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If $n = p_1 p_2$ for primes $p_1 \ne p_2$, $\phi(n)$ is $\phi(p_1)\cdot\phi(p_2)$. Does that mean that the possible range of $\phi(n)$ is somewhat narrow, that $(p_1-1) \cdot (p_2-1)$ (that is, $\phi(n)$) is close to $p_1\cdot p_2$ (that is, $n$)?

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$n - \phi(n) = p + q - 1$. $p$ and $q$ are secret uniform random 1024-bit prime numbers. Is a difference of near $2^{1024}$ ‘close’?

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  • $\begingroup$ what I mean is, phi(n) should be much much closer to n than to 0? $\endgroup$ – asterism May 22 at 17:33
  • $\begingroup$ Well, what's your metric and what do you consider close? The absolute distance from $\phi(n)$to $n$ is $p + q - 1 \approx 2^{1024}$; of course, $n$ and $\phi(n)$ are near $2^{2048}$, so, in that sense, yes, $\phi(n)$ is closer to $n$ than to $0$. $\endgroup$ – Squeamish Ossifrage May 22 at 17:37
  • $\begingroup$ largest difference would be if either prime is 2 then I guess $\endgroup$ – asterism May 22 at 17:49
  • $\begingroup$ If either prime is 2, then you are in trouble for RSA! $\endgroup$ – Squeamish Ossifrage May 22 at 17:50
  • $\begingroup$ yes but, like, I mean, largest difference would be if either prime is as small as possible? or, what you said is exactly that, so if 2 does not work would you not have to adjust the range you gave? $\endgroup$ – asterism May 22 at 17:52

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