Approximating good simultaneous Diophantine approximations is almost NP-hard

Abstract

Given a real vector α=(α ₁,..., α _d ) and a real number ε>0 a good Diophantine approximation to α is a number Q such that ∥Qα mod ℤ∥_∞ ≤ε, where ∥ · ∥_∞ denotes the ℓ_∞-norm ∥x∥_t8 ≔ max_{1 ≤i≤d} ¦ x _i¦ for x=(x ₁,..., x_d).

Lagarias [12] proved the NP-completeness of the corresponding decision problem, i.e., given a vector α ∈ ℚ ^d, a rational number ε>0 and a number N ∈ n ₊, decide whether there exists a number Q with 1 ≤Q ≤N and ¦|Qα mod ℤ∥_t8≤ε.

We prove that, unless NP⊑DTIME(n ^{poly(log n)}), there exists no polynomial-time algorithm which computes on inputs α ∈ ℚ ^d and N ∈ n ₊a number Q ^* with

and

where γ is an arbitrary small positive constant. To put it in other words, it is almost NP-hard to approximate a minimum good Diophantine approximation to α in polynomial-time within a factor

for an arbitrary small positive constant γ.

We also investigate the nonhomogeneous variant of the good Diophantine approximation problem, i.e., given vectors α, β ∈ ℚ^d, a rational number ε > 0 and a number N ∈ n ₊, decide whether there exists a number Q with 1 ≤Q ≤ N and ∥Qα-β mod ℤ∥_t8, ≤ε.

This problem is particularly interesting since finding good nonhomogeneous Diophantine approximations enables us to factor integers and compute discrete logarithms (see Schnorr [17]).

We prove that the problem Good Nonhomogeneous Diophantine Approximation is NP-complete and even approximating it in polynomial-time within a factor

for an arbitrary small positive constant γ is almost NP-hard.

Our results follow from recent work in the theory of probabilistically checkable proofs [4] and 2-prover 1-round interactive proof-systems [7, 14].

Supported by DFG under grant DFG-Leibniz-Programm Schn 143/5-1