Factoring two RSA modulus with known $|p_1 - p_2| < \ell $

Question

I have two 1024bit RSA modulus $N_1 = p_1q_1$ and $N_2 = p_2q_2$ such that $0 <|p_1 - p_2| < \ell$, and $\ell$ is at most 64bit integer. Can I factorise $N_1$ and $N_2$? What's the answer when $p_1$ and $p_2$ are same $\alpha$-bit primes?

Answer 1

Outline of a elementary attack:

if $\mid p_i-q_i \leq 2^s$, we can write for each $p_i=a.2^s+\tau_i$, with $\tau_i \leq 2^s$, and we can imagine that if we determine the integer a, we can factor $N_i$ by a brute force attack. We know that $p_i$ have exactly the same bit size and if $N_i$ have exactly the same bit size (n=1024, 2048, ... s=64), the sizes of $q_i$ could differ by at most one.

Let $r=\frac{a.2^s+\tau_1}{a.2^s+\tau_2} \approx 1 + \frac{1}{a} \pm \epsilon$. And $\rho=\frac{N_1}{N_2} \approx (1 + \frac{1}{a} \pm f(\epsilon))\times \frac{q_1}{q_2}$ is a rational number which can be determined with a infinite precision.

Then by the examination of the bit of this ratio we can get information on the unknow number a, and hope to success in the factorisation Pb, in complexity less than the general know attacks.

Probably with the help of LLL algo we can enhance the attack.