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I have a protocol where I have a random one-time number message $m \in \mathbb{Z}_p$, where $p$ is a 256 bit prime from an Elliptic Curve (EDIT: The order of an elliptic curve prime order group). I also have a textbook RSA ciphertext $m^e \mod n$, encrypted using a 2048 bit RSA key. I am using for its homomorphic properties (I can't pad it or the protocol will not work).

Is it safe to use using a message from a large message space that is smaller than the RSA group size?

(EDIT in response to comment): I am not stuck with any particular value for $e$, but my program does scale with $\log_2{(e)}$, so I would prefer something smaller.

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  • $\begingroup$ Here the value of $e$ is important. If $e=3$ a direct cube-root attack is possible. What is the value of $e$? $\endgroup$
    – kelalaka
    Nov 24, 2023 at 22:34
  • $\begingroup$ I'm not bound to any particular value for $e$, though I would prefer for it to not use too many bits. $e = 65537$ should be fine, but I can also go larger if that would be safer. $\endgroup$
    – Zarquan
    Nov 24, 2023 at 23:03
  • $\begingroup$ what's a "prime p from an Elliptic curve?" $\endgroup$
    – kodlu
    Nov 25, 2023 at 4:38
  • $\begingroup$ What I meant is the prime order from an elliptic curve. $\endgroup$
    – Zarquan
    Nov 25, 2023 at 4:43
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    $\begingroup$ The present question seems to be a small variation of this one. $\endgroup$
    – fgrieu
    Nov 25, 2023 at 6:39

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