▁computational ▁resources ) ▁or ▁perfectly ▁conce aling ▁( it ▁is ▁impossible ▁for ▁Bob ▁to ▁find ▁out ▁the ▁commit ment ▁without ▁Alice ▁reve aling ▁it , ▁even ▁if ▁he ▁has ▁un bounded ▁computational ▁resources ) ▁but ▁not ▁both . ▁ ▁Bit - commit ment ▁in ▁the ▁random ▁oracle ▁model ▁ ▁Bit - commit ment ▁schemes ▁are ▁trivial ▁to ▁construct ▁in ▁the ▁random ▁oracle ▁model . ▁Given ▁a ▁hash ▁function ▁H ▁with ▁a ▁ 3 k ▁bit ▁output , ▁to ▁commit ▁the ▁k - bit ▁message ▁m , ▁Alice ▁generates ▁a ▁random ▁k ▁bit ▁string ▁R ▁and ▁sends ▁Bob ▁H ( R || m ). ▁The ▁probability ▁that ▁any ▁R , ▁m ▁exist ▁where ▁m ▁ ≠ ▁m ▁such ▁that ▁H ( R || m ) ▁= ▁H ( R || m ) ▁is ▁ ≈ ▁ 2 − k , ▁but ▁to ▁test ▁any ▁guess ▁at ▁the ▁message ▁m ▁Bob ▁will ▁need ▁to ▁make ▁ 2 k ▁( for ▁an ▁incorrect ▁guess ) ▁or ▁ 2 k - 1 ▁( on ▁average , ▁for ▁a ▁correct ▁guess ) ▁queries ▁to ▁the ▁random ▁oracle . ▁We ▁note ▁that ▁earlier ▁schemes ▁based ▁on ▁hash ▁functions , ▁essentially ▁can ▁be ▁thought ▁of ▁schemes ▁based ▁on ▁ideal ization ▁of ▁these ▁hash ▁functions ▁as ▁random ▁oracle . ▁ ▁Bit - commit ment ▁from ▁any ▁one - way ▁permut ation ▁ ▁One ▁can ▁create ▁a ▁bit - commit ment ▁scheme ▁from ▁any ▁one - way ▁function ▁that ▁is ▁inject ive . ▁The ▁scheme ▁re lies ▁on ▁the ▁fact ▁that ▁every