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I've been wondering about the 'small' part of the attacks on RSA, like the small public key exponent and the small private key exponent and what's not really clear is how small are we talking about? Like is it how small with respect to the large prime numbers or what exactly?

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Exponentiation by squaring takes time approximately proportional to the number of bits in the exponent. 3 has been used but allows attacks on some weak padding schemes. 65537 ($=2^{16}+1$) is the one used widely in practice.

Cf. Impacts of not using RSA exponent of 65537

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