# Relational $\star$-Liftings for Differential Privacy

## TL;DR

This paper introduces a new existential approximate lifting called $ ext{ extsterling}$-lifting, unifying existing notions and enhancing the tools for proving differential privacy through cleaner properties and more general constructions.

## Contribution

It proposes a novel $ ext{ extsterling}$-lifting that unifies existing approximate liftings, providing more precise composition theorems and cleaner properties for differential privacy proofs.

## Key findings

- $ ext{ extsterling}$-lifting is equivalent to Sato's universal construction for discrete measures
- Unifies all known notions of approximate lifting
- Enables richer and more precise differential privacy proofs

## Abstract

Recent developments in formal verification have identified approximate liftings (also known as approximate couplings) as a clean, compositional abstraction for proving differential privacy. This construction can be defined in two styles. Earlier definitions require the existence of one or more witness distributions, while a recent definition by Sato uses universal quantification over all sets of samples. These notions have each have their own strengths: the universal version is more general than the existential ones, while existential liftings are known to satisfy more precise composition principles. We propose a novel, existential version of approximate lifting, called $\star$-lifting, and show that it is equivalent to Sato's construction for discrete probability measures. Our work unifies all known notions of approximate lifting, yielding cleaner properties, more general constructions, and more precise composition theorems for both styles of lifting, enabling richer proofs of differential privacy. We also clarify the relation between existing definitions of approximate lifting, and consider more general approximate liftings based on $f$-divergences.