Approximating Good Simultaneous Diophantine Approximations Is Almost NP-Hard

Given a real vector =(1; : : : ; d) and a real number " > 0 a good Diophantine approximation to is a number Q such that kQQ mod Zk1 ", where k k1 denotes thè1-norm kxk1 := max 1id jxij for x = (x1; : : : ; x d). Lagarias 12] proved the NP-completeness of the corresponding decision problem, i.e., given a vector 2 Q d , a rational number " > 0 and a number N 2 N+, decide whether there exists a number Q with 1 Q N and kQQ mod Zk1 ". We prove that, unless NP DTIME(n poly(log n)), there exists no polynomial time algorithm which computes on inputs 2 Q d and N 2 N + a number Q with 1 Q 2 log 0:5? d N and kQ mod Zk1 2 log 0:5? d min 1qN jjq mod Zk1; where 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 in polynomial-time within a factor 2 log 0:5? d for an arbitrary small positive constant. We also investigate the nonhomogeneous variant of the good Diophantine approximation problem, i.e., given vectors ; 2 Q d , a rational number " > 0 and a number N 2 N + , decide whether there exists a number Q with 1 Q N and kQQ ? mod Zk1 ". This problem is particularly interesting since nding good nonhomo-geneous 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 2 log 1? d for an arbitrary small positive constant 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].