▁C ▁such ▁that ▁▁ ▁In ▁other ▁words , ▁these ▁are ▁hash ▁functions ▁where ▁the ▁relative ▁distance ▁between ▁the ▁input ▁values ▁is ▁preserved ▁in ▁the ▁relative ▁distance ▁between ▁the ▁output ▁hash ▁values ; ▁input ▁values ▁that ▁are ▁closer ▁to ▁each ▁other ▁will ▁produce ▁output ▁hash ▁values ▁that ▁are ▁closer ▁to ▁each ▁other . ▁ ▁This ▁is ▁in ▁contrast ▁to ▁crypt ographic ▁hash ▁functions ▁and ▁check sum s , ▁which ▁are ▁designed ▁to ▁have ▁random ▁output ▁difference ▁between ▁adjacent ▁inputs . ▁ ▁Loc ality ▁pres erving ▁hash es ▁are ▁related ▁to ▁space - fill ing ▁curves . ▁ ▁Am pl ification ▁ ▁Given ▁a ▁- sens itive ▁family ▁, ▁we ▁can ▁construct ▁new ▁families ▁ ▁by ▁either ▁the ▁AND - const ruction ▁or ▁OR - const ruction ▁of ▁. ▁ ▁To ▁create ▁an ▁AND - const ruction , ▁we ▁define ▁a ▁new ▁family ▁ ▁of ▁hash ▁functions ▁, ▁where ▁each ▁function ▁ ▁is ▁constructed ▁from ▁ ▁random ▁functions ▁ ▁from ▁. ▁ ▁We ▁then ▁say ▁that ▁for ▁a ▁hash ▁function ▁, ▁ ▁if ▁and ▁only ▁if ▁all ▁ ▁for ▁. ▁ ▁Since ▁the ▁members ▁of ▁ ▁are ▁independently ▁chosen ▁for ▁any ▁, ▁ ▁is ▁a ▁- sens itive ▁family . ▁ ▁To ▁create ▁an ▁OR - const ruction , ▁we ▁define ▁a ▁new ▁family ▁ ▁of ▁hash ▁functions ▁, ▁where ▁each ▁function ▁ ▁is ▁constructed ▁from ▁ ▁random ▁functions ▁ ▁from ▁. ▁ ▁We ▁then ▁say ▁that ▁for ▁a ▁hash ▁function ▁, ▁ ▁if ▁and ▁only ▁if ▁ ▁for ▁one ▁or ▁more ▁values ▁of ▁.