# Difference between revisions of "Encryption"

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## 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(
mkInput('p','101'),
mkInput('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'})
);

//raise n to the power e, modulo m
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;
}

//Create an element with tagname and properties
//children can be a string (innerHTML) or a list of elements
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;
}
//Get the value in element with id, convert it to a BigInt
function getbig(id){return BigInt(document.getElementById(id).value)}
//Return a div containing label and input. id is shown in the label
function mkInput(id,value){
return \$m('div',{},[\$m('label',{},id),\$m('input',{id,value})]);
}
```

## Generate public/private key pairs

```document.body.append(
addHiddenInput('p','101'),
addHiddenInput('q','103'),
addInput('e','7'),
\$m('button',{onclick:()=>{
\$i('d').value=modInverse(gb('e'),(gb('p')-1n)*(gb('q')-1n));
\$i('n').value=gb('p')*gb('q');
}},'Generate public/private key'),
addHiddenInput('d',''),
addInput('n',''),
addInput('message','123'),
\$m('button',{onclick:()=>{
\$i('encrypted').value=pow(gb('message'),gb('e'),gb('n'));
}},'encrypt with public key'),
addInput('encrypted',''),
\$m('button',{onclick:()=>{
\$i('decrypted').value=pow(gb('encrypted'),gb('d'),gb('n'));
}},'decrypt with private key'),
addInput('decrypted','')
);

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 \$i(id){return document.getElementById(id);}
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 gb(id){return BigInt(document.getElementById(id).value)}
function addInput(id,value){
return \$m('div',{},[\$m('label',{},`\${id} `),\$m('input',{id,value})]);
}
function addHiddenInput(id,value){
return \$m('div',{},[
\$m('label',{},`\${id} `),
\$m('input',{id,value,type:'password'}),
\$m('span',{onclick:()=>{\$i(id).removeAttribute('type')}},' show')
]);
}
```

## Splitting the exponent

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(
mkInput('p','1000000007'),
mkInput('scramble','112233'),
\$m('button',{onclick:()=>{
document.getElementById('unscramble').value =
modInverse(getbig('scramble'),getbig('p')-1n);
}},'calculate unscramble'),
mkInput('unscramble',''),
mkInput('secret','888'),
\$m('button',{onclick:function(){
document.getElementById('scrambled').value =
pow(getbig('secret'),getbig('scramble'),getbig('p'));
}},`scramble secret`),
mkInput('scrambled',''),
\$m('button',{onclick:function(){
document.getElementById('tada').value =
pow(getbig('scrambled'),getbig('unscramble'),getbig('p'));
}},`unscramble scrambled`),
mkInput('tada','')
);

//raise n to the power e, modulo m
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 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
}

//Create an element with tagname and properties
//children can be a string (innerHTML) or a list of elements
const \$i = document.getElementById;
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;
}
//Get the value in element with id, convert it to a BigInt
function getbig(id){return BigInt(document.getElementById(id).value)}
//Return a div containing label and input. id is shown in the label
function mkInput(id,value){
return \$m('div',{},[\$m('label',{},id),\$m('input',{id,value})]);
}
```