Individual bits of Textbook RSA This question relates to the underlying RSA assumption. Forgetting about the fact that Textbook RSA is deterministic, I am curious about the assumed strength of the RSA problem.

Does RSA hide all bits of the message, or are there some bits that can be easily revealed. Concretely, 

$$m^e \space \bmod \space N$$ 

Can any part of $m$ be learned(with significant probability) without learning the secret key, and if so how?
Under the assumption that $m$ is unknown to the adversary, and random in $[0,N-1]$, then to the best of our knowledge, $m^e\bmod N$ does not reveal anything about any bit of $m$ to an adversary unable to factor $N$.

Update: in the above, "bit of $m$" is in the sense of binary digit of $m$ (for some integer $k≥0$, the value of $⌊m/2^k⌋\bmod 2$), not tiny amount of information. Revealing $m^e\bmod N$ does reveal information abut $m$; most notably the adversary knows the Jacobi symbol $\left(\frac m N\right)$ since that's also $\left(\frac{m^e\bmod N}N\right)$; and can test if an $x$ with $0≤x&lt;N$ is $m$, by checking if $x^e\bmod N$ is $m^e\bmod N$.