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I have two, closely related questions:

  1. Can a really good attacker (hacker) use the public key for his advantage and break RSA encryption faster? My point is that in RSA "communication" public key is moving from server to client… If attacker will get this key and also encrypted data will be there any advantages or it is not important if attacker has public key?

  2. If, for example, there are five users and each user has public key but with different size (2048, 2100, 2120,2250,2310bits etc.) If attacker will break one of these key all other keys are also immediatelly know for this attacker or he must start whole process of breaking again? Is there some advantages to have each RSA key with different bit size?

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Can really good attacker(hacker) use public key for his advantage and break RSA encryption faster?

No; we always assume that the attacker knows the public key (that's what "public" means, after all).

Is there some advantage to have each RSA key with different bit size?

No; the most efficient approach known to attack an RSA key is to use the Number Field Sieve algorithm; which that approach, the attacker picks one RSA key, and attacks it. It doesn't matter if the keys he selects from has different sizes; he's likely to pick the smallest and/or the one that protects data of the most value.

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Can really good attacker(hacker) use public key for his advantage and break RSA encryption faster?

No, the standard attacker models in Cryptography are already way stronger than this assumption and they assert us that you can't break (properly done) RSA encryption given those advanced capabilities (which includes knowledge of the public key).

or it is not important if attacker has public key?

Obviously it's harder for the attacker to break a cryptosystem of which he didn't ever see anything, but in general the name public key is chosen for a reason. By assumption it is always public and thus there's no non-negligible harm in giving it to an attacker.

Is there some advatages to have each RSA key with different bit size?

Except for the varied levels of theoretical security and the varied amount of data that can fit into a single block that come with different key sizes, there's no advantage to this. However, these days, hybrid encryption is mostly used which is why sizes at around 2048 bit are totally fine for that.

If attacker will break one of these key all other keys are also imediatelly know for this attacker or he must start whole process of breaking again?

Yes and no.
There are crypto-systems where breaking one key using the proper method essentially gives you all the other keys with the same parameters with negligble cost (I'm talking about NFS attackable DLOG groups here, using Logjam). There are other cryptosystems where you will experience a "nice" speed-up from breaking multiple keys (I'm talking about generic DLOG groups here), in that breaking the $i$-th key gives you a speed-up of about $\sqrt i$.

For RSA specifically, there is a slight speed-up that can be gained by targeting multiple keys, but it is not nearly enough to make breaking other keys "easy" after having broken one.

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Yes, through factorisation — as alternative to brute force.

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    $\begingroup$ Knowing the public key $(N,e)$ does not help you factorize $N=pq$ faster than brute force, though. $\endgroup$ – user47922 Jun 14 '17 at 21:49

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