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If I sign a .NET assembly and do not give out my private key, how much effort would it take for somebody to crack the private key?

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With which particular method of signing? Strong Naming, or code signing? And of what size key? – Steve Jul 30 '13 at 22:22
    
@SteveS: Strong naming. – anonymous0000000 Jul 31 '13 at 1:35

It would take the same amount of effort as trying to falsify any other signing operation or rather trying to get the private key from any other signature. Generally speaking though, the most common attack is to try to make a different file that collides with the hash in the signature rather than trying to get the private key.

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