it , ▁with ▁the ▁random ness ▁used ▁by ▁Comm it ▁const itut ing ▁the ▁opening ▁information . ▁ ▁If ▁the ▁commit ment ▁C ▁to ▁a ▁value ▁x ▁is ▁computed ▁as ▁C := Com mit ( x , open ) ▁with ▁open ▁the ▁random ness ▁used ▁for ▁computing ▁the ▁commit ment , ▁then ▁Check Re ve al ▁( C , x , open ) ▁simply ▁consists ▁in ▁ver ifying ▁the ▁equation ▁C = Com mit ▁( x , open ). ▁ ▁Using ▁this ▁notation ▁and ▁some ▁knowledge ▁about ▁mathematical ▁functions ▁and ▁probability ▁theory ▁we ▁formal ise ▁different ▁versions ▁of ▁the ▁binding ▁and ▁hiding ▁properties ▁of ▁commit ments . ▁The ▁two ▁most ▁important ▁combinations ▁of ▁these ▁properties ▁are ▁perfectly ▁binding ▁and ▁computation ally ▁hiding ▁commit ment ▁schemes ▁and ▁computation ally ▁binding ▁and ▁perfectly ▁hiding ▁commit ment ▁schemes . ▁Note ▁that ▁no ▁commit ment ▁scheme ▁can ▁be ▁at ▁the ▁same ▁time ▁perfectly ▁binding ▁and ▁perfectly ▁hiding ▁– ▁a ▁computation ally ▁un bounded ▁advers ary ▁can ▁simply ▁generate ▁Comm it ( x , open ) ▁for ▁every ▁value ▁of ▁x ▁and ▁open ▁until ▁finding ▁a ▁pair ▁that ▁outputs ▁C , ▁and ▁in ▁a ▁perfectly ▁binding ▁scheme ▁this ▁uniqu ely ▁ident ifies ▁x . ▁ ▁Comput ational ▁binding ▁Let ▁open ▁be ▁chosen ▁from ▁a ▁set ▁of ▁size ▁, ▁i . e ., ▁it ▁can ▁be ▁represented ▁as ▁a ▁k ▁bit ▁string , ▁and ▁let ▁▁ ▁be ▁the ▁corresponding ▁commit ment ▁scheme . ▁As ▁the ▁size ▁of ▁k ▁determ ines ▁the ▁security ▁of ▁the ▁commit ment ▁scheme ▁it ▁is