A simple constant-probability RP reduction from NP to Parity P

TL;DR

This paper introduces a straightforward algebraic reduction from NP to Parity P with constant success probability, simplifying previous methods that required amplification, and relies on properties of the Legendre symbol.

Contribution

It presents a novel, simple algebraic reduction from NP to Parity P that avoids amplification, using properties of the Legendre symbol in finite fields.

Findings

Achieves constant success probability in NP to Parity P reduction

Simplifies previous randomized reduction methods

Uses properties of the Legendre symbol for analysis

Abstract

The proof of Toda's celebrated theorem that the polynomial hierarchy is contained in $\P^{# P}$ relies on the fact that, under mild technical conditions on the complexity class $C$, we have $\exists C \subset BP \cdot \oplus C$. More concretely, there is a randomized reduction which transforms nonempty sets and the empty set, respectively, into sets of odd or even size. The customary method is to invoke Valiant's and Vazirani's randomized reduction from NP to UP, followed by amplification of the resulting success probability from $1/\poly(n)$ to a constant by combining the parities of $\poly(n)$ trials. Here we give a direct algebraic reduction which achieves constant success probability without the need for amplification. Our reduction is very simple, and its analysis relies on well-known properties of the Legendre symbol in finite fields.