How to compute $m$ value from RSA if $phi(n)$ is not relative prime with the $e$?

Well, if we assume that:

• $e$ is prime (65537 is)
• Only one of the primes minus one has $e$ as a factor; for example, $p-1$ is divisible by $e$, but $q-1$ is not. For this discussion, we'll assume that $p$ is the prime with $p-1 equiv 0 bmod e$ (which might happen to be the size 1023 factor for you)
• $p-1$ is not divisible by $e^2$
• That the ciphertext was actually generated by computing $P^e bmod n$ for some plaintext value $P$.

Then, one way to derive the possible plaintexts is to compute:

$$C^d cdot L^i bmod n$$

where:

• $C$ is the ciphertext
• $d = e^{-1} bmod lambda / e$ . This is well defined, as $lambda/e$ is an integer which is relatively prime to $e$.
• $L = k^{lambda/e} bmod n$, where $k$ is an integer such that $L ne 1$ (and any such value $L$ works); most values of $k$ work
• $lambda = (p-1)(q-1)/gcd(p-1, q-1)$
• $i$ is any integer $0 le i < e$

Now, if we iterate over the possible values of $i$, this will give $e$ possible values for the plaintext (unless $C$ happens to be a multiple of $p$). The original plaintext will be one of these values. All these values, when raised to the power $e$, will result in the ciphertext, hence we cannot distinguish from the ciphertext which one it is.