Encryption

Fermat's Little Theorem

Fermat's little theorem tells us that

xp mod p = x

if p is prime for x<p

document.body.append(
  addInput('p','101'),
  addInput('m','3'),
  $m('button',{onclick:function(){
    document.getElementById('result').value =
      pow(getbig('m'),getbig('p'),getbig('p'));
  }},`m<sup>p</sup> mod p`),
  $m('input',{id:'result'})
);

//Utility functions
function pow(n,e,m){
  if (e<=0) return 1n;
  let r = pow(n,e/2n,m);
  return (r*r*(e%2n===1n?n:1n))%m;
}

function $m(tag,prop,children){
  let ret = document.createElement(tag);
  for(let k in prop)
    ret[k] = prop[k];
  if (typeof(children)==='string')
    ret.innerHTML = children;
  if (Array.isArray(children))
    for(let c of children)
      ret.append(c);
  return ret;
}
function getbig(id){return BigInt(document.getElementById(id).value)}
function addInput(id,value){
  return $m('div',{},[$m('label',{},id),$m('input',{id,value})]);  
}

function pow(n,e,m){
    if (e<=0) return 1n;
    let h = e/2n;
    let r = pow(n,h,m);
    r = (r*r) % m;
    if (e % 2n === 1n){
        return (n * r) % m;
    }
    return r;
}

function modInverse(a, m){
  a = (a % m + m) % m
  if (!a || m < 2n) {
    return NaN // invalid input
  }
  // find the gcd
  const s = []
  let b = m
  while(b) {
    [a, b] = [b, a % b]
    s.push({a, b})
  }
  if (a !== 1n) {
    return NaN // inverse does not exists
  }
  // find the inverse
  let x = 1n
  let y = 0n
  for(let i = s.length - 2; i >= 0; --i) {
    [x, y] = [y,  x - y * (s[i].a / s[i].b)]
  }
  return (y % m + m) % m
}
let getbig = id => BigInt(document.getElementById(id).value);
let addInput = (id,value) => {
  return $m('div',{},[$m('label',{},id),$m('input',{id,value})]);  
}
document.body.append(addInput('p','101'),addInput('q','103'));