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I have been studying additive ElGamal and I think I have the hang of it except the part where the message $M$ must be retrieved by computing the discrete log of $g^M$.

From what I've read, the result of such computations is a function of $k$ e.g. for $g=3, M=4, \pmod{17}$ we have $3^4=13 \pmod{17}$ but computing the discrete log however gives $M=4 + 16k$. How do I know that $M=4$ and not $M=20$?

Note that for $g=3,M=20 \pmod{17}$ we also get $M=4+16k$.

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The answer is simple if you know the answer to this question:

  • Can you use 20 instead of 4 for a plaintext?
  • If you can they will be the same encryption and no problem since 20 and 4 same for yo.
  • If you cannot, therefore, you cannot decide the plaintext after you decrypted.

To solve this issue you have to limit your message space into $\pmod{17}$ and divide your longer message block in an appropriate way. Now, you know, you have to take the smallest positive value where $k=0$.

The problem with adding homomorphically is a different issue. Before the computation, you have to consider the upper bound of your calculations and choose your modulus according to it. For example, if you are going to add 4 and 8 there is no problem, however, if you are going to add 10 and 9 you have to consider a bigger modulus.

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  • $\begingroup$ In my use case I know the maximum amount to sum can reach which means I have to find the next biggest prime after that correct? $\endgroup$ – Konstantine Oct 11 '19 at 10:07
  • $\begingroup$ That is what you need and that is commonly used in FHE implementations. Bigger than the necessary means more unnecessary computation. $\endgroup$ – kelalaka Oct 11 '19 at 10:09

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