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I'm trying to solve this question related to one-time pads and perfect secrecy:

question

My solution is: I assumed that the current message space is $M = \left\{ 0,1 \right\}^l$ and the new keyspace after removing $0^l$ from the set is $K = \left\{ 0,1 \right\}^l - 0^l$. Since clearly $|K| < |M|$, the new scheme is not perfectly secret.

Is my solution correct?

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  • $\begingroup$ Do you expect anything but a simple "yes" as the answer? $\endgroup$
    – SEJPM
    Mar 31, 2018 at 9:12
  • $\begingroup$ Hi Mitch and welcome. Just a small remark: try to make your titles as precise as possible; there are many ways in which the OTP's perfect secrecy can be broken: the title should clearly indicate what is asked. $\endgroup$
    – Maarten Bodewes ♦
    Mar 31, 2018 at 13:46
  • $\begingroup$ Similar question: crypto.stackexchange.com/q/3992/18700 $\endgroup$
    – Andrew T.
    Jun 14, 2021 at 13:15

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Yes your solution is correct; the flawed OTP scheme is no longer perfectly secret if there are fewer keys than messages.

Once you remove the all zero key the property $$ \mathbb{P}[M=m|C=c]=\mathbb{P}[M=m],\qquad \forall m,\forall c,\quad(1) $$ required for perfect secrecy will no longer necessarily

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  • $\begingroup$ I've removed the word "also" from the second part of your answer, could you check if this still good for your answer? $\endgroup$
    – Maarten Bodewes ♦
    Mar 31, 2018 at 13:44

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