It　is　well　known　that　the　inverse　function　of　y　=　x　with　the　derivative　y＇　=　1　is　x　=　y,　the　inverse　function　of　y　=　c　with　the　derivative　y＇　=　0　is　nonexistent,　and　so　on.　Hence,　on　the　assumption　that　the　noninvertibility　of　the　univariate　increasing　function　y　=　(x)　with　x　＞　0　is　in　direct　proportion　to　the　growth　rate　reflected　by　its　derivative,　the　authors　put　forward　a　method　of　comparing　difficulties　in　inverting　two　functions　on　a　continuous　or　discrete　interval　called　asymptotic　granularity　reduction　(AGR)　which　integrates　asymptotic　analysis　with　logarithmic　granularities,　and　is　an　extension　and　a　complement　to　polynomial　time　(Turing)　reduction　(PTR).　Prove　by　AGR　that　inverting　y　equivalent　to　x(x)　(mod　p)　is　computationally　harder　than　inverting　y　equivalent　to　g(x)　(mod　p),　and　inverting　y　equivalent　to　g(xn)　(mod　p)　is　computationally　equivalent　to　inverting　y　equivalent　to　g(x)　(mod　p),　which　are　compatible　with　the　results　from　PTR.　Besides,　apply　AGR　to　the　comparison　of　inverting　y　equivalent　to　x(n)　(mod　p)　with　y　equivalent　to　g(x)　(mod　p),　y　equivalent　to　g(g1x)　(mod　p)　with　y　equivalent　to　g(x)　(mod　p),　and　y　equivalent　to　x(n)　+　x　+　1　(mod　p)　with　y　equivalent　to　x(n)　(mod　p)　in　difficulty,　and　observe　that　the　results　are　consistent　with　existing　facts,　which　further　illustrates　that　AGR　is　suitable　for　comparison　of　inversion　problems　in　difficulty.　Last,　prove　by　AGR　that　inverting　y　equivalent　to　x(n)g(x)　(mod　p)　is　computationally　equivalent　to　inverting　y　equivalent　to　g(x)　(mod　p)　when　PTR　cannot　be　utilized　expediently.　AGR　with　the　assumption　partitions　the　complexities　of　problems　more　detailedly,　and　finds　out　some　new　evidence　for　the　security　of　cryptosystems.　(C)　2011　Elsevier　B.V.　All　rights　reserved.