▁called ▁the ▁security ▁parameter . ▁ ▁Then ▁for ▁all ▁non - uniform ▁probabil istic ▁polynomial ▁time ▁algorithms ▁that ▁output ▁ ▁and ▁ ▁of ▁increasing ▁length ▁k , ▁the ▁probability ▁that ▁ ▁and ▁ ▁is ▁a ▁neg lig ible ▁function ▁in ▁k . ▁ ▁This ▁is ▁a ▁form ▁of ▁asympt otic ▁analysis . ▁It ▁is ▁also ▁possible ▁to ▁state ▁the ▁same ▁requirement ▁using ▁concrete ▁security : ▁A ▁commit ment ▁scheme ▁Comm it ▁is ▁ ▁secure , ▁if ▁for ▁all ▁algorithms ▁that ▁run ▁in ▁time ▁t ▁and ▁output ▁ ▁the ▁probability ▁that ▁ ▁and ▁ ▁is ▁at ▁most ▁. ▁ ▁Per fect , ▁statistical , ▁and ▁computational ▁hiding ▁ ▁Let ▁ ▁be ▁the ▁uniform ▁distribution ▁over ▁the ▁ ▁opening ▁values ▁for ▁security ▁parameter ▁k . ▁A ▁commit ment ▁scheme ▁is ▁respectively ▁perfect , ▁statistical , ▁or ▁computational ▁hiding , ▁if ▁for ▁all ▁ ▁the ▁probability ▁ens emb les ▁ ▁and ▁ ▁are ▁equal , ▁statist ically ▁close , ▁or ▁computation ally ▁ind ist ingu ish able . ▁ ▁Im poss ibility ▁of ▁univers ally ▁compos able ▁commit ment ▁schemes ▁ ▁It ▁is ▁impossible ▁to ▁realize ▁commit ment ▁schemes ▁in ▁the ▁universal ▁compos ability ▁( UC ) ▁framework . ▁The ▁reason ▁is ▁that ▁U C ▁commit ment ▁has ▁to ▁be ▁extract able , ▁as ▁shown ▁by ▁Can etti ▁and ▁F isch lin ▁and ▁explained ▁below . ▁ ▁The ▁ideal ▁commit ment ▁functionality , ▁denoted ▁here ▁by ▁F , ▁works ▁roughly ▁as ▁follows . ▁Comm itter ▁C ▁sends ▁value ▁m ▁to ▁F , ▁which ▁stores ▁it ▁and