In　this　paper,　the　authors　derive　a　new　sufficient　and　necessary　condition　for　the　high　degree　congruence　xn　≡=　a　(mod　p)　to　have　solutions　from　a　known　discriminant.　Put　forward　the　definition　of　the　trivial　solution　to　the　congruence,　elaborate　two　methods　of　computing　the　trivial　solution　in　deterministic　polynomial　time,　argue　that　non-trivial　solutions　to　the　congruence　can　not　be　obtained　cyclically　from　the　trivial　solution,　and　infer　the　two　new　methods　of　seeking　the　trivial　solution.　The　paper　analyzes　the　time　complexity　of　a　probabilistic　algorithm　for　random　solutions　to　the　congruence　without　especially　large　p　and　n,　and　gives　the　discrete　logarithm　time　algorithm　for　non-trivial　solutions　to　the　congruence　with　especially　large　p　or　n.　Therefore,　the　paper　discusses　roundly　the　way　how　to　seek　solutions　to　the　congruence　xn　≡　a　(mod　p),　which　is　helpful　to　designing　new　public-key　cryptosystems.　©　2007　IEEE.