual protocol. (For this implies anything the colluding users can learn from their transcripts can be learnt from their inputs and outputs alone.)

• Malicious users: the adversary controls a fixed set of $t$ users. The remaining $u-t$ users are honest. A protocol is $t$-secure if the adversary learns nothing about the $u-t$ user inputs beyond the outputs of the $t$ corrupt parties.

Usually, the goal is to construct a $t$-secure, $t'$-private protocol for some $t' \ge t$.

• Dynamic adversary: in this case, at any time period, the adversary can corrupt any $t$ users.

## Example

Suppose we have three users, who’s secrets are $x_1, x_2, x_3 \in \mathbb{F}_p$, and their functions are $F_1 = F_2 = F_3 = x_1 + x_2 + x_3$.

Trivially, any valid protocol is 2-private because if two parties collude, they can determine the third party’s secret.

A 1-private protocol can be constructed by using secret sharing:

User 1: $r_1,s_1\leftarrow\mathbb{F}_p, 1\rightarrow 2 : r_1, 1\rightarrow 3:s_1$

User 2: $r_2,s_2\leftarrow\mathbb{F}_p, 2\rightarrow 1 : r_2, 2\rightarrow 3:s_2$

User 3: $r_3,s_3\leftarrow\mathbb{F}_p, 3\rightarrow 1 : r_3, 3\rightarrow 2:s_3$

(can be done in parallel)

User 1: publishes $y_1 = (x_1-r_1-s_1)+r_2+r_3$

User 2: publishes $y_2 = (x_1-r_2-s_2)+r_1+s_3$

User 3: publishes $y_3 = (x_1-r_3-s_3)+s_1+s_2$

Then each user computes $y_1+y_2+y_3 = x_1+x_2+x_3$.

1-privacy proof: user 1’s transcript is $[x_1,r_1,s_1,r_2,r_3,y_2,y_3,x_1+x_2+x3]$. Then we construct a simulator as follows: given $x_1, z=x_1+x_2+x_3$, we generate the transcript by picking $r_1,s_1,r_2,r_3,y_2\leftarrow \mathbb{F}_p$, setting $y_1 = (x_1-r_1-s_2)+r_2+r_3$, and outputing $[x_1,r_1,x_1,r_2,r_3,y_2,z-y_1-y_2,z]$. From user 1’s view, $y_2$ is random because user 1 never sees $s_3$. We can construct simulators for the other users in a similar fashion.

This protocol generalizes to $n$ parties and any linear combination, and becomes a $(n-2)$-private protocol. It is sometimes referred to as Benaloh’s protocol.

## Modeling Cryptographic Protocols

Practically any cryptographic protocol can be described in terms of SFE. For example:

• Identification: $A$ has a secret key $x$, and a public key $f(x)$ for some one-way function $f$, and wishes to prove possession of $x$ to $B$.

In SFE terms: $A$'s input is $x$, $F_A = 0$, $B$'s input is $f(x) = y$, $F_B(x,y) = (y = f(x)) ? 1 : 0$.

(The SFE model captures the fact that $B$ should not learn anything about $x$.)

• Key exchange: (secure against eavesdropping). Three parties, Alice, Bob, Eve. $x_A = r, x_B = 0, x_E = 0, F_A = 0, F_B = r, F_E = 0$, and Eve is passive, i.e. does not send any messages.

• Voting:$x_i \in \{0,1\}$ for $i=1,...,u$, $F_i =...=F_u = MAJORITY(x_1,...,x_u)$.

• Threshold signatures: Let $PK, SK$ be a public/private key pair for some signature scheme. Take $SK = SK_1 \oplus ... \oplus SK_u$. $x_i = SK_i$ for $i=1,...,u$, $F_i =...=F_u = Sign(SK_1 \oplus ... \oplus SK_u, M)$.

• Private auctions: (sealed bid, 2nd-price auction) $x_i$ = bid of user $i$. Let $S = 2ND-MAX(x_1,...,x_u)$. $F_1 = ... =F_u = (x_i = MAX(x_1,...,x_u))? S : 0$.

## Results

1. [Yao’82,Yao’86,GMW’87,G’97] 2-party SFE (using complexity assumptions)

2. [BGW’87] $n$-party SFE for $n\gt 2$, $\lfloor n/2 \rfloor - 1$-private (information theoretic result)