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Suppose a common message ($M$) is XOR'ed with two different keys $K_1,K_2$ producing two ciphertexts $C_1,C_2$. Thus,

$$M \oplus K_1 = C_1\\ M \oplus K_2 = C_2$$

On observing $C_1$ and $C_2$, an adversary can obtain the value of $C_1 \oplus C_2 = K_1 \oplus K_2$. Is this scheme semantically secure? Note that $|K_1|=|K_2|=|M|$ and the keys are randomly chosen from a uniformly distributed large key space ($\approx 2^{255}$).

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  • $\begingroup$ In the OTP, what is the aim ? Hide the key or hide the message ? Also remember that the key are uniformly distributed. $\endgroup$ – Biv May 9 '16 at 13:28
  • $\begingroup$ Is $|M| = 255 bits$? $\endgroup$ – Henrick Hellström May 9 '16 at 13:31
  • $\begingroup$ @Biv The aim is to determine whether $C1$ and $C2$ gives any extra information about $M$. Thus is- $E(M|C1,C2)=E(M)$\\ $K_1$ and $K_2$ are two different keys which have same length of that of $M$ and are chosen randomly from a uniformly distributed key space. $\endgroup$ – Debanjan Sadhya May 9 '16 at 13:32
  • $\begingroup$ @HenrickHellström: yes $|M| = 255 bits$ $\endgroup$ – Debanjan Sadhya May 9 '16 at 13:35
  • $\begingroup$ @DebanjanSadhya then I would advise you to, simplify your scheme: consider that $M$ is only 1 bit long, idem for $C$ and $K$. What is the probability that $K_1 = 1$, that $K_1 = 0$ idem for $K_2$. Then add $M$ in the mix. What pieces of information did you gain ? What pieces of information do you want to gain ? Try a case study. $\endgroup$ – Biv May 9 '16 at 13:47
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As long as the keys $K_i$ are only used once, this is semantically secure. To see it, observe that if $K_i$ is a uniformly random value in $\{0,1\}^{|M|}$ then so is $C_i = M \oplus K_i$.

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