▁one - way ▁function ▁can ▁be ▁modified ▁( via ▁the ▁Gold reich - Le vin ▁theorem ) ▁to ▁possess ▁a ▁computation ally ▁hard - core ▁predicate ▁( while ▁retain ing ▁the ▁inject ive ▁property ). ▁Let ▁f ▁be ▁an ▁inject ive ▁one - way ▁function , ▁with ▁h ▁a ▁hard - core ▁predicate . ▁Then ▁to ▁commit ▁to ▁a ▁bit ▁b ▁Alice ▁pick s ▁a ▁random ▁input ▁x ▁and ▁sends ▁the ▁triple ▁ ▁to ▁Bob , ▁where ▁ ▁denotes ▁X OR , ▁i . e ., ▁bit wise ▁addition ▁mod ulo ▁ 2 . ▁To ▁de commit , ▁Alice ▁simply ▁sends ▁x ▁to ▁Bob . ▁Bob ▁ver ifies ▁by ▁computing ▁f ( x ) ▁and ▁comparing ▁to ▁the ▁committed ▁value . ▁This ▁scheme ▁is ▁conce aling ▁because ▁for ▁Bob ▁to ▁recover ▁b ▁he ▁must ▁recover ▁h ( x ). ▁Since ▁h ▁is ▁a ▁computation ally ▁hard - core ▁predicate , ▁recover ing ▁h ( x ) ▁from ▁ ▁f ( x ) ▁with ▁probability ▁greater ▁than ▁one - half ▁is ▁as ▁hard ▁as ▁in ver ting ▁f . ▁Per fect ▁binding ▁follows ▁from ▁the ▁fact ▁that ▁f ▁is ▁inject ive ▁and ▁thus ▁f ( x ) ▁has ▁exactly ▁one ▁pre image . ▁ ▁Bit - commit ment ▁from ▁a ▁pseudo - random ▁generator ▁ ▁Note ▁that ▁since ▁we ▁do ▁not ▁know ▁how ▁to ▁construct ▁a ▁one - way ▁permut ation ▁from ▁any ▁one - way ▁function , ▁this ▁section ▁reduces ▁the ▁strength ▁of ▁the ▁crypt ographic ▁assumption ▁necessary ▁to ▁construct ▁a ▁bit -