Prime Numbers

8.1 Cryptography 389
is, if an oracle could tell you g ab on input of g a and g b , could you use this
oracle to quickly solve for discrete logarithms?
8.1.2 RSA cryptosystem
Soon after the Diffie–Hellman ideas, the now prevalent RSA cryptosystem was
invented by Rivest, Shamir, and Adleman [Rivest et al. 1978].
Algorithm 8.1.2 (RSA private/public key generation). In this algorithm
we generate an individual’s private and associated public keys for the RSA
cryptosystem.
1. [Choose primes]
Choose two distinct primes p, q under prevailing safety criteria (see text);
2. [Generate public key]
N = pq;
ϕ =(p − 1)(q − 1); // Euler totient of N.
Choose random integer E ∈ [3,N − 2] coprime to ϕ;
Report public key as (N,E); // User publishes this key.
3. [Generate private key]
D = E −1 mod ϕ;
Report private key as D; // User keeps D secret.
The primary observation is that because of the difficulty of factoring N = pq,
the public integer N does not give an easy prescription for the private primes
p, q. Furthermore, it is known that if one knows integers D, E in [1,n− 1]
with DE ≡ 1(modϕ), then one can factor N in (probabilistic) polynomial
time [Long 1981] (cf. Exercise 5.27). In the above algorithm it is fashionable
to choose approximately equal private primes p, q, but some cryptographers
suggest further safety tests. In fact, one can locate in the literature a host of
potential drawbacks for certain p, q choices. There is a brief but illuminating
listing of possible security flaws that depend on the magnitudes and other
number-theoretical properties of p, q in [Williams 1998, p. 391]. The reference
[Bressoud and Wagon 2000, p. 249] also lists RSA pitfalls. See also Exercise
8.2 for a variety of RSA security issues.
Having adopted the notion that the public key is the hard-to-break (i.e.,
difficult to factor) composite integer N = pq, we can proceed with actual
encryption of messages, as follows:
Algorithm 8.1.3 (RSA encryption/decryption). We assume that Alice possesses
a private key DA and public key (NA,EA) from Algorithm 8.1.2. Here we
show how another individual (Bob) can encrypt a message x (thought of as an
integer in [0,NA)) to Alice, and how Alice can decrypt said message.
1. [Bob encrypts]
y = x EA mod NA; // Bob is using Alice’s public key.
Bob then sends y to Alice;
2. [Alice decrypts]