Quantum　computation　is　a　new　computational　model　based　on　quantum　mechanical　principle.　Shor　invented　the　polynomial　time　algorithms　for　the　prime　factorization　and　discrete　logarithm　problem,　which　indicated　that　the　cryptosystems　based　on　them　are　totally　unsafe　in　the　quantum　world.　Grover　constructed　an　algorithm　that　finds　a　solution　in　only　O(√2　n　)steps　whereas　the　exhaustive　search　algorithm　needs　O(2　n　)　steps　on　average.　In　this　paper　we　investigate　the　cryptanalysis　of　a　new　cryptography　problem-multivariate　permutation　problem　(MPP),　which　could　be　used　to　design　public-key　cryptosystem,　with　the　help　of　the　two　quantum　algorithms.　Specially,　we　discuss　the　strength　of　a　private　key　of　the　REESSE1+　public-key　cryptosystem,　whose　security　is　based　on　the　hardness　of　MPP.　Besides,　some　suggestions　are　also　given　about　the　implementation　of　the　REESSE1+.