Diffie-Hellman in JavaScript
Variable Naming

It took me a while to figure out what are p1, p2 and n (and you don't mention them in your example).

String operation

You could use math operation to do the computation:

function KeyGen(base, modulo, exponent) {
  var result = 1;
  while(exponent &gt; 0){
    if(exponent % 2 == 1){
        result = (result * base) % modulo;
    }
    base = (base * base) % modulo;
    exponent = exponent &gt;&gt;&gt; 1;
  }
  return result;
}


The memory footprint is then very small (no array required).

Big numbers

Issues appear when you try to use big integers (more than 54 bits - like in your comment).
To handle them correctly, I would recommand an external library like BigInt:



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t=e.abs();if(t.isUnit())return!1;if(t.equals(2)||t.equals(3)||t.equals(5))return!0;if(t.isEven()||t.isDivisibleBy(3)||t.isDivisibleBy(5))return!1;if(t.lesser(25))return!0}function H(e){return(typeof e=="number"||typeof e=="string")&amp;&amp;+Math.abs(e)&lt;=t||e instanceof o&amp;&amp;e.value.length&lt;=1}function B(e,t,n){t=Q(t);var r=e.isNegative(),i=t.isNegative(),s=r?e.not():e,o=i?t.not():t,u=[],a=[],f=!1,l=!1;while(!f||!l)s.isZero()?(f=!0,u.push(r?1:0)):r?u.push(s.isEven()?1:0):u.push(s.isEven()?0:1),o.isZero()?(l=!0,a.push(i?1:0)):i?a.push(o.isEven()?1:0):a.push(o.isEven()?0:1),s=s.over(2),o=o.over(2);var c=[];for(var h=0;h&lt;u.length;h++)c.push(n(u[h],a[h]));var p=bigInt(c.pop()).negate().times(bigInt(2).pow(c.length));while(c.length)p=p.add(bigInt(c.pop()).times(bigInt(2).pow(c.length)));return p}function I(e){var n=e.value,r=typeof n=="number"?n|j:n[0]+n[1]*t|F;return r&amp;-r}function q(e,t){return e=Q(e),t=Q(t),e.greater(t)?e:t}function R(e,t){return 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this.compare(e)===0},u.prototype.eq=u.prototype.equals=o.prototype.eq=o.prototype.equals,o.prototype.notEquals=function(e){return this.compare(e)!==0},u.prototype.neq=u.prototype.notEquals=o.prototype.neq=o.prototype.notEquals,o.prototype.greater=function(e){return this.compare(e)&gt;0},u.prototype.gt=u.prototype.greater=o.prototype.gt=o.prototype.greater,o.prototype.lesser=function(e){return this.compare(e)&lt;0},u.prototype.lt=u.prototype.lesser=o.prototype.lt=o.prototype.lesser,o.prototype.greaterOrEquals=function(e){return this.compare(e)&gt;=0},u.prototype.geq=u.prototype.greaterOrEquals=o.prototype.geq=o.prototype.greaterOrEquals,o.prototype.lesserOrEquals=function(e){return 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t=M(this);if(t!==e)return t;var n=this.abs(),r=n.prev(),i=[2,3,5,7,11,13,17,19],s=r,o,u,a,f;while(s.isEven())s=s.divide(2);for(a=0;a&lt;i.length;a++){f=bigInt(i[a]).modPow(s,n);if(f.equals(G[1])||f.equals(r))continue;for(u=!0,o=s;u&amp;&amp;o.lesser(r);o=o.multiply(2))f=f.square().mod(n),f.equals(r)&amp;&amp;(u=!1);if(u)return!1}return!0},u.prototype.isPrime=o.prototype.isPrime,o.prototype.isProbablePrime=function(t){var n=M(this);if(n!==e)return n;var r=this.abs(),i=t===e?5:t;for(var s=0;s&lt;i;s++){var o=bigInt.randBetween(2,r.minus(2));if(!o.modPow(r.prev(),r).isUnit())return!1}return!0},u.prototype.isProbablePrime=o.prototype.isProbablePrime,o.prototype.next=function(){var e=this.value;return this.sign?b(e,1,this.sign):new o(m(e,1),this.sign)},u.prototype.next=function(){var e=this.value;return e+1&lt;r?new u(e+1):new o(i,!1)},o.prototype.prev=function(){var e=this.value;return this.sign?new o(m(e,1),!0):b(e,1,this.sign)},u.prototype.prev=function(){var e=this.value;return e-1&gt;-r?new u(e-1):new o(i,!0)};var _=[1];while(_[_.length-1]&lt;=t)_.push(2*_[_.length-1]);var D=_.length,P=_[D-1];o.prototype.shiftLeft=function(e){if(!H(e))return e.isNegative()?this.shiftRight(e.abs()):this.times(G[2].pow(e));e=+e;if(e&lt;0)return this.shiftRight(-e);var t=this;while(e&gt;=D)t=t.multiply(P),e-=D-1;return t.multiply(_[e])},u.prototype.shiftLeft=o.prototype.shiftLeft,o.prototype.shiftRight=function(e){var t;if(!H(e))return e.isNegative()?this.shiftLeft(e.abs()):(t=this.divmod(G[2].pow(e)),t.remainder.isNegative()?t.quotient.prev():t.quotient);e=+e;if(e&lt;0)return this.shiftLeft(-e);var n=this;while(e&gt;=D){if(n.isZero())return n;t=A(n,P),n=t[1].isNegative()?t[0].prev():t[0],e-=D-1}return t=A(n,_[e]),t[1].isNegative()?t[0].prev():t[0]},u.prototype.shiftRight=o.prototype.shiftRight,o.prototype.not=function(){return this.negate().prev()},u.prototype.not=o.prototype.not,o.prototype.and=function(e){return B(this,e,function(e,t){return e&amp;t})},u.prototype.and=o.prototype.and,o.prototype.or=function(e){return B(this,e,function(e,t){return e|t})},u.prototype.or=o.prototype.or,o.prototype.xor=function(e){return B(this,e,function(e,t){return e^t})},u.prototype.xor=o.prototype.xor;var j=1&lt;&lt;30,F=(t&amp;-t)*(t&amp;-t)|j,X=function(e,t){var n=G[0],r=G[1],i=e.length;if(2&lt;=t&amp;&amp;t&lt;=36&amp;&amp;i&lt;=s/Math.log(t))return new u(parseInt(e,t));t=Q(t);var o=[],a,f=e[0]==="-";for(a=f?1:0;a&lt;e.length;a++){var l=e[a].toLowerCase(),c=l.charCodeAt(0);if(48&lt;=c&amp;&amp;c&lt;=57)o.push(Q(l));else if(97&lt;=c&amp;&amp;c&lt;=122)o.push(Q(l.charCodeAt(0)-87));else{if(l!=="&lt;")throw new Error(l+" is not a valid character");var h=a;do a++;while(e[a]!=="&gt;");o.push(Q(e.slice(h+1,a)))}}o.reverse();for(a=0;a&lt;o.length;a++)n=n.add(o[a].times(r)),r=r.times(t);return f?n.negate():n};o.prototype.toString=function(t){t===e&amp;&amp;(t=10);if(t!==10)return $(this,t);var n=this.value,r=n.length,i=String(n[--r]),s="0000000",o;while(--r&gt;=0)o=String(n[r]),i+=s.slice(o.length)+o;var u=this.sign?"-":"";return u+i},u.prototype.toString=function(t){return t===e&amp;&amp;(t=10),t!=10?$(this,t):String(this.value)},o.prototype.valueOf=function(){return+this.toString()},o.prototype.toJSNumber=o.prototype.valueOf,u.prototype.valueOf=function(){return this.value},u.prototype.toJSNumber=u.prototype.valueOf;var G=function(e,t){return typeof e=="undefined"?G[0]:typeof t!="undefined"?+t===10?Q(e):X(e,t):Q(e)};for(var Y=0;Y&lt;1e3;Y++)G[Y]=new u(Y),Y&gt;0&amp;&amp;(G[-Y]=new u(-Y));return G.one=G[1],G.zero=G[0],G.minusOne=G[-1],G.max=q,G.min=R,G.gcd=U,G.lcm=z,G.isInstance=function(e){return e instanceof o||e instanceof u},G.randBetween=W,G}();typeof module!="undefined"&amp;&amp;module.hasOwnProperty("exports")&amp;&amp;(module.exports=bigInt);


function compute(){
  var base = bigInt(document.getElementById('base').value);
  var exponent = bigInt(document.getElementById('exponent').value);
  var modulo = bigInt(document.getElementById('modulo').value);

  var res = base.modPow(exponent, modulo);
  document.getElementById('result').value = res;
}
&lt;label&gt;Base:&lt;input id="base" value="3"&gt;&lt;/label&gt;&lt;br&gt;
&lt;label&gt;Exponent:&lt;input id="exponent" value="23984081230374123749712934791249172394719237497219347129374498"&gt;&lt;/label&gt;&lt;br&gt;
&lt;label&gt;Modulo:&lt;input id="modulo" value="112457129983317064494133258034491756790943511028023366901014968560410379195027"&gt;&lt;/label&gt;&lt;br&gt;
&lt;input type="button" value="Compute" onclick="compute()"&gt;&lt;br&gt;
Result: &lt;input id="result"&gt;




The result seems to be exact.

PS: The modPow implementation is actually quite readable.