# Fast systematic encoding of multiplicity codes

## TL;DR

This paper introduces efficient systematic encoding algorithms for multiplicity codes, enabling practical use in private information retrieval protocols by building on advanced multivariate interpolation techniques.

## Contribution

It generalizes existing fast interpolation algorithms to Hermite-type problems, providing the first quasi-linear time encoding methods for multiplicity codes.

## Key findings

- Achieves quasi-linear encoding time for multiplicity codes
- Enables practical application of private information retrieval protocols
- Extends multivariate interpolation algorithms to Hermite-type problems

## Abstract

We present quasi-linear time systematic encoding algorithms for multiplicity codes. The algorithms have their origins in the fast multivariate interpolation and evaluation algorithms of van der Hoeven and Schost (2013), which we generalise to address certain Hermite-type interpolation and evaluation problems. By providing fast encoding algorithms for multiplicity codes, we remove an obstruction on the road to the practical application of the private information retrieval protocol of Augot, Levy-dit-Vehel and Shikfa (2014).