Example of RSA Encryption

1.

p = 41, q = 59

2.

n = pq = 2419

3.

$\phi$(n) = (p-1)(q-1) = 40*58 = 2320
Find e such that gcd(e, 2320) = 1 and e is small and odd
e = 3 works

4.

d = e^-1 mod $\phi$(n)
= 3^-1 mod 2320
d = 1547
d*e mod $\phi$(n) = 1547*3 mod 2320 = 1

5.

P = (e, n) = (3, 2419)

6.

S = (d, n) = (1547, 2419)
P(M) = M³ (mod 2419)
S(M) = M¹⁵⁴⁷ (mod 2419)
Note: Only 2419 different messages are possible.