# How to solve high exponent attack for same message in RSA

I know Håstad's broadcast attack when e = 3, but what if e = getPrime(randint(350))?

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If you have $$e$$ ciphertexts for the same message, then the attack is the same, you just have to apply the CRT with $$e$$ values, then computing the $$e$$'th root of the resulting value might need some work. And this is assuming that all moduli are relatively prime of course. If that's not the case, there is $$i,j$$ and $$gcd(N_i, N_j) \neq 1$$. So you can factor one of $$N_i$$ or $$N_j$$ and decrypt the message. Lastly, depending on the size of $$e$$ and that of the $$N_i$$ you might want to investigate attacks on "small private exponents" such as Wiener's attack or from Boneh et al.