If decryption key is hacked then is it safe to change encryption and decryption key?

Question

Suppose the decryption key $d$ of a RSA cryptosystem is hacked and instead of choosing a new $n$ if encryption and decryption keys are changed with same $n$,then is this safe ?

My attempt: Previous encryption key $e$ and decryption key $d$ are known. Then $ed \equiv 1$(mod $\varphi(n)$)

i.e, $\;ed-1 = k \cdot \varphi(n)$, for some natural number $k$.

$\implies \varphi(n) =\frac{ed-1}{k}$

So we get $\varphi(n)$ and could get prime factors of n.

Is this right?

Answer 1

The vulnerability as mentioned by @Randall, here is additional detail.

From the paper, "Solving Systems of Modular Equations in One Variable: How Many RSA-Encrypted Messages Does Eve Need to Know?" by Alexander May and Maike Ritzenhofen:

G. Simmons [16] has presented a neat attack for this special setting with running time polynomial in the bitlength of $(e_1, e_2)$. Namely, one computes integers $u_1, u_2$ such that $u_1e_1+u_2e_2 = 1$ with the help of the Extended Euclidean Algorithm. This gives us $m = (m^{e_1})^{u_1} (m^{e_2})^{u_2} \pmod{N}$.

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References

[16]: Simmons, G.: A “Weak” Privacy Protocol Using the RSA Crypto Algorithm. Cryptologia 7(2), 180–182 (1983)

The original Simmons paper is here.