The　paper　discusses　the　selection　of　orders　of　elements　in　one　generator　set　for　a　specified　group,　and　analyzes　multi-discrete　logarithm　problems　(MDLP)　and　a　public　key　encryption　scheme　based　on　the　MDLP.　The　paper　points　out　that　under　the　circumstances　described　by　the　original　paper,　the　MDLP　may　be　transformed　into　a　discrete　logarithm　problem,　which　manifests　that　there　exists　a　sub-exponential　time　solution　for　the　MDLP,　and　causes　a　related　private　key　insecure　in　sub-exponential　time　in　most　cases.　Further,　in　almost　any　case,　a　ciphertext　inversion　problem　may　be　transformed　into　a　discrete　logorithm　problem,　which　illustrates　that　there　also　exists　a　sub-exponential　time　solution　to　the　ciphertext.　Therefore,　to　convert　a　discrete　logarithm　and　the　ElGamal　cryptosystem　into　those　which　are　resistant　to　the　Shor　quantum　alg.　orithm　attack,　the　people　still　need　to　make　deeper　and　longer　explorations.　©　2018,　Chinese　Institute　of　Electronics.　All　right　reserved.