Your IP : 172.28.240.42
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<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1">
<style>
body {
background-color:#ffffff;
}
body, .cfsbdyfnt {
font-family: 'Oswald', sans-serif;
font-size: 18px;
}
h1, h2, h3, h4, h5, h5, .cfsttlfnt {
font-family: 'Playfair Display', serif;
}
.panel-title { font-family: 'Oswald', sans-serif; }
</style>
<title></title>
<style id="sitestyles">
@import url( solid #1b2a29}#outhdr .lr-borders{border-left:1px solid #609892;border-right:1px solid #609892;height:100%;max-height:3em;margin:15px 0}@media (max-width:767px){#outhdr .lr-borders{border-left:0 solid #609892}}a,a:hover{color:#379078;text-decoration:none}h2{color:#426965}.pagetitle h1{color:#00a097}#innersite{padding:0}.container-body{background:transparent!important}.btn-default{color:#fff!important;border-color:#426965!important;background-color:#426965!important}.btn-default:hover{color:#426965!important;background-color:#fff!important;border-color:#fff!important}.btn-primary{color:#426965!important;border-color:#426965!important;background-color:rgba(255,255,255,0)!important}.btn-primary:hover{color:rgba(255,255,255,0)!important;background-color:#426965!important;border-color:#426965!important}.btn-info{color:#fff!important;border-color:#000!important;background-color:#000!important}.btn-info:hover{color:#000!important;background-color:#fff!important;border-color:#fff!important}.btn-success{color:#000!important;border-color:#000!important;background-color:light!important}.btn-success:hover{color:light!important;background-color:#000!important;border-color:#000!important}.btn-white{color:#fff!important;border-color:#fff!important;background-color:rgba(255,255,255,0)!important}.btn-white:hover{color:rgba(255,255,255,0)!important;background-color:#fff!important;border-color:#fff!important}#inbdy .btn{border:2px solid;line-height:1.2;margin-left:10px;margin-right:10px}.btn-primary:hover{color:#fff!important}#site button,#site .btn,#site .btn-small,#site .btn-lg,#site .tmslider .btn{transition:all .8s ease;border-radius:25px;font-size:;padding:.5em .7em;letter-spacing:1px}#site .zonetools .btn,#site .edimg{transition:initial;border-radius:initial;font-size:14px;padding:2px 5px;letter-spacing:initial}#inbdy{max-width:1366px}.topstrip{color:#fff;background:#1b2a29;border-bottom:0 solid #379078}.topstrip .row{max-width:1366px;float:none;margin:auto}.topstrip a{color:#000}.topstrip a:hover{color:rgba(66,105,101,.85)}.topstrip .txttkn a{color:#426965}.topstrip .txttkn a:hover{color:rgba(66,105,101,.85)}.topstrip .addressitem{margin:20px 5px}@media (min-width:992px){.topstrip .addressitem .lbtel,.topstrip .addressitem .number,.topstrip .addressitem .vsep{display:none}.topstrip .addressitem [itemprop="streetAddress"]:after{content:" | "}}.topstrip [data-typeid="TextBlock"]{animation:slideInDown 2s ease}@media (max-width:767px){.topstrip [data-typeid="TextBlock"]{font-size:.7em}}.topstrip [data-typeid="TextBlock"] p{margin:15px 5px}@media (max-width:767px){.topstrip [data-typeid="TextBlock"] p{font-size:;padding-top:8px}}@media (max-width:767px){#block-inhdr .navbar-toggle,#block-outhdr .navbar-toggle{padding:4px 4px}#block-inhdr .navbar-toggle .icon-bar,#block-outhdr .navbar-toggle .icon-bar{width:18px}}#block-inhdr .btn-social,#block-outhdr .btn-social{color:#fff!important;background-color:transparent;transition:all .5s ease}#block-inhdr .btn-social:hover,#block-outhdr .btn-social:hover{transform:scale(1.5);background-color:transparent;color:#6da49e!important}.img-thumbnail{border:none;border-radius:0;padding:0}#inbdy .form-control{border-radius:0;background:rgba(255,255,255,.5);border:1px solid #609892;margin-top:.35em;margin-bottom:.35em}#inbdy [data-type="formblocks"] .fmname{display:none}#inbdy [data-type="formblocks"] .well{box-shadow:none;background:rgba(0,160,151,.1);border:none}.navbar-brand{color:#fff!important}.navbar-brand{display:none!important}.cfshznav a{letter-spacing:1px;color:#fff!important;border-top:4px solid transparent}.cfshznav a:hover{color:#fff!important;background:#609892!important;border-top:4px solid #1b2a29}.cfshznav .open a{color:#fff!important;background:#609892!important;border-top:4px solid #1b2a29}.cfshznav .open a:hover{border-top:4px solid #1b2a29}.cfshznav .dropdown-menu{padding-top:0;padding-bottom:0;background:rgba(255,255,255,.95)!important}.cfshznav .dropdown-menu li a{color:#426965!important;background:transparent!important;font-size:.9em;padding-left:20px;padding-right:20px;padding-top:12px!important;padding-bottom:10px!important;text-transform:uppercase;border-top:0 solid transparent;border-left:-1px solid transparent;border-right:1px solid transparent;transition:background-color .2s}.cfshznav .dropdown-menu li a:hover{color:#426965!important;box-shadow:unset;border-left:5px solid #00a097;padding-left:15px;border-top:0 solid #609892}.navbar{background-color:#fff!important;border:0 solid #fff!important}.navbox{background-color:transparent!important}.js-clingify-locked .navbar{background-color:#fff!important;border:0 solid #fff!important}.js-clingify-locked .navbox{background-color:transparent!important}.navbarlocked{height:unset!important}.navbarlocked .dropdown-menu li a{background:#fff}#inhdr .upperbanner img{max-height:80px}@media (max-width:767px){#inhdr .upperbanner img{max-height:50px}}#strip{background:#fff!important}#strip [data-type="image"]{max-height:10em;overflow:hidden}#strip .page-title{text-shadow:none;background:rgba(66,105,101,.6)}#strip .page-title h1{color:#fff;margin:auto auto}@media (max-width:767px){#strip .page-title h1{font-size:}}.section-strip-item{color:#00a097!important}.section-strip-item a{color:#00a097!important}[data-typeid="inlinesearch"] input{border:1px solid #426965;border-radius:20px;height:40px;box-shadow:none;background:#3afff4;max-width:420px;float:right;margin:auto;margin-bottom:10px}[data-typeid="inlinesearch"] input .form-control{color:#fff!important}.homeobit-box{color:#000;padding-top:5px;padding-bottom:5px;max-width:1366px;float:none;margin:auto}.homeobit-box a,.homeobit-box a:hover,.homeobit-box p,.homeobit-box h1,.homeobit-box h2,.homeobit-box h3,.homeobit-box h4{color:#1b2a29!important}.homeobit-box .obpgimg{transition:all 2s ease!important;border-radius:10px!important}.homeobit-box .obphlst{transition:all 2s ease!important;border-radius:10px!important;padding:0px!important;margin-left:0;margin-right:0;box-shadow:0 0 0 #888!important;border:0 solid!important}.homeobit-box .obphlst:hover{transform:scale(1.2)}.homeobit-box .{padding-bottom:100%;padding-left:92%;margin:auto;border-radius:10px!important}.homeobit-box .form-control{background:rgba(255,255,255,.9)!important}.obslide{background:rgba(0,0,0,.1)}.obslide .details .obitdate{color:#fff!important}.obslide .details .obitdate a{color:#fff!important}.obitname,.obitdate{color:#379078}.obitname{font-weight:700;text-transform:uppercase}.horizobits{margin:0 }.glyphicon-chevron-right,.glyphicon-chevron-left{color:}.glyphicon-chevron-right:hover,.glyphicon-chevron-left:hover{color:}[data-typeid="locationmap"]{background:#609892}[data-typeid="locationmap"] iframe{border:none;filter:grayscale(1.9) sepia(2%) opacity(.85);transition:all 2s ease}[data-typeid="locationmap"] iframe:hover{filter:unset}[data-typeid="multimap"]{background:transparent}[data-typeid="multimap"] .multimap{border:0 solid #ccc;background:#609892}[data-typeid="multimap"] .multimap .leaflet-tile-pane{-webkit-filter:opacity(.85) grayscale(60%) brightness(1.1);-moz-filter:opacity(.85) grayscale(60%) brightness(1.1);filter:opacity(.85) grayscale(60%) brightness(1.1);transition:all .5s ease}[data-typeid="multimap"] .multimap:hover .leaflet-tile-pane{-webkit-filter:opacity(1) grayscale(0%) brightness();-moz-filter:opacity(1) grayscale(0%) brightness();filter:opacity(1) grayscale(0%) brightness()}[data-typeid="multimap"] .multimap .leaflet-marker-pane .leaflet-marker-icon:hover{filter:brightness()}[data-typeid="multimap"] .multimap .leaflet-popup{border:2px solid mediumblue}[data-typeid="multimap"] .multimap .leaflet-popup h4{color:mediumblue;font-weight:700;font-size:;text-align:center}[data-typeid="multimap"] .multimap .leaflet-popup .leaflet-popup-content-wrapper{background:linear-gradient(rgba(255,255,255,.7),white);border-radius:0;box-shadow:none}[data-typeid="multimap"] .multimap .leaflet-popup .leaflet-popup-tip{background:rgba(255,255,255,.8);border-bottom:2px solid mediumblue;border-right:2px solid mediumblue;display:none}[data-typeid="multimap"] .multimap button{background:#888}[data-typeid="multimap"] .multimap button:hover{background:mediumblue}[data-typeid="multimap"] .multimap-location{border:none;border-top:4px solid #ccc;border-radius:0;background:#eee;margin-top:5px}[data-typeid="multimap"] .multimap-location h4{color:#000;font-weight:700}[data-typeid="multimap"] .multimap-location:hover{background:radial-gradient(#fff,#eee);border-top:4px solid #888}[data-typeid="multimap"] .{background:rgba(238,238,238,.5);border-top:4px solid #c00}[data-typeid="multimap"] .multimap-location button{color:white;background:#888;border-radius:0;margin-bottom:10px}[data-typeid="multimap"] .multimap-location button:hover{background:mediumblue}.edgetoedge{margin-left:-100vw;margin-right:-100vw;margin-bottom:0;padding-left:100vw;padding-right:100vw;padding-top:5px;padding-bottom:5px}.edgetoedge .tools{margin-left:100vw;margin-right:100vw}.edgetoedge .inner .tools{margin-left:0vw;margin-right:0vw}.edgetoedge2{margin-left:-100vw;margin-right:-100vw;margin-bottom:0;padding-left:100vw;padding-right:100vw}.edgetoedge2 .tools{margin-left:100vw;margin-right:100vw}.edgetoedge2 .inner .tools{margin-left:0vw;margin-right:0vw}.pale-col{color:#000;background-color:!important}.color-col{background-color:#426965!important}.color-col p,.color-col h1,.color-col h2,.color-col h3,.color-col h4{color:#fff}.footer{background-color:#1b2a29!important}.footer [data-typeid="sitemap"] div a:nth-child(4){display:none}.footer p,.footer .addressitem{color:#fff}.footer h1,.footer h2,.footer h3,.footer h4,.footer .form-group{color:#b2e1d5}.footer a{color:#fff}.footer-box .row{padding:0}.footer-box .semiopaque{background-color:rgba(66,105,101,0);min-height:300px;animation:slideInUp 2s ease}.footer-box .semiopaque p,.footer-box .semiopaque h1,.footer-box .semiopaque h2,.footer-box .semiopaque h3,.footer-box .semiopaque h4{color:#fff}.sitemapsubitem{display:none}.sitemapitem{display:inline;padding:0}.panel-success .panel-heading{background-color:#426965!important}.panel-success .panel-title{color:#fff}.panel-success .panel-body{border-left:1px solid #426965!important;border-right:1px solid #426965!important;border-bottom:1px solid #426965!important}.cfsacdn .panel-title{background:transparent}.cfsacdn .panel-title a{color:#fff!important}.cfsacdn .panel-heading{background:#379078!important}.cfsacdn .panel{border-color:#379078!important}.cfsacdn .panel font{color:!important}.blackbg{background:#609892}.max1570{max-width:1570px!important;float:none!important;margin:auto!important}.max1470{max-width:1470px!important;float:none!important;margin:auto!important}.max1370{max-width:1370px!important;float:none!important;margin:auto!important}.max1270{max-width:1270px!important;float:none!important;margin:auto!important}.max1170{max-width:1170px!important;float:none!important;margin:auto!important}.site-credit .credit-text,.site-credit .credit-text a{background-color:transparent;color:#000}.obitlist-title a{color:#000}{color:}{color:#000}{color:#000}#popout-add h4,#popout-settings h4{color:#fff}.btn-danger{color:#fff!important;border-color:#5cb85c!important;background-color:#5cb85c!important}.btn-danger:hover{color:#5cb85c!important;background-color:#fff!important;border-color:#fff!important}.max1570{max-width:1570px!important;float:none!important;margin:auto!important}.max1470{max-width:1470px!important;float:none!important;margin:auto!important}.max1370{max-width:1370px!important;float:none!important;margin:auto!important}.max1270{max-width:1270px!important;float:none!important;margin:auto!important}.max1170{max-width:1170px!important;float:none!important;margin:auto!important}.upperbanner{background-color:#fff;padding-top:0;padding-bottom:5px;border-top:0 solid #379078;border-bottom:0 solid #379078}.upperbanner p{color:#000;animation:slideInLeft 2s ease}.upperbanner a{color:#426965}.upperbanner a:hover{color:rgba(66,105,101,.7)}.cta-box{background:#2e4a47!important}.cta-box p{color:#fff}.cta-box a{color:#fff}.cta-box a:hover{color:#379078}.js-clingify-locked .upperbanner{background-color:#fff;max-width:100vw;float:none;margin:auto}#outhdr .navbar{background:#fff;background:transparent}#outhdr .navbar a{color:#fff!important;border:0 solid transparent;transition:background-color .4s;transition:all .4s ease-in-out;padding-top:!important;padding-bottom:!important}#outhdr .navbar {font-weight:bold!important;letter-spacing:1px}@media (max-width:991px){#outhdr .navbar a{font-size:.75em!important}#outhdr .navbar {padding:25px 10px 20px 10px!important}}@media (max-width:767px){#outhdr .navbar a{padding-top:14px!important}}#outhdr .navbar a:hover{color:#426965!important;background:#d6f0e9!important;border:0 solid #379078}#outhdr .navbar .open a:hover{background-color:#fff!important}#outhdr .navbar .open {color:#426965!important;background-color:#d6f0e9!important}#outhdr .navbar .dropdown-menu{padding-top:0;padding-bottom:0;background:rgba(255,255,255,.95)!important}#outhdr .navbar .dropdown-menu li a{color:#426965!important;background:transparent!important;font-family:helvetica,sans-serif;font-size:.8em;padding-left:20px;padding-right:20px;padding-top:12px!important;padding-bottom:10px!important;text-transform:uppercase;border:0 solid #379078;border-left:0 solid transparent;transition:background-color .2s}#outhdr .navbar .dropdown-menu li a:hover{color:#fff!important;background:#8dd3c0!important;border:0 solid #379078;border-left:5px solid #379078;padding-left:15px}#outhdr .navbar {background:none!important;border:#fff!important;outline:#fff!important}#outhdr .navbar-brand{display:none!important}#outhdr .cfshznav{background:#426965}#outhdr .cfshznav .nav{padding:0 0 0 0}@media (max-width:991px){#outhdr .cfshznav .nav>:nth-child(4){display:none}}#outhdr .cfshznav .nav>:nth-child(4) a{color:rgba(255,255,255,0)!important;background:url();background-repeat:no-repeat;background-size:84%;width:240px;height:155px;color:rgba(255,255,255,0);font-size:0;background-position:center;padding-bottom:30px}#outhdr .cfshznav .nav>:nth-child(4) a:hover{background:url(),transparent!important;background-size:89%!important;background-repeat:no-repeat!important;background-position:center!important}#outhdr .cfshznav .nav>:nth-child(4):hover{background:transparent!important}#outhdr .js-clingify-locked{background:#426965!important}#outhdr .js-clingify-locked .navbar{background:#426965!important}#outhdr .js-clingify-locked .nav{padding:5px 0 0 0}#outhdr .js-clingify-locked .nav a{color:#fff!important;padding-top:2em!important;padding-bottom:!important;margin-bottom:0px!important}@media (max-width:991px){#outhdr .js-clingify-locked .nav>:nth-child(4){display:none}}#outhdr .js-clingify-locked .nav>:nth-child(4) a{color:rgba(255,255,255,0)!important;background:url(background:url();background-repeat:no-repeat;background-size:contain;width:150px;height:60px;color:rgba(255,255,255,0);font-size:0;margin-top:10px;background-position:center;margin-bottom:5px;border-radius:0%;bottom:0;padding-bottom:0}#outhdr .js-clingify-locked .nav>:nth-child(4):hover{background:transparent!important}.mobile-logo{background:#426965}@media (max-width:991px){.sidr-inner .sidr-class-nav>:nth-child(5){display:none}}.cta-box{background-color:#426965}.cta-box p{color:#fff}.cta-box a{color:#fff}.cta-box a:hover{color:#379078}[data-typeid="popoutnotice"] .popout-notice .widget-label{background:yellow;color:green;padding:10px}[data-typeid="popoutnotice"] .popout-notice .widget-label:after{content:""}.cfs-popout{background:linear-gradient(120deg,#2e4a47,#568883 120%)!important;color:#fff;max-width:280px;padding:10px;border:0;border-left:8px solid #379078;outline:0 solid rgba(255,255,255,.2);outline-offset:0;box-shadow: .25em 1em rgba(0,0,0,.1)}.cfs-popout .close{width:;height:;text-shadow:none;color:#fff;opacity:1;padding:5px;margin:3px;background:#90374f;border-radius:100%;border:1px solid rgba(255,255,255,.3);font-family:raleway,sans-serif;font-size:75%;z-index:1}.cfs-popout .content-area .title{border-bottom:1px solid rgba(255,255,255,.2);padding-bottom:10px;margin-top:2px;margin-bottom:10px;line-height:auto;opacity:1}.cfs-popout .content-area h3{font-weight:700;transition:all 1s ease;animation:pulse ease-in-out;animation-delay:3s}.cfs-popout .content-area h3:hover{text-shadow:0 0 2em #000}.cfs-popout .clickable{font-style:italic;border:1px solid #fff;display:inline-block;padding:4px 10px 6px;opacity:.5;transition:all .5s ease}.cfs-popout .clickable:hover{opacity:1}
#obitlist .row {
border: 0px;
border-bottom: 1px solid #a0fffa;
border-radius: 0px;
padding: 2em;
}
#obitlist .obphlst {
border-radius: 0px;
border: 0px solid #E0D9D9 !important;
padding: 0px;
box-shadow: 1px 1px 1px 1px rgba(50,50,50,0) !important;
background: #fff;
}
</style>
<style> #smart2881336973111-1 { color: !important; background-color: } #smart2881336973111-1:hover { color: !important; background-color: } #smart2881336973111-2 { color: !important; background-color: } #smart2881336973111-2:hover { color: !important; background-color: } #smart2881336973111-3 { color: !important; background-color: } #smart2881336973111-3:hover { color: !important; background-color: } </style>
<style scoped="">
#smart138401661026 .toplevel {
font-size: 15px;
padding: 20px 18px;
font-weight: normal;
}
#smart138401661026 .navbar-default .navbar-nav > li > a {
text-transform: uppercase;
}
</style>
<style>
/* Default arrow for menu items with submenus */
.sidr-class-dropdown > a::after {
content: '\25B6'; /* Unicode for a right-pointing triangle */
position: absolute;
right: 30px;
color: white;
transition: transform ;
}
/* Arrow rotates down when the submenu is open */
. > a::after {
content: '\25BC'; /* Unicode for a down-pointing triangle */
transform: rotate(0deg); /* Reset rotation */
}
/* Hide Sidr menu if the screen width is greater than 768px */
@media (min-width: 769px) {
#sidr-main-mn966128 {
display: none !important;
}
}
</style>
<style scoped="">
#smart3739698360101 .toplevel {
font-size: 15px;
padding: 20px 18px;
font-weight: normal;
}
#smart3739698360101 .navbar-default .navbar-nav > li > a {
text-transform: uppercase;
}
</style>
<style>
/* Default arrow for menu items with submenus */
.sidr-class-dropdown > a::after {
content: '\25B6'; /* Unicode for a right-pointing triangle */
position: absolute;
right: 30px;
color: white;
transition: transform ;
}
/* Arrow rotates down when the submenu is open */
. > a::after {
content: '\25BC'; /* Unicode for a down-pointing triangle */
transform: rotate(0deg); /* Reset rotation */
}
/* Hide Sidr menu if the screen width is greater than 768px */
@media (min-width: 769px) {
#sidr-main-mn184060 {
display: none !important;
}
}
</style>
<style> #smart2333938227047-1 { color: !important; background-color: } #smart2333938227047-1:hover { color: !important; background-color: } #smart2333938227047-2 { color: !important; background-color: } #smart2333938227047-2:hover { color: !important; background-color: } #smart2333938227047-3 { color: !important; background-color: } #smart2333938227047-3:hover { color: !important; background-color: } </style>
</head>
<body class="cs56-229">
<br>
<div id="site" class="container-fluid">
<div id="innersite" class="row">
<div id="block-outhdr" class="container-header dropzone">
<div class="row stockrow">
<div id="outhdr" class="col-xs-12 column zone">
<div class="inplace top-border" data-type="struct" data-typeid="FullCol" data-desc="Full Col" data-exec="1" id="struct1326593510923" data-o-bgid="" data-o-bgname="" data-o-src="">
<div class="row">
<div class="col-sm-12 column ui-sortable">
<div class="inplace cta-box" data-type="struct" data-typeid="FullCol" data-desc="Full Col" data-exec="1" id="struct735952154750">
<div class="row">
<div class="col-sm-12 column ui-sortable">
<div class="inplace" data-type="struct" data-typeid="Thirds2-1" data-desc="Thirds 2-1" data-exec="1" id="struct5203190405039">
<div class="row">
<div class="col-xs-4 column ui-sortable">
<div class="inplace pad-left pad-right smallmedia text-center pad-top pad-bottom" data-type="smart" data-typeid="socialmedia" data-desc="Social Media & Links" data-rtag="socialmedia" id="smart2881336973111" data-itemlabel="">
<div class="smbuttons"> <span class="btn btn-social btn-facebook"></span>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="inplace hidden-md hidden-lg mobile-logo" data-type="struct" data-typeid="ThreeCols" data-desc="Three Cols" data-exec="1" id="struct361897052728" data-o-bgid="" data-o-bgname="" data-o-src="" style="">
<div class="row">
<div class="col-sm-4 column ui-sortable"></div>
<div class="col-sm-4 col-xs-4 column ui-sortable">
<div class="inplace pad-left pad-right hidden-md hidden-lg pad-top pad-bottom" data-type="image" data-typeid="site" data-desc="Site Image" id="image3805680664636" style="" data-itemlabel=""><img alt="site image" class="img-responsive" src="" style="">
<div contenteditable="false" style="height: 0px;"></div>
</div>
</div>
<div class="col-sm-4 col-xs-8 column ui-sortable">
<div class="inplace menu-ip hidden-sm hidden-md hidden-lg transparent-menu" data-type="smart" data-typeid="menu" data-desc="Menu Bar" data-exec="1" data-rtag="menu" id="smart138401661026" data-itemlabel="" style="position: relative; z-index: 30; left: 0px; top: 0px;" data-rsttrans="1">
<div style="position: relative; z-index: 3;">
<div class="cfshznav" id="navbar-mn966128">
<div class="navbar cfsbdyfnt navbar-default" role="navigation"><br>
<div id="mn966128" class="navbar-collapse collapse mnujst centered">
<ul class="nav navbar-nav mnujst centered">
<li id="li-1-2" class="dropdown navbox"><span class="dropdown-toggle toplevel navlink ln-listings"></span>
<ul class="dropdown-menu">
<li class="navbox" id="li-1-2-0">
<span class="navlink ln-listings">Modular exponentiation recursive. As a result, we get (a b .</span>
</li>
<li class="navbox" id="li-1-2-1">
<span class="navlink ln-listings"><br>
</span>
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<span class="dropdown-toggle toplevel navlink ln-about-us">Modular exponentiation recursive We present a fast algorithm for modular exponentiation when the factorization of the modulus is known. Mar 13, 2012 · This is known as Exponentiation by repeated squaring (see also Modular exponentiation) It deserves to be better known that this arises simply from writing the exponent in binary radix in Horner polynomial form, i. 2 11 = 2 5 x 2 5 x 2. #Modular Arithmetic In this Sep 21, 2015 · When I first got my hands on this code, I had no idea how it worked. The modular exponentiation is useful before the size of the result is bounded. Important practical applications include public-key cryptography (RSA). The Steps are as follows: Declare a function modularExponentiation which has three arguments ‘X’, ‘N’, and ‘M’. As a result, we get (a b Modular Exponentiation - Using Recursion - Coding With Mr. $\rm\ d_0 + 2\, (d_1 + 2\, (d_2\ +\:\cdots)). Solving the problem taught me some fundamental facts about modulus, cryptography and recursion. For example, let’s see what happens for each of the possible bases modulo 7: Because exponentiation by positive integers corresponds to repeated multiplication, which behaves “nicely” with modular arithmetic, the list below covers all possible integers. For a negative exponent b , the definition can be extended by finding the modular multiplicative inverse d of a modulo m , that is Oct 4, 2018 · I see that you're implementing binary exponentiation, with the extra feature that it's reduced mod r. **Recursive Case**: The ongoing call to the function with a simpler input. This method for computing x n {\displaystyle x^{n}} in group G , where n is a natural integer, whose algorithm is given below, is using the following equality recursively: May 28, 2017 · Quick tutorial on doing modular exponentiation in Java in O(log(b)) time Rate Like Subscribe This page covers computational techniques for modular arithmetic operations and efficient exponentiation algorithms, with a particular focus on binary exponentiation (also known as exponentiation by s Explanation of this recursive function of Fast_Modular_Exponentiation 原文 2020-08-01 18:05:21 4 2 c / performance / recursion / exponentiation / modular Dec 9, 2021 · Recursive Approach. Jérôme Richard. Math; Advanced Math; Advanced Math questions and answers; Q4. Thus, every 2 recursive calls result in dividing the exponent by 2. Follow edited Aug 1, 2020 at 18:24. I would like to present my Check out TUF+:https://takeuforward. Given three numbers a, b and c, we need to find (ab) % cNow why do ?% c? after exponentiation, because ab will be really large even for relatively small values of a, b and that is a problem because the data type of the language that we try to code the problem, will most probably not let us store such a large number. Related article — Modulo 1⁰⁹+7 — [Notes] Fast Modular Exponentiation - Recursive Implementation is a free tutorial by Andrei Chiriac from Coding Interview courseLink to this course(Special Discount) May 14, 2024 · Even though, both the iterative and recursive approach have the same time complexity, the iterative approach is still faster because there are no overheads for recursive calls. In/2), m) mod m. CSES - Easy. Iterative modular binary exponentiation. Apr 18, 2024 · Modular exponentiation (Recursive) Given three numbers a, b and c, we need to find (ab) % cNow why do “% c†after exponentiation, because ab will be really large even for relatively small values of a, b and that is a problem because the data type of the language that we try to code the problem, will most probably not let us store suc Fast modular exponentiation efficiently computes $\mod{a^k}{m}$. Induction lets us prove statements about all natural numbers. Property 1: The Euclidean method was first introduced in Efficient exponentiation using precomputation and vector addition chains by P. It will never produce a number larger than the modulus. C++ source code to compute modular exponetiation recursively. 50. 2 2 = 2 1 x 2 1. Using induction Using induction in formal and English proofs. Finding a^b mod m is the Modular Exponentiation. h> Hi! I have been tasked with writing a MIPS program that prompts the user for three positive numbers x, n, and p, and outputs x^n mod p. 2 1 = 2 0 x 2 0 x 2. Fast Modular Exponentiation (top down recursive way) 255 = (227)2 x 2 = ((213)2 x 2)2 x 2 By Hand… 255 = (210)525 fundamentally, by hand, you can break this down any which way you want, such that the exponents add up to the proper value. Let a,n,m be positive integers and suppose m factors canon-ically as Q k i=1 p e i i. Recursive modular binary exponentation. Abstract. Follow. Thus, the stack depth is at most O(log exp) which is quite manageable. Mathematical induction A method for proving statements about all natural numbers. Jul 24, 2023--Listen. Follow the steps below : If N = 0, the result is always 1 because any non zero number raised to the power of 0 is 1. Resources. If ‘N’ == 0 then return 1. Sep 11, 2013 · Modular exponentiation only gives you the remainder of x to the y over z, Recursive exponentiation. It turns out that one prevalent method for encryption of data (such as credit card numbers) involves modular exponentiation, with very big exponents. There are two approaches for this - recursive and iterative. Division Input two positive integers a and b. See complete series on recursion herehttp://www. procedure mpower(b, n, m: integers with b > 0 and m > 2, n > 0) if n O then return 1 else if n is even then return mpower(b, n/2, m)2 mod m else return (mpower(b, [n/2), m)2 mod m·b Question: Q4. A recursive approach is one in which the recursive function calls itself with slightly smaller parameter, until the base case is reached. e b in a b a^b a b, turns 0. In the fast exponentiation strategy developed in this section we write any powers such that it can be computed as a product of powers obtained with repeated squaring. Then we can compute the modular exponentiation an (mod m) in O(max(e i/t i) + P k i=1 t i Aug 1, 2020 · recursion; exponentiation; modular; Share. Recursive exponentiation is a method used to efficiently compute A N, where A & N are integers. Choose integer parameters t i ∈[0,e i] for 1 ≤i ≤k. cpp Non-recursive modular exponentiation PRNGs using counters I really hope you found a helpful solution! ♡The Content is licensed under CC BY-SA(https://meta Aug 1, 2024 · Exponentiation, or raising a number to a power, is a fundamental operation in mathematics and computer science. I found it in a forum with a title, “Faster Approach to Modular Exponentiation”. The reason why such “clever” and more optimized approaches are necessary is because naive methods, like the one listed below, simply take too much memory. Trace Algorithm 4 to compute 316 mod 13 ALGORITHM. AN ELEMENTARY METHOD FOR FAST MODULAR EXPONENTIATION WITH FACTORED MODULUS ANAY AGGARWAL, MANU ISAACS Abstract. The modular multiplicative inverse of a is an integer 'x' such that. }\) Jan 8, 2021 · So we will use recursion to calculate ‘X’ ^ ‘N’ by dividing ‘N’ into two equal parts in each recursive call. It leverages recursion to break down the problem into smaller subproblems. , Recursive Version: Time Complexity: O(logN) & Space Complexity: O(logN) as recursion takes stack space. Lecture 14 ak mod m 2. AshIn this lesson, we will see an efficient recursive algorithm to calculate (x^n)%M - (x to powe Oct 3, 2023 · Modular exponentiation (Recursive) Given three numbers a, b and c, we need to find (ab) % cNow why do “% c†after exponentiation, because ab will be really large even for relatively small values of a, b and that is a problem because the data type of the language that we try to code the problem, will most probably not let us store suc Notice that each time the exponent is odd, it spurs a recursive call to an even exponent, and then each time a recursive call is even, it creates a recursive call to an exponent that is half in value. This is a common requirement in cryptography problems. In Section 11. Feb 25, 2025 · Modular Exponentiation. Engineering; Computer Science; Computer Science questions and answers; Q4. e. This led to the development of modular exponentiation, which allows a more efficient calculation procedure for this situation. Oct 12, 2023 · Recently I solved a problem called modular exponentiation at CodeAbbey. There is often a need to efficiently calculate the value of x n mod m. Since then I have been using this code. In such a case, we simply return 1 irrespective of a. We need to know about bitwise operations before we learn Repeated Squaring Apr 16, 2014 · I have implemented a non-recursive modular exponentiation typedef long long uii; uii modularExponentiation(uii base,uii exponent,uii p) { int result= 1; base = base % p; while( exponen Modular exponentiation is the process of repeatedly squaring and reducing a number modulo some integer, and then combining the results to find the required answer. b mod m) mod m (output is b'mod m) Apr 18, 2024 · Another efficient approach : Recursive exponentiation. One of the most effective methods to compute large powers efficiently is Fast Exponentiation, also known […] Oct 15, 2012 · See complete series on recursion herehttp://www. Output two integers q and r such that a = bq + r, and 0 <= r < b Binary Exponentiation Binary Exponentiation Table of contents Modular arithmetic Modular The following recursive approach expresses the same idea: Feb 6, 2022 · Modular Exponentiation means finding the exponentiation under modulo i. Improve this question. D Rooij. Use Cases of Binary Exponentiation in Competitive Programming: 1. Example 3. Share. In *fast exponentiation*, recursion helps significantly reduce the time complexity by simplifying the problem iteratively instead of tackling the whole problem at once. 1. (We can use the same trick when exponentiating integers, but then the multiplications are not modular multiplications, and each multiplication takes at least twice as long as the previous one. 2 5 = 2 2 x 2 2 x 2. Sep 20, 2023 · The parameters x and N are misleading because they don't change between recursive calls to the function, so they can be thought of as global variables or inline constants that aren't actually impacting how many recursive calls are made. ) UVA- 374 (Big Mod) Recursive Modular Exponentiation bigmod , uva Edit At first check it : Big Mod Tutorial in Bangla #include <iostream> #include <bits/stdc++. For binary exponentiation, our base case will be when our exponent, i. Trace Algorithm 4 to compute 316 mod 13 ALGORITHM 4 Recursive Modular Exponentiation. This can be done in O(logn) time using the following recursion: For a positive exponent b, the modular exponentiation c is defined as c = a b mod m . \, $ Below is an example of computing $\ x^{25}\ $ by repeated squaring. Here’s a simple, typed example. Apr 15, 2024 · Modular exponentiation (Recursive) Given three numbers a, b and c, we need to find (ab) % cNow why do “% c†after exponentiation, because ab will be really large even for relatively small values of a, b and that is a problem because the data type of the language that we try to code the problem, will most probably not let us store suc Fast Modular Power¶ The modular exponentiation of a number is the result of computing an exponent followed by getting the remainder from division. com/playlist?list=PL2_aWCzGMAwLz3g66WrxFGSXvSsvyfzCOIn this lesson, we have described two different r In modular arithmetic (and computational algebra in general), you often need to raise a number to the n n n-th power — to do modular division, perform primality tests, or compute some combinatorial values — ­and you usually want to spend fewer than Θ (n) \Theta(n) Θ (n) operations calculating it. 4 . a x ? 1 (mod prime) Examples: Input : n = 10, prime = 17 Output : 1 9 6 13 7 3 5 15 2 12 Ex Modular exponentiation is exponentiation performed over a modulus. Repeated Squaring Algorithm has the same complexity as D&C, but it is iterative, thus does not suffer from recursion overhead. As far as the assembly code is concerned, recursion (in such an environment with a call stack) is just one function calling another function (which could be the same function or could be a different function). What you may want to do is take a normal (tail recursive) binary exponentiation algorithm and simply change the 2-ary functions + and * to your own user defined 3-ary functions +/mod and */mode which also take r and reduce the result mod r before returning it. Modular exponentiation A fast algorithm for computing . It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie–Hellman key exchange and RSA public/private keys. procedure mpower(b, n, m: integers with b > 0 and m > 2, n > 0) if n = ( then return 1 else if n is even then return mpower(b, n/2, m)2 mod m else return (mpower(b, [n/2 Jun 27, 2014 · Simple modular multiplication. We computer the value of a x/2 each time till x becomes 0. Trace Algorithm 4 to compute 290 mod 13 ALGORITHM 4 Recursive Modular Exponentiation. Modular equations A quick review of . Let a,n,m be positive integers and suppose m factors canonically as Q k i=1 p e i i. Example: a = 5, b = 2, m = 7 (5 ^ 2) % 7 = 25 % 7 = 4. Mar 6, 2023 · Give a positive integer n, find modular multiplicative inverse of all integer from 1 to n with respect to a big prime number, say, 'prime'. why Recusrsive modular exponentiation not equals iterative? 1. Jul 18, 2023 · · Recurrence relation ∘ Example — Reverse an array (Decrease by a constant) · Methods to solve recurrence relations ∘ Recursion Tree Method ∘ TL;DR ∘ Master Theorem · Fibonacci Sequence complexity analysis · Pow(x,n) using recursion and complexity analysis — [Notes] · Modular exponentiation using Recursion and complexity Answer to Q4. procedure mpowerb, n,m:integers with b > 0 and m > 2,120) if n=0 then return 1 else if n is even then return mpower b. org/plus?source=youtubeFind DSA, LLD, OOPs, Core Subjects, 1000+ Premium Questions company wise, Aptitude, SQL, AI doubt The modular exponentiation with divide-and-conquer approach is the top choice because it’s highly efficient—O(n) time, where n is the length of the exponent array—and uses O(1) space beyond recursion, leveraging Euler’s theorem and modular properties to handle large exponents. Both ways are fine. Example proofs by induction Example proofs about sums and divisibility. 2 on binary numbers, we saw that every natural number can be written as a sum of powers of \(2\text{. Exponentiation. 5. Fast modular exponentiation can be carried out recursively based on the fact that: a e = a e/2 x a e/2 (when e is even) and; a e = a e - 1 * a (when e is not even) The recursive method breaks the exponentiation down into simpler and simpler computations: Define function modExp(base, exponent, modulus) Set base case - if exponent = 1, return 1 Jun 1, 2023 · Extremely fast method for modular exponentiation with modulus and exponent of several million digits Feb 17, 2025 · Modular exponentiation (Recursive) Given three numbers a, b and c, we need to find (ab) % cNow why do “% c†after exponentiation, because ab will be really large even for relatively small values of a, b and that is a problem because the data type of the language that we try to code the problem, will most probably not let us store suc (a b) % m in O(log 2 b) time complexity (Modular Exponentiation): Recursive approach: Let’s try to analyze the situation with an example. youtube. A recursive algorithm for fast modular exponentiation in MIPS - marcusm009/fast-modular-exponentiation Jul 24, 2023 · Modular exponentiation using Recursion — [Notes] Tarun Jain. Choose integer parameters t i ∈[1,e i] for This is a really useful function that I thought needed to be explained. For a = 2, b = 22, 2 22 = 2 11 x 2 11. Trace Algorithm 4 to compute 290 mod 13 ALGORITHM. When dealing with security methods, like RSA or the Diffie-Hellman, or finding primes Fast Exponentiation Algorithm 81453 in binary is 10011101000101101 81453 = 2 16 + 2 13 + 2 12 + 2 11 + 2 9 + 2 5 + 2 3 + 2 2 + 2 0 The fast exponentiation algorithm computes ? mod ˛ using ≤ 2log &multiplications mod ˛ a81453 = a 216 +2 13 +2 12 +2 11 +2 9+2 5+2 3+2 2+2 0 = a 216 · a 213 · a 212 · a 211 · a 29 · a 25 · a 23 · a 22 Modular Exponentiation. By using the Repeated Squaring algorithm, we can find the Modular Exponentiation faster than D&C. Using the original recursive We can do a modular exponentiation calculation by hand, by working out the sequence of values of A, and then calculating g A mod n for each of the A, starting with the smallest (which is g 0 = 1). com/playlist?list=PL2_aWCzGMAwLz3g66WrxFGSXvSsvyfzCOIn this lesson, we will see an efficient recursiv Fast Modular Exponentiation The first recursive version of exponentiation shown works fine, but is very slow for very large exponents. n/2m) mod m else return (mpowerb. Examples:The idea is based on below properties. forthrigth48 – Modular Exponentiation; forthright48 – Bit Manipulation; algorithmist – Repeated Squaring; Wiki – Exponentiation by This trick, known as repeated squaring, allows us to compute \(a^k\) mod \(n\) using only \(O(\log k)\) modular multiplications. - ModularExponentiation2. Whether it’s calculating large powers for cryptographic algorithms or scientific computations, the efficiency of exponentiation plays a critical role in various applications. Accordingly, it would probably be faster on a computer for most inputs because of branch prediction. 8k 6 6 gold badges 46 46 silver badges Answer to Q4. Fast Computation of N th Fibonacci Number: Feb 15, 2023 · FYI, there's nothing to recursion on MIPS or any other environment where there is a call stack. The binary way is a little bit cleaner, because you don't have this case work to do at each iteration depending on whether the next exponent is even or odd, or the analogous case work in the recursive implementation. #Introduction When given a calculation like A B % C, computing the value causes difficulty due to the large number incorporated with the program. Some key aspects of recursion include: **Base Case**: The condition where recursion stops. The call to mod_exp(x, y//2, N) is the most interesting part. Whenever I'm running my code in QTSpim, I'm able to enter the values for these variables, but the program never outputs the desired result. Focus Problem – try your best to solve this problem before continuing! 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