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Suppose I implement RSA myself in C and I use just the formulas from number theory. What attacks would my implementation be vulnerable to? Can anyone explain this to me like I'm 5?

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RSA is a trapdoor permutation used in various protocols and cryptographic schemes. The attacks are on the protocol/scheme and the primitive together. What scheme are you implementing? –  pg1989 Aug 22 '13 at 0:55
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Shall we assume textbook RSA with no message padding / formatting? –  rath Aug 22 '13 at 2:51
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Not my answer, but a quick search turned up members.tripod.com/irish_ronan/rsa/attacks.html which is about as simple an explanation as I've seen. –  John Deters Aug 22 '13 at 3:40
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Additionally, there are sidechannel attacks. E.g. the decryption is done with square-and-multiply exponentiation, then the timing is a linear function of the Hamminh weight of the key (n times squaring and the Hamming weight in multiplications). If the 0s and 1s are almost uniform, this is not a problem, but otherwise you have a problem. If the attacker can even identify the order of multiplications and squaring (the operations take different amounts of time and cache access), he can just grab $d$ directly. –  tylo Aug 22 '13 at 11:13
    
@rath Yes, textbook RSA with no message padding or formatting. –  bodacydo Aug 22 '13 at 21:50
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