less_retarded_wiki/rsa.md
2024-04-29 21:44:17 +02:00

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# RSA
TODO
generating keys:
1. *p := large random prime*
2. *q := large random prime*
3. *n := p * q*
4. *f := (p - 1) * (q - 1)* (this step may differ in other versions)
5. *e := 65537* (most common, other constants exist)
6. *d := solve for x: 1 = (x * e) mod f*
7. *public key := (n,e)*
8. *private key := d*
message encryption:
1. *m := message encoded as a number < n*
2. *encrypted := m^e mod n*
message decryption:
1. *m := encrypted^d mod n*
2. *decrypted := decode message from number m*
## Code Example
Here is a stupidly simple [C](c.md) code demonstrating the algorithm, for simplicity we use laughably small primes, we only consider 4 character string as a message and make other simplifications.
```
#include <stdio.h>
#define e 65537 // often used constant
typedef unsigned long long u64;
void generateKeys(u64 prime1, u64 prime2, u64 *privateKey, u64 *publicKey)
{
u64 f = (prime1 - 1) * (prime2 - 1);
*publicKey = prime1 * prime2;
*privateKey = 1;
while (*privateKey) // brute force solve the equation
{
if (((*privateKey) * e) % f == 1)
break;
(*privateKey)++;
}
}
u64 powerMod(u64 n, u64 power, u64 mod) // helper func, returns n^power % mod
{
u64 result = 1;
for (int i = 0; i < power; ++i)
result = (result * n) % mod;
return result;
}
u64 encryptNum(u64 n, u64 publicKey)
{
return powerMod(n,e,publicKey);
}
u64 decryptNum(u64 n, u64 publicKey, u64 privateKey)
{
return powerMod(n,privateKey,publicKey);
}
int main(void)
{
u64 priv, pub,
prime1 = 33461,
prime2 = 17977;
const char *str = "bich"; // we'll only consider 4 char string
puts("generating keys, wait...");
generateKeys(prime1,prime2,&priv,&pub);
printf("prime1 = %llu\nprime2 = %llu\nprivate key = %llu\npublic key = %llu\n",
prime1,prime2,priv,pub);
u64 data = str[0] | (str[1] << 7) | (str[2] << 14) | (str[3] << 21);
printf("string to encode: \"%s\"\n",str);
printf("string as numeric data: %lld\n",data);
if (data >= pub)
{
puts("Data is too big, choose bigger primes or smaller data.");
return 1;
}
puts("encrypting...");
u64 encrypted = encryptNum(data,pub);
printf("encrypted: %lld\n",encrypted);
puts("decrypting...");
data = decryptNum(encrypted,pub,priv);
printf("decrypted: %lld\n",data);
printf("retrieved string: \"%c%c%c%c\"\n",data & 0x7f,(data >> 7) & 0x7f,
(data >> 14) & 0x7f,(data >> 21) & 0x7f);
return 0;
}
```
The program prints out:
```
generating keys, wait...
prime1 = 33461
prime2 = 17977
private key = 323099873
public key = 601528397
string to encode: "bich"
string as numeric data: 219739362
encrypting...
encrypted: 233361060
decrypting...
decrypted: 219739362
retrieved string: "bich"
```