  We consider the average-case complexity of some otherwise undecidable or open
Diophantine problems. More precisely, consider the following: (I) Given a
polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y
f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given
polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a
rational solution to f_1=...=f_m=0. We show that, for almost all inputs,
problem (I) can be done within coNP. The decidability of problem (I), over N
and Z, was previously unknown. We also show that the Generalized Riemann
Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done
via within the complexity class PP^{NP^NP}, i.e., within the third level of the
polynomial hierarchy. The decidability of problem (II), even in the case m=n=2,
remains open in general.
  Along the way, we prove results relating polynomial system solving over C, Q,
and Z/pZ. We also prove a result on Galois groups associated to sparse
polynomial systems which may be of independent interest. A practical
observation is that the aforementioned Diophantine problems should perhaps be
avoided in the construction of crypto-systems.
