I want to calculate $d$ from $e$ when generating RSA keys.

What is faster?

  • Calculating $\operatorname{xgcd}(e,p)$ and $\operatorname{xgcd}(e,q)$ and CRT.
  • Or calculating $e^{p-2}\bmod p$ and $e^{q-2}\bmod q$ and CRT.

I'm having a hard time understanding how to calculate the complexity here.

  • 3
    $\begingroup$ Typically, when creating an RSA private/public key pair, almost all the time is taken searching for primes; the tiny amount of time taken computing $d$ (or $dp$ and $dq$) is negligible; either method you cite would work. $\endgroup$
    – poncho
    Dec 11 '14 at 15:35
  • The extended Euclidean algorithm ($\operatorname{xgcd}$), when applied to $p$ and $e<p$, uses at most $\lceil\log_\phi(\sqrt5(p+1))\rceil-2\in\mathcal O(\log p)$ divisions in $\mathbb Z$. (This is Knuth's corollary to Lamé's theorem.)
  • Using Fermat's little theorem and square-and-multiply exponentiation on $e$ modulo $p$ takes at most $2\cdot\lfloor\log_2(p-2)\rfloor\in\mathcal O(\log p)$ multiplications in $\mathbb F_p$.

Therefore, both methods are asymptotically equivalent. However, there are some practical benefits when using $\operatorname{xgcd}$:

  • The involved numbers' bit length decreases very quickly.
  • A division with remainder is generally cheaper than a multiplication followed by a modulo operation.
  • $\begingroup$ Could you explain a bit more how you got to these complexities? $\endgroup$ Dec 12 '14 at 20:52

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