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

The proof of Toda’s celebrated theorem that the polynomial hierarchy is contained in P relies on the fact that, under mild technical conditions on the complexity class C, we have ∃ C ⊂ BP · ⊕ 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. Valiant and Vazirani [VV86] gave a clever randomized reduction from NP to UP, the class of promise problems which have either a unique solution or no solution at all. Their reduction works as follows. Given, say, a 3-SAT formula φ on n variables, we begin choose an integer k uniformly from {1, . . . , n}. We then add the additional constraint that a hash function h takes the value zero, where h is chosen from a pairwise independent family of hash functions, and where a given truth assignment x obeys h(x) = 0 with probability 2. If φ is satisfiable, then with probability Ω(1/n) this additional constraint makes the solution unique. So long as the complexity class C is expressive enough to compute the hash function h and is closed under intersection, this reduction asserts that: ∃ C ⊆ RPpoly · ∃! C , where RPpoly denotes one-sided error with a 1/poly(n) probability of success and where ∃! denotes unique existence. Since 1 is odd, we can also write ∃ C ⊆ RPpoly · ⊕ C , where, for instance, ⊕P is the class of decision problems which ask whether the number of witnesses for a problem in NP is odd. For the case of ⊕P, we can amplify the probability of success as follows: if we perform m = Ω(n) independent trials of this reduction, then with probability Ω(1) at least one trial will yield a formula φ with a unique solution (assuming φ is satisfiable). Since the expression a = 1 + m ∏ i=1 (ai + 1) is odd if and only if at least one of the ai is odd, and it is easy to implement such expressions within ⊕P by constructing m-tuples of witnesses, we conclude ∃P ⊆ RP · ⊕P 1 where now the reduction works with probability Ω(1). (Of course, by taking, say, m = n, we can make the probability of success exponentially close to 1.) By showing that the operators BP and ⊕ can be commuted, we obtain Toda’s result [Tod91] that PH ⊆ BP · ⊕P ⊆ P . The purpose of this note is to give an alternate reduction from NP to RP which works with constant probability without the need for amplification. Our reduction is quite simple, and may be of independent interest. First, let p be a prime, let Fp denote the field of order p, and for a ∈ Fp let χ(a) denote the Legendre symbol χ(a) = 