Reductions in PPP

Highlights

• We introduce two new problems, Minkowski and Dirichlet.

• We show that these two problems belong in the complexity class PPP.

• We show that Equal Sums and Dirichlet are reducible to Minkowski.

Abstract

We show several reductions between problems in the complexity class PPP related to the pigeonhole principle, and harboring several intriguing problems relevant to Cryptography. We define a problem related to Minkowski's theorem and another related to Dirichlet's theorem, and we show them to belong to this class. We also show that Minkowski is very expressive, in the sense that all other non-generic problems in PPP considered here can be reduced to it. We conjecture that Minkowski is PPP-complete.

Introduction

Total search problems [7], [9] still constitute an exotic domain of complexity theory — exotic in both senses “strange, unusual” and “remote, unexplored.” Take the class PPP, for example. It is known to include all of PPAD, and is defined in terms of the generic complete problem

Pigeon: Given a Boolean circuit C with n inputs and n outputs, find $x\ne y\in {\{0,1\}}^n$ such that either $C(x)=0^n$ or $C(x)=C(y)$.

The class PPP consists of all search problems reducible to Pigeon. As for other problems known to be in PPP, [9] only mentions

Equal sums: Given positive integers $a_1,\dots ,a_n$ such that ${\sum }_i{a}_i<2^n-1$, find two subsets $S\ne T\subseteq \{1,\dots ,n\}$ such that ${\sum }_{i\in S}{a}_i={\sum }_{i\in T}{a}_i$.

and these two problems, Pigeon and Equal sums, are to our knowledge¹ the only problems discussed in the literature that are known to be in PPP and not known to be in included classes, e.g. Nash.

Over the past decade, the class PPAD and its complete problems have substantially informed and advanced our algorithmic understanding of Game Theory [8]. Can PPP serve the same role for Cryptography? The generic problem Pigeon is loosely about collisions of hash functions; furthermore, it was recently pointed out that Factoring belongs to PWPP (the class of problems whose totality is proved through a weak pigeonhole principle), a subclass of PPP, via randomized reductions [6]. The two new problems that we introduce here, Minkowski and Dirichlet, are also motivated by Cryptography, since they are both the complexity renderings of mathematical results which have been used in the foundations of lattice-based crypto systems [1]. There may be important dividends in understanding better this class and its complete problems.

Incidentally, other classes of total problems from [9] have recently been connected to Cryptography: Factoring was shown to also belong in the class PPA, again via randomized reductions, and also in the class PWPP [6]. Note that the only other problems known to be in PPP ∩ PPA are the problems in PPAD, such as Nash [3], [4]. Determining whether Factoring is in PPAD (via randomized reductions) is a most important open question. Recall also the recent insight that PPAD is intractable under standard cryptographic assumptions [2] (standard in the sense that they are adopted in parts of the mainstream literature).

As we mentioned, Pigeon and Equal sums are, as far as we know, the only problems in PPP discussed in the literature.² In other words, the picture of PPP prior to this note has been as shown in Fig. 1(a). In this note we introduce two new problems:

Minkowski: Given an $n\times n$ matrix A with $|\det \left(A\right)|<1$, find a nontrivial integer combination of its rows with $l_{\infty }$ norm less than one.

Dirichlet: Given n rational numbers $a_1,\dots ,a_n$ and an integer N, find integers $q,p_1,\dots ,p_n$ such that $|a_i-\frac{p_i}{q}|<\frac{1}{qN}$ for all i, and $1\le q\le N^n$.

We show that these two problems are in PPP. In fact, we show that Minkowski is remarkably expressive, in the sense that the two currently known non-generic problems in PPP (Equal sums and Dirichlet, leaving out problems, such as Nash, which belong to subclasses) are reducible to Minkowski.

Theorem 1

Minkowski and Dirichlet are reducible to Pigeon.

Theorem 2

Equal sums and Dirichlet are reducible to Minkowski.

None of these results is trivial, but the reduction from Minkowski to Pigeon is, surprisingly, the hardest to prove. In other words, the new picture of problems in PPP is as shown in Fig. 1(b). These results suggest that Minkowski is a natural candidate for a non-generic complete problem for PPP. Thus, the following important open problem is the main message of this work:

Conjecture 1

Pigeon is reducible to Minkowski.

When writing this note the authors were not aware of the following two, very relevant works (or the works didn't exist in the case of the second paper by Sotiraki et al. [10]).

Hoberg et al. [5]

In a recent work, Hoberg et al. [5] show (independently of this work) that the problem Number balancing is equivalent to polynomial approximations of Minkowski's theorem. The result most relevant to this note is the following. Given an algorithm that takes as input a lattice $\mathrm{\Lambda }\subseteq {\mathbb{R}}^n$ with $|\det \left(\mathrm{\Lambda }\right)|<1$ and finds a non-zero vector $\mathbf{x}\in \mathrm{\Lambda }$ such that ${\Vert \mathbf{x}\Vert }_{\infty }\le \rho$, there exists a δ-approximation algorithm for the number balancing problem, where $\delta =2^{-n^{\mathrm{\Theta }(1/\rho )}}$. Their proof is similar to the reduction of Equal sums to Minkowski presented in this note.

Sotiraki et al. [10]

In a recent unpublished manuscript, Sotiraki et al. [10] identify two new problems in PPP: a computational problem associated with Blichfeldt's fundamental theorem, and a generalized version of the Short Integer Solution problem from lattice based cryptography. They prove that both problems are in PPP, and furthermore that both problems are PPP-hard. This breakthrough therefore gives us the first natural PPP-complete problems. The authors provide a new proof that Minkowski is in PPP, via a reduction to Blichfeldt. Their reduction of Blichfeldt to Pigeon is similar in spirit to the reduction of Minkowski to Pigeon presented in this note.