<p>In this paper, the authors give quantum algorithms for two fundamental computation problems: Solving polynomial systems over finite fields and optimization where the arguments of the objective function and constraints take values from a finite field or a bounded interval of integers. The quantum algorithms can solve these problems with any given success probability and have polynomial runtime complexities in the size of the input, the degree of the inequality constraints, and the condition number of the associated matrices of the problem. So, the authors achieve exponential speedup for these problems when their condition numbers are small. As applications, quantum algorithms are given to three basic computational problems in cryptography: The short integer solution problem, the shortest vector problem, the polynomial system with noise problem, and cryptanalysis for the lattice-based NTRU cryptosystem. It is shown that these problems and NTRU can against quantum computer attacks only if their condition numbers are large, so the condition number could be used as a new criterion for lattice-based post-quantum cryptosystems.</p>