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Adi Shamir, back in 1995, proposed "RSA for paranoids" in RSA's CryptoBytes newsletter: ftp://ftp.rsasecurity.com/pub/cryptobytes/crypto1n3.pdf. The idea is that you have p of 500 bits, q of 4500 bits, n of 500+4500 = 5000 bits, and constrain m to be 500 bits or less. Then you can encrypt with an exponent of around 20, and decrypt by calculating only m mod p ...


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No, there is no specific vulnerability associated to choosing $p$ and $q$ with size differing by $i$ bits (or $2\cdot i$ bits as in the statement) for small $i$. However, if $i$ gets too big: That improves the odds that ECM will manage to factor $n$ for some fixed size of $n$, and at some point ECM will become the best algorithm; this is the case if $i$ is ...



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