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Recently I have few experiences with Questions in RSA which e is 2^n instead of 2^n+1, and that leads to gcd(e, phi) is not equal to 1... Won't this make the private key impossible to get? Is the Rabin cryptosystem the only way out?

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    $\begingroup$ What have you tried? $\endgroup$
    – kelalaka
    Jun 2, 2022 at 16:26
  • $\begingroup$ I tried the Rabin cryptosystem, to find the two square roots (only one prime number is used so 2 square roots instead 4) and do it repeatedly as my e is a multiple of 2. $\endgroup$
    – dlfls
    Jun 2, 2022 at 16:36
  • $\begingroup$ And I used N as public key, and p as the private key in this case. Since d won't generate and phi won't be helpful. (but please correct me if this is incorrect!) $\endgroup$
    – dlfls
    Jun 2, 2022 at 16:37

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In RSA, what does $\gcd(e,\operatorname{phi})\ne1$ means?

RSA encryption $m\mapsto m^e\bmod n$ is a reversible encryption of $[0,n)$ if and only if

  1. $n$ is the product of distinct prime factors $p_i$ (which is met if $n=p\,q$ for two large distinct primes $p$ and $q$, the most common setup)
  2. and the public exponent $e$ has an inverse $d_i$ modulo each $p_i-1$, that is $\exists d_i\in\mathbb N: e\,d_i\equiv1\pmod{p_i-1}$. Equivalently: and it holds $\gcd(e,p_i-1)=1$ for each $p_i$. This condition is to ensure that $m\mapsto\left(m^e\right)^{d_i}\equiv m\pmod{p_i}$ for every $m\in\mathbb N$.

When (1) holds, $\operatorname{phi}(n)=\prod(p_i-1)$, therefore the condition $\gcd(e,\operatorname{phi}(n))$ is equivalent to (2).

Why always choose $e=2^k+1$ not $2^k$?

We don't always choose $e$ of the form $2^k+1$. For example $e=37$ is quite common (see this). We always choose $e$ odd in RSA because otherwise the condition $\gcd(e,p_i-1)=1$ can not be met for $p_i>2$, because $2$ is the only even prime.

If one uses even $e$, that's not RSA. That can be the Rabin cryptosystem.

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