The　authors　give　the　definition　and　property　of　a　bit-pair　shadow,　and　design　the　algorithms　of　a　public　key　cryptoscheme　based　on　a　multivariate　permutation　problem　and　an　anomalous　subset　product　problem　to　which　no　subexponential　time　solutions　are　found　so　far,　and　regards　a　bit-pair　as　an　operation　unit.　Further,　demonstrate　that　the　decryption　algorithm　is　correct,　deduce　the　probability　that　a　plaintext　solution　is　nonunique　is　nearly　zero,　analyze　the　security　of　the　new　scheme　against　extracting　a　private　key　from　a　public　key　and　recovering　a　plaintext　from　a　ciphertext　on　the　assumption　that　an　integer　factorization　problem,　a　discrete　logarithm　problem,　and　a　low-density　subset　sum　problem　can　be　solved　efficiently,　and　prove　that　new　scheme　using　random　padding　and　permutation　is　semantically　secure.　The　analysis　shows　that　the　bit-pair　method　increases　the　density　D　of　a　related　knapsack　to　1+,　and　decreases　the　modulus　length　inverted　right　perpendicular　lg　M　inverted　left　perpendicular　of　the　new　scheme　to　464,　544,　or　640.