# Secure Computation of Randomized Functions: Further Results

## TL;DR

This paper investigates the optimal rates for secure computation of randomized functions between two users under different privacy constraints, providing explicit characterizations and simplified expressions for certain cases.

## Contribution

It offers single-letter formulas for optimal rates in secure randomized function computation and characterizes when functions are securely computable with full support inputs.

## Key findings

- Explicit single-letter expressions for optimal rates.
- Simplified rate expressions for functions with full support inputs.
- Characterization of securely computable functions under privacy constraints.

## Abstract

We consider secure computation of randomized functions between two users, where both the users (Alice and Bob) have inputs, Alice sends a message to Bob over a rate-limited, noise-free link, and then Bob produces the output. We study two cases: (i) when privacy condition is required only against Bob, who tries to learn more about Alice's input from the message than what can be inferred by his own input and output, and (ii) when there is no privacy requirement. For both the problems, we give single-letter expressions for the optimal rates. For the first problem, we also explicitly characterize securely computable randomized functions when input has full support, which leads to a much simpler expression for the optimal rate. Recently, Data (ISIT 2016) studied the remaining two cases (first, when privacy conditions are against both the users; and second, when privacy condition is only against Alice) and obtained single-letter expressions for optimal rates in both the scenarios.