# Encryption

## Fermat's Little Theorem

Fermat's little theorem tells us that

```xp mod p = x
```

if `p` is prime for `x<p`

Verify this by trying prime and non-prime values for p. You can can generate prime numbers from https://bigprimes.org/

```document.body.append(
\$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)}
return \$m('div',{},[\$m('label',{},id),\$m('input',{id,value})]);
}
```

## Generate public/private key pairs

```document.body.append(
\$m('button',{},'Generate public/private key'),
);

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
}
//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)}