In　this　paper,　the　authors　give　the　definitions　of　a　coprime　sequence　and　a　lever　function,　and　describe　the　five　algorithms　and　six　characteristics　of　a　prototypal　public　key　cryptosystem　which　is　used　for　encryption　and　signature,　and　is　based　on　three　new　problems　and　one　existent　problem:　the　multivariate　permutation　problem　(MPP),　the　anomalous　subset　product　problem　(ASPP),　the　transcendental　logarithm　problem　(TLP),　and　the　polynomial　root　finding　problem　(PRFP).　Prove　by　reduction　that　MPP,　ASPP,　and　TLP　are　computationally　at　least　equivalent　to　the　discrete　logarithm　problem　(DLP)　in　the　same　prime　field,　and　meanwhile　find　some　evidence　which　inclines　people　to　believe　that　the　new　problems　are　harder　than　DLP　each,　namely　unsolvable　in　DLP　subexponential　time.　Demonstrate　the　correctness　of　the　decryption　and　the　verification,　deduce　the　probability　of　a　plaintext　solution　being　nonunique　is　nearly　zero,　and　analyze　the　exact　securities　of　the　cryptosystem　against　recovering　a　plaintext　from　a　ciphertext,　extracting　a　private　key　from　a　public　key　or　a　signature,　and　forging　a　signature　through　known　signatures,　public　keys,　and　messages　on　the　assumption　that　IFP,　DLP,　and　LSSP　can　be　solved.　Studies　manifest　that　the　running　times　of　effectual　attack　tasks　are　greater　than　or　equal　to　O(2(n))　so　far　when　n　=　80,　96,　112,　or　128　with　Ig　M　approximate　to　696,　864,　1030,　or　1216.　As　viewed　from　utility,　it　should　be　researched　further　how　to　decrease　the　length　of　a　modulus　and　to　increase　the　speed　of　the　decryption.　(C)　2011　Elsevier　B.V.　All　rights　reserved.