Efficiently factor N when e and d become known to you.

The problem is:

In the RSA cryptosystem, Alice’s public key (N, e) is available to everyone. Suppose that her private key d is compromised and becomes known to Eve. Show that if e = 3 (a common choice) then Eve can efficiently factor N .

We need to find the two primes p and q, and

N = p * q

M = (p - 1) * (q - 1)

e * d mod M = 1

Steps:

1. because d < M < N, 2 <= N / M <= e;

2. e * d = k * M + 1, so k = 2 or k = 3( but it is impossible), so M = (e * d - 1) / 2;

3. so we get such a equation group:

    N = p *q

    M = p * q - p - q + 1 => p + q - 1 = N - M

4. solve the equation group.