# Private Polynomial Computation from Lagrange Encoding

## TL;DR

This paper introduces a private polynomial computation scheme using Lagrange encoding that ensures data and function privacy, robustness against server failures, and extends private computation to high-degree polynomials.

## Contribution

It demonstrates that Lagrange encoding enables private computation of high-degree polynomials with robustness and data privacy, connecting private and coded computation.

## Key findings

- Supports private computation of any degree polynomial
- Robust to Byzantine and straggling servers
- Ensures data privacy using Shamir secret sharing

## Abstract

Private computation is a generalization of private information retrieval, in which a user is able to compute a function on a distributed dataset without revealing the identity of that function to the servers. In this paper it is shown that Lagrange encoding, a powerful technique for encoding Reed-Solomon codes, enables private computation in many cases of interest. In particular, we present a scheme that enables private computation of polynomials of any degree on Lagrange encoded data, while being robust to Byzantine and straggling servers, and to servers colluding to attempt to deduce the identities of the functions to be evaluated. Moreover, incorporating ideas from the well-known Shamir secret sharing scheme allows the data itself to be concealed from the servers as well. Our results extend private computation to high degree polynomials and to data-privacy, and reveal a tight connection between private computation and coded computation.