On Tuesday, September 1, 2020 at 10:30:23 AM UTC-4 star...@codev.com wrote:
> Good day to everyone,
> I just would like to ask something with regards to RSA Key generation.
> I know everyone knows that we can initialize or create a private key using
> pre-generated modulus, public exponent and private exponent.
> Is there a way that we can recompute the private key or the modulus given
> only the private exponent and the public exponent or, private exponent and
> the modulus?
>
The RSA private key is either {n,e,d} (short form) or {n,e,d,p,q,dp-1,invq}
(long form). If you have the short form you can recalculate the long form
parameters using the Chinese Remainder Theorem (CRT).
The Crypto++ code to calculate the long form parameters given short form is
at https://github.com/weidai11/cryptopp/blob/master/rsa.cpp. Checkout the
function InvertibleRSAFunction::Initialize().
You may be able to recover a private key given {n,d}. That's because you
can sometimes guess the public exponent -- it is usually 3, 17 or 65537.
There's no requirement e has to be 3, 17 or 65537. I think the only
requirement is e and Phi(n) must be co-prime, where Phi is Euler's
Phi-function. Phi(n) = (p-1)(q-1). So you can select a random e, coprime to
Phi(n), with a low hamming weight, and RSA will work just fine.
You will probably have trouble recovering a private key given {e,d}. You
have to guess the modulus.
Jeff
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