How to generate private public key in JAVA using no external libraries

Question

We have been given an assignment part of which requires us to generate a public private key pair. These do not need to be particularly secure, as it is just for demonstration of the concept.

We can not use any sort of cryptography libraries or external tools.

How would I go about generating these?

Edit: Found a pretty nice explanation of RSA here: https://www.educative.io/edpresso/what-is-the-rsa-algorithm

Answer 1

I use JShell to demonstrate the basic public-private key generation just using Java's BigInteger :

jshell> import java.math.BigInteger;
jshell> var rnd = new java.security.SecureRandom();
rnd ==> Hash_DRBG,SHA-256,128,reseed_only

First we need 2 primes

jshell> var p1 = BigInteger.probablePrime(512, rnd);
p1 ==> 1176110601168217581401499298469596353224364190716 ... 72507270343325790065694831

jshell> var p2 = BigInteger.probablePrime(512, rnd);
p2 ==> 1001341560055006431459083188828513502474297271020 ... 34378293673605844490263567

Next we calculate the public key. 0x10001 is a common exponent for the public key.

jshell> var n = p1.multiply(p2);
n ==> 1177688424171014462551464978852125044384293220079 ... 24824881562893076179522177

jshell> var e = BigInteger.valueOf(0x10001);
e ==> 65537

The public key is e and n . Now the private part.

jshell> var phi = p1.subtract(BigInteger.ONE).multiply(p2.subtract(BigInteger.ONE));
phi ==> 1177688424171014462551464978852125044384293220079 ... 17939317545961441623563780

jshell> var d = e.modInverse(phi);
d ==> 7023685818262702180949167670691999860354377649273 ... 38390163809778429090416313

The private key is now d and n .
Let's test it:

jshell> var secret = BigInteger.valueOf(1337);
secret ==> 1337

jshell> var enc = secret.modPow(e, n);
enc ==> 1059982071031392497566614763259148320406936402012 ... 39171914529632475117049800

jshell> enc.modPow(d, n);
$11 ==> 1337

We could send enc over the wire, and nobody could decrypt it without the knowledge of the private key. Well, at least in theory. In practice, you should pad your messages.